# Picosecond pump pulses probe the relevance of hot electrons for the laser-
induced phase transition in FeRh

M. Mattern  Institut für Physik und Astronomie, Universität Potsdam, 14476
Potsdam, Germany  S. P. Zeuschner  Institut für Physik und Astronomie,
Universität Potsdam, 14476 Potsdam, Germany  J. A. Arregi  CEITEC BUT, Brno
University of Technology, 61200 Brno, Czech Republic  V. Uhlíř  CEITEC BUT,
Brno University of Technology, 61200 Brno, Czech Republic  Institute of
Physical Engineering, Brno University of Technology, 61669 Brno, Czech
Republic  M. Bargheer  Institut für Physik und Astronomie, Universität
Potsdam, 14476 Potsdam, Germany  Helmholtz-Zentrum Berlin für Materialien und
Energie GmbH, Wilhelm-Conrad-Röntgen Campus, BESSY II, 12489 Berlin, Germany
<EMAIL_ADDRESS>
###### Abstract

Recent ultrafast photoemission experiments showed signatures of an ultrafast
modification of the electronic band structure in FeRh indicative of a
ferromagnetic (FM) state that is initiated by a non-equilibrium occupation of
the electronic states upon femtosecond laser excitation. We use ultrafast
x-ray diffraction to examine the impact of hot electrons on the
antiferromagnetic (AFM) to FM phase transition. By increasing the pump-pulse
duration up to $10.5\,\text{ps}$, we eliminate hot electrons and see that the
nucleation of FM domains still proceeds at the intrinsic timescale of
$8\,\text{ps}$, which starts when the deposited energy surpasses the threshold
energy. For long pulses, the phase transition proceeds considerably faster
than predicted by a convolution of the dynamics observed for ultrafast
excitation with the long pump pulse duration. We predict that quite generally,
slow photoexcitation can result in a fast response, if the non-linear
threshold behavior of a first-order phase transition is involved.

First-order phase transitions are characterized by an abrupt change of
structural, electronic or/and magnetic properties and a co-existence of
multiple phases when the deposited energy in thermal equilibrium[1, 2, 3, 4]
or on ultrafast timescales[5, 6, 7, 8, 9, 10, 11, 12] exceeds a threshold. The
abruptly emerging phase is a consequence of a fine interplay of spin, charge
and lattice degrees of freedom[6, 13]. Since the optical excitation often
affects only the electrons directly, they heat up far beyond the transition
temperature, rendering the driving mechanism of the laser-induced phase
transition a formidable question.

Since the discovery of the first-order magneto-structural antiferromagnetic-
to-ferromagnetic (AFM-FM) phase transition of FeRh at $370\,\text{K}$
different mechanisms such as expansion-induced sign change of the exchange
constant[14], excitation of spin waves[15] and dominant FM exchange of the Fe
moments mediated by an induced Rh moment[16, 8, 17, 18, 13, 19] were proposed.

On ultrafast timescales, direct signatures of the ferromagnetic phase
transition of FeRh are the rise of the increased FM lattice constant[10, 20,
11, 21] and the emergence of a net magnetization[8, 22, 9, 23, 24]. While the
rise of the magnetization is most significant after the coalescence of the
nucleated FM domains[22, 23, 24], probing the structural order parameter via
ultrafast x-ray diffraction (UXRD) directly yields insights into their
nucleation and growth[10, 20, 11, 21], where the intrinsic nucleation time[11]
of $8\,\text{ps}$ applies for a wide range of fluences, external magnetic
fields and even for FeRh nanostructures[21].

The local configuration of magnetic moments has been described as a
competition between bilinear and higher-order four spin exchange terms in
atomistic spin dynamics [18], a combination of Heisenberg exchange of Fe and a
Stoner model for Rh [8] and a modification of the Rh-Fe hybridization[17, 13,
19]. Recently, time-resolved photoelectron spectroscopy identified a sub-
picosecond formation of an electronic ferromagnetic state[19], related to a
photo-induced change of the band structure: Charge transfer from Rh to Fe and
an intersite spin transfer between Fe sites was found to induce a Rh moment
during the relaxation of optically excited non-equilibrium electrons. However,
it remains unclear if the non-equilibrium character of the photoexcited
electrons is required for the laser-induced phase transition.

Here, we investigate the role of hot electrons concomitant with ultrashort
laser pulse excitation for this prototypical phase transition. By ultrafast
x-ray diffraction (UXRD), we directly measure how the kinetics of domain
nucleation and the threshold of the AFM-FM phase transition depend on the
duration of optical pump-pulses. For $10.5\,\text{ps}$-long pulses, the
pronounced electron-phonon non-equilibrium present upon femtosecond laser
excitation is effectively suppressed. Thus, we gradually deposit the energy in
all subsystems and recover the energy threshold for the laser-induced phase
transition known from equilibrium. The transient FM volume fraction extracted
from the laser-induced strain response is found to exclusively depend on the
total deposited energy. It reaches the same final value at the intrinsic
nucleation timescale irrespective of the pump-pulse duration. As a consequence
of the non-linear threshold behavior, the rise of the FM phase is delayed but
faster than the convolution of the pump pulse duration with the signal upon
femtosecond pulse excitation. In total, a laser-induced hot Fermi-distribution
is not necessary to drive the phase transition on the intrinsic nucleation
timescale.

Figure 1(a) sketches the epitaxial $12.6\,\text{nm}$ thick FeRh(001) film
grown by magnetron sputtering from an equiatomic FeRh target[25] on an
MgO(001) substrate. We used synchrotron radiation from the KMC-3 XPP
endstation at BESSY II[26] to determine the film thickness via x-ray
reflectivity (XRR) and to characterize the first order AFM-FM phase transition
via the concomitant change of the mean out-of-plane lattice constant $d$
(symbols in Fig. 1(b)). The hysteresis for this locally probed lattice
constant is narrower than the global temperature-dependent magnetization
$M_{\text{FeRh}}$ (solid line) determined by Vibrating sample magnetometry
(VSM) using a QuantumDesign VersaLab magnetometer. The magnetization data
indicates the presence of a residual FM phase of around $20\,\%$ originating
from interface effects[27, 28, 29] consistent with the reduced out-of-plane
expansion of $\eta^{\text{thin}}_{\text{AFM-FM}}=0.48\,\%$ compared to
$\eta^{\text{thick}}_{\text{AFM-FM}}=0.6\,\%$ observed in thicker films[11,
21].

Figure 1(c) illustrates the influence of increasing the pump-pulse duration on
the optically induced electron-phonon non-equilibrium. We model the transient
mean electron $T_{\text{el}}$ and phonon $T_{\text{ph}}$ temperature of the
FeRh film as function of the pump-pulse duration in the framework of a
diffusive two-temperature model[30]. We use the modular Python library
udkm1Dsim [31] and literature values for the Sommerfeld constant[32]
$\gamma^{\text{S}}=0.06\,\text{J}\text{kg}^{-1}\text{K}^{-2}$, the phononic
heat capacity[33] $C_{\text{ph}}=350\,\text{Jkg}^{-1}\text{K}^{-1}$ and the
electron-phonon coupling constant [34] $g^{\text{el-ph}}=9\cdot
10^{13}\,\text{Jkg}^{-1}\text{K}^{-1}$. The thermophysical properties of the
MgO substrate were used as reported previously[11]. The excitation by a
$60\,\text{fs}$ pump pulse increases the electron temperature that stays
significantly higher than the slowly rising phonon temperature within the
first $3\,\text{ps}$, with the maximum $\Delta
T^{\text{max}}_{\text{el}}\approx 7\Delta T^{\text{max}}_{\text{ph}}$.
Increasing the pump pulse duration to $2.9\,\text{ps}$ drastically reduces the
maximum electron temperature, as a considerable amount of energy already
dissipated to the phonons during the optical deposition. For a pump pulse of
duration $10.5\,\text{ps}$, the electron temperature barely exceeds the phonon
temperature. A strong electron-phonon non-equilibrium with a substantial
amount of hot electrons is absent.

In the following, we experimentally apply such pump-pulses to investigate the
kinetics of the laser-induced phase transition in FeRh by UXRD. The thin film
is excited by p-polarized pump pulses with a central wavelength of
$800\,\text{nm}$ that are incident under $40^{\circ}$ with respect to the
sample normal. We probe the transient out-of-plane strain response of the FeRh
layer via symmetric $\theta-2\theta$ scans [35] around the FeRh(002) Bragg
peak at our table-top laser-driven plasma x-ray source [36] providing
$200\,\text{fs}$ hard x-ray pulses with a photon energy of approx.
$8\,\text{keV}$. The Bragg peak position along the reciprocal space coordinate
$q_{z}$ encodes the mean out-of-plane lattice constant $d$ of the FeRh films
via $q_{\text{z}}=4\pi/d$. The lattice strain $\eta_{\text{FeRh}}=\Delta
d/d_{0}$ is the relative change $\Delta d$ of the lattice constant with
respect to its value $d_{0}$ before excitation. We independently determined
the pump-probe overlap and calibrated the duration of the pump-pulse by the
strictly linear laser-induced response of a metal-insulator superlattice
serving as reference sample[36].

Figure 2(a) displays the laser-induced strain response of the thin FeRh film
for a fluence of $2.0\,\text{mJ}\,\text{cm}^{-2}$ and different pump pulse
durations ranging from $60\,\text{fs}$ to $10.5\,\text{ps}$. The excitation
fluence exceeds the previously identified critical threshold[11] of
$F_{\text{th}}=0.6\,\text{mJ}\,\text{cm}^{-2}$ for this sample at room
temperature and drives the magnetostructural phase transition that is
associated with an out-of-plane expansion of FeRh. In total, the strain
response is the superposition of an expansion due to the phase transition, a
quasi-static expansion due to heating and a propagating strain pulse reflected
at the surface and the FeRh-MgO interface, where it is partially transmitted
into the substrate. In case of the excitation by a $60\,\text{fs}$ pulse, this
results in a a decaying oscillation [30] with a period of
$2L_{\text{FeRh}}/v_{\text{s}}=5\,\text{ps}$ given by the layer thickness
$L_{\text{FeRh}}$ and the sound velocity
$v_{\text{s}}=5\,\text{nm}\,\text{ps}^{-1}$[37] (see Figure 2(a)). Increasing
the pump-pulse duration successively suppresses this oscillation of the mean
out-of-plane strain until $10.5\,\text{ps}$ pump-pulses only drive a slow
expansion. The strain at $40\,\text{ps}$ is identical for all pump-pulse
durations. This indicates that all strain contributions including the
expansion associated with the AFM-FM phase transition exclusively depend on
the deposited energy and not on the details of the optical excitation.

Figure 1:  Long pump-pulses suppress electron-phonon non-equilibrium: (a)
Sketch of the thin film sample and the UXRD experiment, mapping the reciprocal
space via $\theta-2\theta$ scans. (b) Characterization of the AFM-FM phase
transition in equilibrium by the magnetization (solid line) and the mean out-
of-plane lattice constant $d$ (symbols). (c) Modelled transient electron and
phonon temperature upon excitation by pump-pulses of various durations for a
fluence of $2.0\,\text{mJ}\,\text{cm}^{-2}$.

Fig. 2(b) compares the transient strain for pump pulse durations of
$5\,\text{ps}$ (symbols) and $60\,\text{fs}$ (grey solid curves) for various
super- and sub-threshold fluences. At first sight, the long pump pulses seem
just to smear out the oscillations resulting from coherent longitudinal
acoustic phonons. We obtain better insight by disentangling the linear
acoustic response of the sample from the expansion driven by the phase
transition, which depends non-linearly on the excitation fluence. The sub-
threshold fluence of $F_{\text{bt}}=0.4\,\text{mJ}\,\text{cm}^{-2}$ does not
induce the AFM-FM phase transition[11]. This simplifies the strain response to
a superposition of a quasi-static expansion and propagating strain pulses,
which both depend linearly on the deposited pulse energy. Therefore, the
acoustic strain contribution upon a super-threshold excitation is given by
this sub-threshold strain response $\eta^{\text{bt}}_{\text{FeRh}}$ (grey
solid line) scaled by the ratio of the fluences
$s=F_{\text{at}}/F_{\text{bt}}$ and convoluted with the respective pump-pulse
duration.

Figure 2:  Transient strain response contains expansion from phase transition:
The laser-induced out-of-plane expansion of FeRh for various pump-pulse
durations at the constant fluence $2\,\text{mJ}\,\text{cm}^{-2}$ (a) and as
function of the fluence for the constant pulse duration $5.2\,\text{ps}$ pump-
pulses (b). The grey solid lines denote the modelled strain response for the
respective fluence driven by femtosecond laser excitation reproduced as
reference from a previous publication[11]. The colored solid lines in (a) and
(b) display the model described in the text, which adds the fluence-scaled
strain response to $F_{\text{st}}=0.4\,\text{mJ}\,\text{cm}^{-2}$ convoluted
with the respective pump-pulse to the expansion
$\eta^{\text{thin}}_{\text{AFM-FM}}\Delta V_{\text{FM}}(t)$ associated with
the phase transition. $\Delta V_{\text{FM}}(t)$ is plotted in Fig. 3. The
results are off-set for clarity.

The dotted line in Fig. 2(b) depicts the acoustic strain contribution for
$F_{\text{at}}=2\,\text{mJ}\,\text{cm}^{-2}$ given by
$s\cdot\eta^{\text{bt}}_{\text{FeRh}}$ with $s=5$. The deviation
$\Delta\eta(t)=\eta^{\text{at}}_{\text{FeRh}}-s\cdot\eta^{\text{bt}}_{\text{FeRh}}$
from the strain response to the super-threshold excitation
$\eta^{\text{at}}_{\text{FeRh}}$ is essentially due to the additional
expansion induced by the phase transition, which directly provides insights
into the rise of the FM volume fraction $V_{\text{FM}}(t)$. To determine its
absolute value, we consider the residual FM phase $V^{0}_{\text{FM}}=0.2$
present before the laser excitation due to interface effects and the latent
heat $C_{\text{AFM-FM}}$ of the first-order phase transition. The energy
$Q_{\text{AFM-FM}}=\int C_{\text{AFM-FM}}dT$ required for transforming FeRh
into the FM phase reduces the local temperature[38] by $\Delta T=Q_{\text{AFM-
FM}}/C_{\text{ph}}$, which reduces the quasi-static expansion of FeRh by
$\eta_{\text{AFM}}=\alpha_{\text{AFM}}\Delta T$ with the expansion coefficient
$\alpha_{\text{AFM}}$ in the AFM phase. This relates the deviation in the
transient strain $\Delta\eta(t)$ to the laser-induced FM volume fraction
$\Delta V_{\text{FM}}$ via:

$\Delta\eta(t)=(\eta^{\text{thick}}_{\text{AFM-FM}}-\eta_{\text{AFM}})\Delta
V_{\text{FM}}(t)\,.$ (1)

Figure 3 displays the laser-induced volume fraction $\Delta V_{\text{FM}}$
derived from the measured transient strain via Eq. 1. The variation of the
excitation fluence in Fig. 3(b) exemplifies the non-linear response at the
phase transition: A tiny increase of the fluence from $1.8$ to
$2.1\,\text{mJ}\,\text{cm}^{-2}$ changes $\Delta V_{\text{FM}}$ from $60\%$ to
a complete phase transition, which corresponds to $\Delta V_{\text{FM}}=80\%$.
The data for $60\,\text{fs}$ pulse excitation in Fig. 3(a) is very well
reproduced by the nucleation of domains on a $\tau=8\,\text{ps}$ timescale by
[30]:

$\displaystyle\Delta V_{\text{FM}}(t)$
$\displaystyle=\mathcal{H}(t)V_{\text{FM}}^{*}\cdot\left(1-e^{-t/\tau}\right)\;,$
(2)

with the Heaviside function $\mathcal{H}(t)$ and the final FM volume fraction
increase $V_{\text{FM}}^{*}$ that depends on the fluence with
$V_{\text{FM}}^{*}>0$ if $F>F_{\text{th}}$ and $V_{\text{FM}}^{*}=0.8$ if
$F=2.1\,\text{mJ}\,\text{cm}^{-2}$ as reported previously[11].

Figure 3:  Laser-induced phase transition: (a) Laser-induced rise of the FM
volume fraction $\Delta V_{\text{FM}}$ for different pump-pulse durations
(pulse profile indicated by grey solid lines) and a fluence of
$2\,\text{mJ}\,\text{cm}^{-2}$. (b) Laser-induced rise of $\Delta
V_{\text{FM}}$ for various super-threshold fluences and $5.2\,\text{ps}$ pump-
pulse duration. The coloured areas denote the fraction of the film that
remains in the AFM phase. For $2.1\,\text{mJ}\,\text{cm}^{-2}$ the complete
film ends up in the FM phase. In both panels, the dashed lines represent the
FM volume fraction according to Eq. 2, i.e. for ultrashort pulse excitation,
convoluted with a Gaussian representing the duration of the respective pump-
pulse. The data rise faster but starts delayed compared to the dashed lines,
which assume linear response via the convolution. The solid lines show the
appropriate model for the FM volume fraction according to Eq. 3, which
accounts for the non-linear threshold behavior. The plots are off-set for
clarity.

Equation 2 yields excellent agreement with $\Delta V_{\text{FM}}$ for
$60\,\text{fs}$ in Fig. 3(a). As a first attempt, the dashed lines for the
longer pump-pulses in Fig. 3 represent $\Delta V_{\text{FM}}(t)$ from Eq. 2
convoluted with a Gaussian representing the pump-pulse in the experiment.
However, the deviation of the measured transient FM volume fraction from this
simple estimation becomes larger with increasing pump-pulse duration (see Fig.
3(a)). Interestingly, the data rise faster than the convolution although the
rise starts later, which must be a consequence of the non-linearity associated
with the threshold for the phase transition.

As an improved model of the rising $\Delta V_{\text{FM}}$ for long pump-
pulses, we explicitly consider the successively deposited energy that leads to
the unlocking of the AFM-FM phase transition in an increasing volume fraction
of the film during the pump-pulse. This explicit treatment of the threshold
character of first-order phase transitions extends Eq. 2 in the case of
picosecond pump-pulses to:

$\displaystyle\Delta V_{\text{FM}}(t)$
$\displaystyle=\int\mathcal{H}(t-t^{\prime})V_{\text{FM}}^{*}(t^{\prime})\cdot\left(1-e^{-(t-t^{\prime})/\tau}\right)dt^{\prime}\;.$
(3)

Here, the FM volume fraction $V_{\text{FM}}^{*}(t^{\prime})$ is unlocked at
delay $t^{\prime}$ by the increase of the deposited energy and rises on the
$8\,\text{ps}$ nucleation timescale. This transiently unlocked phase
transition adds to the already present FM phase driven at delays
$t<t^{\prime}$. The Heaviside function $\mathcal{H}(t-t^{\prime})$ ensures a
start of the phase transition at $t^{\prime}$. To model the transient FM
volume fraction we assume a linear increase of $V_{\text{FM}}^{*}$ from $0$ to
$1$ between the fluences $F_{\text{th}}=0.6\,\text{mJ}\,\text{cm}^{-2}$ and
$2.1\,\text{mJ}\,\text{cm}^{-2}$, which corresponds to a full phase transition
as characterized previously[11].

Under this assumption, Eq. 3 yields excellent agreement (see solid lines) with
the experimentally determined transients $\Delta V_{\text{FM}}(t)$ in Fig. 3(a
and b) for various pump-pulse durations and super-threshold fluences. Our
model including the critical threshold of the first-order phase transition
reproduces both the delayed start of the domain nucleation relative to the
beginning of the pump pulse for longer pump-pulses and the earlier start of
the domain nucleation with increasing fluence, which highlights the central
role of the threshold for the laser-induced phase transition.

In summary, we studied the laser-induced magnetostructural AFM-FM phase
transition in FeRh driven by picosecond pump-pulses via UXRD. The extracted
transient FM volume fraction highlights the crucial role of the threshold for
the first-order phase transition in FeRh. The insensitivity of the final FM
volume fraction on varying the pump-pulse duration from 60 fs up to
$10.5\,\text{ps}$ reveals that the threshold is exclusively determined by the
amount of deposited energy and that the laser-induced AFM-FM phase transition
does not need to proceed though the generation of hot non-equilibrium
electrons. With increasing pump-pulse duration we observe an increasing
deviation of the FM volume fraction from a linear response model. We
successfully model the data using the intrinsic nucleation of FM domains on an
$8\,\text{ps}$ timescale, by simply considering the slow deposition of energy
by picosecond pump-pulses, which successively overcomes the critical threshold
that makes the phase transition dynamics nonlinear.

We acknowledge the DFG for financial support via Project-No. 328545488 – TRR
227, project A10 and the BMBF for funding via 05K22IP1. Access to the CEITEC
Nano Research Infrastructure was supported by the Ministry of Education, Youth
and Sports (MEYS) of the Czech Republic under the project CzechNanoLab
(LM2023051).

## References

  * Qazilbash  _et al._ [2007] M. M. Qazilbash, M. Brehm, B.-G. Chae, P.-C. Ho, G. O. Andreev, B.-J. Kim, S. J. Yun, A. Balatsky, M. Maple, F. Keilmann, _et al._ , “Mott transition in vo2 revealed by infrared spectroscopy and nano-imaging,” Science 318, 1750–1753 (2007).
  * Roy  _et al._ [2004] S. Roy, G. Perkins, M. Chattopadhyay, A. Nigam, K. Sokhey, P. Chaddah, A. Caplin, and L. Cohen, “First order magnetic transition in doped cefe 2 alloys: Phase coexistence and metastability,” Physical review letters 92, 147203 (2004).
  * Uhlíř, Arregi, and Fullerton [2016] V. Uhlíř, J. A. Arregi, and E. E. Fullerton, “Colossal magnetic phase transition asymmetry in mesoscale ferh stripes,” Nature communications 7, 13113 (2016).
  * Baldasseroni  _et al._ [2012] C. Baldasseroni, C. Bordel, A. Gray, A. Kaiser, F. Kronast, J. Herrero-Albillos, C. Schneider, C. Fadley, and F. Hellman, “Temperature-driven nucleation of ferromagnetic domains in ferh thin films,” Applied Physics Letters 100, 262401 (2012).
  * Randi  _et al._ [2016] F. Randi, I. Vergara, F. Novelli, M. Esposito, M. Dell’Angela, V. Brabers, P. Metcalf, R. Kukreja, H. A. Dürr, D. Fausti, _et al._ , “Phase separation in the nonequilibrium verwey transition in magnetite,” Physical Review B 93, 054305 (2016).
  * De Jong  _et al._ [2013] S. De Jong, R. Kukreja, C. Trabant, N. Pontius, C. Chang, T. Kachel, M. Beye, F. Sorgenfrei, C. Back, B. Bräuer, _et al._ , “Speed limit of the insulator–metal transition in magnetite,” Nature materials 12, 882–886 (2013).
  * Wegkamp  _et al._ [2014] D. Wegkamp, M. Herzog, L. Xian, M. Gatti, P. Cudazzo, C. L. McGahan, R. E. Marvel, R. F. Haglund Jr, A. Rubio, M. Wolf, _et al._ , “Instantaneous band gap collapse in photoexcited monoclinic vo 2 due to photocarrier doping,” Physical review letters 113, 216401 (2014).
  * Ju  _et al._ [2004] G. Ju, J. Hohlfeld, B. Bergman, R. J. van de Veerdonk, O. N. Mryasov, J.-Y. Kim, X. Wu, D. Weller, and B. Koopmans, “Ultrafast generation of ferromagnetic order via a laser-induced phase transformation in ferh thin films,” Physical review letters 93, 197403 (2004).
  * Radu  _et al._ [2010] I. Radu, C. Stamm, N. Pontius, T. Kachel, P. Ramm, J.-U. Thiele, H. Dürr, and C. Back, “Laser-induced generation and quenching of magnetization on ferh studied with time-resolved x-ray magnetic circular dichroism,” Physical Review B 81, 104415 (2010).
  * Mariager  _et al._ [2012] S. O. Mariager, F. Pressacco, G. Ingold, A. Caviezel, E. Möhr-Vorobeva, P. Beaud, S. Johnson, C. Milne, E. Mancini, S. Moyerman, _et al._ , “Structural and magnetic dynamics of a laser induced phase transition in ferh,” Physical Review Letters 108, 087201 (2012).
  * Mattern  _et al._ [2023a] M. Mattern, J. Jarecki, J. A. Arregi, V. Uhlíř, M. Rössle, and M. Bargheer, “Disentangling nucleation and domain growth during a laser-induced phase transition,”  (2023a), 10.48550/arXiv.2305.02094.
  * Mattern  _et al._ [2023b] M. Mattern, J.-E. Pudell, K. Dumesnil, A. von Reppert, and M. Bargheer, “Towards shaping picosecond strain pulses via magnetostrictive transducers,” Photoacoustics 30, 100463 (2023b).
  * Polesya  _et al._ [2016] S. Polesya, S. Mankovsky, D. Ködderitzsch, J. Minár, and H. Ebert, “Finite-temperature magnetism of ferh compounds,” Physical Review B 93, 024423 (2016).
  * Kittel [1960] C. Kittel, “Model of exchange-inversion magnetization,” Physical Review 120, 335 (1960).
  * Gu and Antropov [2005] R. Gu and V. Antropov, “Dominance of the spin-wave contribution to the magnetic phase transition in ferh,” Physical Review B 72, 012403 (2005).
  * Gruner, Hoffmann, and Entel [2003] M. Gruner, E. Hoffmann, and P. Entel, “Instability of the rhodium magnetic moment as the origin of the metamagnetic phase transition in $\alpha$\- ferh,” Physical Review B 67, 064415 (2003).
  * Sandratskii and Mavropoulos [2011] L. M. Sandratskii and P. Mavropoulos, “Magnetic excitations and femtomagnetism of ferh: A first-principles study,” Physical Review B 83, 174408 (2011).
  * Barker and Chantrell [2015] J. Barker and R. W. Chantrell, “Higher-order exchange interactions leading to metamagnetism in ferh,” Physical Review B 92, 094402 (2015).
  * Pressacco  _et al._ [2021] F. Pressacco, D. Sangalli, V. Uhlíř, D. Kutnyakhov, J. A. Arregi, S. Y. Agustsson, G. Brenner, H. Redlin, M. Heber, D. Vasilyev, _et al._ , “Subpicosecond metamagnetic phase transition in ferh driven by non-equilibrium electron dynamics,” Nature Communications 12, 5088 (2021).
  * Quirin  _et al._ [2012] F. Quirin, M. Vattilana, U. Shymanovich, A.-E. El-Kamhawy, A. Tarasevitch, J. Hohlfeld, D. von der Linde, and K. Sokolowski-Tinten, “Structural dynamics in ferh during a laser-induced metamagnetic phase transition,” Physical Review B 85, 020103 (2012).
  * Mattern  _et al._ [2023c] M. Mattern, J.-E. Pudell, J. A. Arregi, J. Zlámal, R. Kalousek, V. Uhlíř, M. Rössle, and M. Bargheer, “Accelerating the laser-induced phase transition in nanostructured ferh via plasmonic absorption,”  (2023c), 10.48550/arXiv.2309.12683.
  * Bergman  _et al._ [2006] B. Bergman, G. Ju, J. Hohlfeld, R. J. van de Veerdonk, J.-Y. Kim, X. Wu, D. Weller, and B. Koopmans, “Identifying growth mechanisms for laser-induced magnetization in ferh,” Physical Review B 73, 060407 (2006).
  * Li  _et al._ [2022] G. Li, R. Medapalli, J. Mentink, R. Mikhaylovskiy, T. Blank, S. Patel, A. Zvezdin, T. Rasing, E. Fullerton, and A. Kimel, “Ultrafast kinetics of the antiferromagnetic-ferromagnetic phase transition in ferh,” Nature Communications 13, 2998 (2022).
  * Dolgikh  _et al._ [2022] I. Dolgikh, T. Blank, G. Li, K. Prabhakara, S. Patel, A. Buzdakov, R. Medapalli, E. Fullerton, O. Koplak, J. Mentink, _et al._ , “Ultrafast emergence of ferromagnetism in antiferromagnetic ferh in high magnetic fields,”  (2022), 10.48550/arXiv.2202.03931.
  * Arregi, Caha, and Uhlíř [2020] J. A. Arregi, O. Caha, and V. Uhlíř, “Evolution of strain across the magnetostructural phase transition in epitaxial ferh films on different substrates,” Physical Review B 101, 174413 (2020).
  * Rössle  _et al._ [2021] M. Rössle, W. Leitenberger, M. Reinhardt, A. Koç, J. Pudell, C. Kwamen, and M. Bargheer, “The time-resolved hard x-ray diffraction endstation kmc-3 xpp at bessy ii,” Journal of Synchrotron Radiation 28, 948–960 (2021).
  * Pressacco  _et al._ [2016] F. Pressacco, V. Uhl$\acute{\iota}$ř, M. Gatti, A. Bendounan, E. E. Fullerton, and F. Sirotti, “Stable room-temperature ferromagnetic phase at the ferh (100) surface,” Scientific reports 6, 22383 (2016).
  * Fan  _et al._ [2010] R. Fan, C. J. Kinane, T. Charlton, R. Dorner, M. Ali, M. De Vries, R. M. Brydson, C. H. Marrows, B. J. Hickey, D. A. Arena, _et al._ , “Ferromagnetism at the interfaces of antiferromagnetic ferh epilayers,” Physical Review B 82, 184418 (2010).
  * Chen  _et al._ [2017] X. Chen, J. Feng, Z. Wang, J. Zhang, X. Zhong, C. Song, L. Jin, B. Zhang, F. Li, M. Jiang, _et al._ , “Tunneling anisotropic magnetoresistance driven by magnetic phase transition,” Nature Communications 8, 449 (2017).
  * Mattern  _et al._ [2023d] M. Mattern, A. von Reppert, S. P. Zeuschner, M. Herzog, J.-E. Pudell, and M. Bargheer, “Concepts and use cases for picosecond ultrasonics with x-rays,” Photoacoustics , 100503 (2023d).
  * Schick [2021] D. Schick, “udkm1dsim–a python toolbox for simulating 1d ultrafast dynamics in condensed matter,” Computer Physics Communications 266, 108031 (2021).
  * Tu  _et al._ [1969] P. Tu, A. Heeger, J. Kouvel, and J. Comly, “Mechanism for the first-order magnetic transition in the ferh system,” Journal of Applied Physics 40, 1368–1369 (1969).
  * Richardson, Melville, and Ricodeau [1973] M. Richardson, D. Melville, and J. Ricodeau, “Specific heat measurements on an fe rh alloy,” Physics Letters A 46, 153–154 (1973).
  * Günther  _et al._ [2014] S. Günther, C. Spezzani, R. Ciprian, C. Grazioli, B. Ressel, M. Coreno, L. Poletto, P. Miotti, M. Sacchi, G. Panaccione, _et al._ , “Testing spin-flip scattering as a possible mechanism of ultrafast demagnetization in ordered magnetic alloys,” Physical Review B 90, 180407 (2014).
  * Schick  _et al._ [2013] D. Schick, R. Shayduk, A. Bojahr, M. Herzog, C. v. Korff Schmising, P. Gaal, and M. Bargheer, “Ultrafast reciprocal-space mapping with a convergent beam,” Journal of Applied Crystallography 46, 1372–1377 (2013).
  * Schick  _et al._ [2012] D. Schick, A. Bojahr, M. Herzog, C. v. K. Schmising, R. Shayduk, W. Leitenberger, P. Gaal, and M. Bargheer, “Normalization schemes for ultrafast x-ray diffraction using a table-top laser-driven plasma source,” Review of Scientific Instruments 83, 025104 (2012).
  * Palmer, Dentschuk, and Melville [1975] S. Palmer, P. Dentschuk, and D. Melville, “Elastic properties of an iron-rhodium alloy,” physica status solidi (a) 32, 503–508 (1975).
  * Ahn  _et al._ [2022] Y. Ahn, M. J. Cherukara, Z. Cai, M. Bartlein, T. Zhou, A. DiChiara, D. A. Walko, M. Holt, E. E. Fullerton, P. G. Evans, _et al._ , “X-ray nanodiffraction imaging reveals distinct nanoscopic dynamics of an ultrafast phase transition,” Proceedings of the National Academy of Sciences 119, e2118597119 (2022).


# On the best constant in fractional $p$-Poincaré inequalities on cylindrical
domains

Kaushik Mohanta1 and Firoj Sk2 Indian Institute of Technology Kanpur, India
<EMAIL_ADDRESS>(Corresponding author<EMAIL_ADDRESS>
###### Abstract.

We investigate the best constants for the regional fractional $p$-Poincaré
inequality and the fractional $p$-Poincaré inequality in cylindrical domains.
For the special case $p=2$, the result was already known due to Chowdhury-
Csató-Roy-Sk [Study of fractional Poincaré inequalities on unbounded domains,
Discrete Contin. Dyn. Syst., 41(6), 2021]. We addressed the asymptotic
behaviour of the first eigenvalue of the nonlocal Dirichlet p-Laplacian
eigenvalue problem when the domain is becoming unbounded in several
directions.

###### 2020 Mathematics Subject Classification:

26D10; 35R09; 46E35; 49J40.

##  1\. introduction

In the theory of partial differential equations, Poincaré inequality has
always played an important role. In recent years, the study of various
nonlocal analogues of Poincaré inequality has seen a steep surge. For the
particular case $p=2$, Chowdhury-Csató-Roy-Sk [CCRS21] have found the best
constants for fractional Poincaré inequalities in certain unbounded domains.
The aim of this article is to generalise [CCRS21] for any $p\in(1,\infty)$.

For any open set $\Omega\subseteq\mathbb{R}^{n}$, $0<s<1$, $1\leq p<\infty$,
we define the fractional Sobolev space

$W^{s,p}(\Omega):=\left\\{u\in
L^{p}(\Omega):[u]_{s,p,\Omega}<\infty\right\\},$

where

$[u]_{s,p,\Omega}:=\left(\frac{C_{n,s,p}}{2}\int_{\Omega}\int_{\Omega}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}dxdy\right)^{\frac{1}{p}},$
(1.1)

is the so called Gagliardo seminorm. The constant $C_{n,s,p}$ [AW19] is given
by

$C_{n,s,p}=\frac{sp\;2^{2s-1}\Gamma\left(\frac{n+sp}{2}\right)}{2\pi^{\frac{n-1}{2}}\Gamma(1-s)\Gamma\left(\frac{p+1}{2}\right)}.$
(1.2)

We endow this space with the so-called fractional Sobolev norm, given by

$||u||_{s,p,\Omega}:=\left(\|u\|_{L^{p}(\Omega)}^{p}+[u]_{s,p,\Omega}^{p}\right)^{\frac{1}{p}}.$

At this point, we would like to introduce two more Banach spaces, directly
related to the fractional Sobolev spaces $W^{s,p}(\Omega)$ defined above,
which will be useful in framing the problem dealt in this article. The spaces
$W^{s,p}_{\Omega}(\mathbb{R}^{n})$ and $W^{s,p}_{0}(\Omega)$ denote the
closures of $C_{c}^{\infty}(\Omega)$ with respect to the norms
$\left(\|u\|_{L^{p}(\Omega)}^{p}+[u]_{s,p,\mathbb{R}^{n}}^{p}\right)^{\frac{1}{p}}$
and $||\cdot||_{s,p,\Omega}$ respectively. The Gagliardo seminorm and the
fractional Sobolev norm are also important tools for studying the fractional
$p$-Laplacian operator, defined by

$(-\Delta_{n,p})^{s}u(x):=C_{n,s,p}\lim_{\begin{subarray}{c}\epsilon\to
0\end{subarray}}\int_{\mathbb{R}^{n}\setminus
B_{\epsilon}(x)}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{n+sp}}dy,\;\;\;x\in\mathbb{R}^{n}.$

Note that while defining the Gagliardo seminorm and the fractional
$p$-Laplacian operator, in the existing literature, the constant $C_{n,s,p}$
is often ignored. However, we shall take this constant into account as this
will be convenient while studying the best constants in fractional Poincaŕe
inequalities (see remark 1.3 bellow). We refer the reader to [AF03, BS19,
DNPV12, FSV15, FP14, LL14, DPFBLR18] for basic results regarding the
fractional Sobolev spaces and the fractional $p$-Laplacian operator.

The regional fractional Poincaré constant $P^{1}_{n,s,p}(\Omega)$, and
fractional Poincaré constant $P^{2}_{n,s,p}(\Omega)$ are defined as follows:

$P^{1}_{n,s,p}(\Omega):=\inf_{\begin{subarray}{c}u\in W_{0}^{s,p}(\Omega)\\\
u\neq
0\end{subarray}}\frac{[u]_{s,p,\Omega}^{p}}{\displaystyle\int_{\Omega}|u|^{p}},\;\text{
and }\;P^{2}_{n,s,p}(\Omega):=\inf_{\begin{subarray}{c}u\in
W^{s,p}_{\Omega}(\mathbb{R}^{n})\\\ u\neq
0\end{subarray}}\frac{[u]_{s,p,\mathbb{R}^{n}}^{p}}{\displaystyle\int_{\Omega}|u|^{p}}.$

######  Definition 1.1.

Let $\Omega$ be an open set in $\mathbb{R}^{n}$. We say that

  * •

the regional fractional Poincaré inequality (RFPI) holds if
$P^{1}_{n,s,p}(\Omega)>0$.

  * •

the fractional Poincaré inequality (FPI) holds if $P^{2}_{n,s,p}(\Omega)>0$.

For $\ell>0$, set $\Omega_{\ell}:=\ell\omega_{1}\times\omega$, where
$\omega_{1}$ and $\omega$ are bounded open subsets of $\mathbb{R}^{m}$ and
$\mathbb{R}^{n-m}$ respectively, and consider the nonlocal Dirichlet
$p$-Laplacian eigenvalue problem on $\Omega_{\ell}$:

$\begin{cases}(-\Delta_{n,p})^{s}u_{\ell}=P_{n,s,p}^{2}(\Omega_{\ell})|u_{\ell}|^{p-2}u_{\ell}\text{
in }\Omega_{\ell},\\\ u_{\ell}=0\text{ in
}\mathbb{R}^{n}\setminus\Omega_{\ell},\end{cases}$ (1.3)

and the corresponding cross section eigenvalue problem of eq. 1.3

$\begin{cases}(-\Delta_{n-m,p})^{s}u=P_{n-m,s,p}^{2}(\omega)|u|^{p-2}u\text{
in }\omega,\\\ u=0\text{ in }\mathbb{R}^{n-m}\setminus\omega.\end{cases}$
(1.4)

In the existing literature, the RFPI and FPI, in unbounded domains, are not
much explored yet. However, it is well-known that the FPI holds true that is
$P^{2}_{n,s,p}(\Omega)>0$, when $\Omega$ is a bounded domain [BLP14]. In
[BC18], the authors have shown that the FPI holds true for any domain which is
bounded in one direction, although the question of best fractional Poincaré
constant remained unattended (see also [Yer14]). In the special case $p=2$, it
is known (see [CCRS21]) that the best fractional Poincaré constants
$P_{n,s,2}^{1}(\mathbb{R}^{n-1}\times(-1,1))$ and
$P^{2}_{n,s,2}(\mathbb{R}^{m}\times\omega)$ are equal to that of the cross
sections, that is to $P^{1}_{1,s,2}((-1,1))$ and $P^{2}_{n-m,s,2}(\omega)$
respectively, where $\omega$ is a bounded domain in $\mathbb{R}^{n-m}$. The
first work in this direction, to the best of our knowledge, was done in
[CCRS21, AFM20]. Later, some of the results of [CCRS21] were generalized in
the Orlicz fractional Sobolev setup in [BMRS20]. Regarding the RFPI, it is
known that when $\Omega$ is a bounded domain with Lipschitz boundary, RFPI
does not hold, that is $P^{1}_{n,s,p}(\Omega)=0$ if $0<s\leq\frac{1}{p}$
[AW19, War15]. The RFPI, however, holds true, that is
$P^{1}_{n,s,p}(\Omega)>0$ if $\frac{1}{p}<s<1$, when $\Omega$ is any bounded
domain in $\mathbb{R}^{n}$. We refer the reader to [AW19, War15] for other
related results regarding regional fractional $p$-Laplacian operator.
Regarding the asymptotic behaviour of the first eigenvalue
$P^{2}_{n,s,p}(\Omega_{\ell})$ of eq. 1.3, when $\ell\to\infty$, in the linear
case, that is for $p=2$, Chowdhury-Roy [CR17] proved that the first eigenvalue
$P^{2}_{n,s,2}(\Omega_{\ell})$ of eq. 1.3 converges to the first eigenvalue
$P^{2}_{n-m,s,2}(\omega)$ of eq. 1.4, when $\ell\to\infty$. Different kind of
problems were studied regarding the asymptotic behaviour of
$P^{2}_{n-m,s,2}(\Omega_{\ell})$ as $\ell\to\infty$; we refer the readers to
[CRS13, ERS21, CR17, Yer14] and the references therein for more relevant
information in this direction.

Our first result depicts, for a cylindrical domain, that the best constant for
FPI of the domain and that of the cross-section are the same. Indeed, we have
the following result:

######  Theorem 1.2.

Let $0<s<1$, $1<p<\infty$ and $\Omega_{\infty}=\mathbb{R}^{m}\times\omega$ in
$\mathbb{R}^{n}$ with $1\leq m<n$, where $\omega$ is a bounded open subset of
$\mathbb{R}^{n-m}$. Then we have

$P^{2}_{n,s,p}(\Omega_{\infty})=P^{2}_{n-m,s,p}(\omega).$ (1.5)

Furthermore, the best fractional Poincaré constant
$P^{2}_{n,s,p}(\Omega_{\infty})$, is never achvied.

######  Remark 1.3.

Recall that we took into account, the constant $\frac{C_{n,s,p}}{2}$ (defined
in eq. 1.2) while defining the seminorm in eq. 1.1. If one ignores this
constant in the definition of the seminorm, then one would get an additional
multiplicative constant in eq. 1.5 in place of plain equality in theorem 1.2.

The strategy for the proof of theorem 1.2, done in section 3, is somewhat
analogous to the local case. It is as follows: the constant
$P^{2}_{n,s,p}(\Omega_{\infty})$ is bounded above by the constant
$P^{2}_{n,s,p}(\omega)$ (see (2) of proposition 3.2). For the special case
$p=2$, the regularity of the first eigenfunction of eq. 1.4 is extensively
used for the bounded below case. However, for general $p>1$, we do not have
such regularity theory of the first eigenfunction of eq. 1.4. We use simple
approximation argument and discrete Picone inequality (see, lemma 2.5) to
prove this part. The last part of the theorem is proven via contradiction,
where the geometry of the domain has played an important role. Note that the
use of discrete Picone inequality forces us to exclude the case $p=1$ from the
statement of the theorem 1.2.

Next, we deal with the case of RFPI. As above, we show that the best constant
for RFPI for a strip is equal to that of its cross-section.

######  Theorem 1.4.

Let $0<s<1$, $1\leq p<\infty$ and
$\Omega_{\infty}=(-1,1)\times\mathbb{R}^{n-1}\subset\mathbb{R}^{n}$, then we
have the following:

  1. (1)

$P^{1}_{n,s,p}(\Omega_{\infty})=P^{1}_{1,s,p}((-1,1))=0$, if
$0<s\leq\frac{1}{p}$.

  2. (2)

$P^{1}_{n,s,p}(\Omega_{\infty})=P^{1}_{1,s,p}((-1,1)).$ Consequently,
$P^{1}_{n,s,p}(\Omega_{\infty})>0$, if $\frac{1}{p}<s<1$.

The proof of theorem 1.4 goes along the same line as it is done in [CCRS21]
for $p=2$ but with necessary modifications. However, the method of the proof
differs significantly from that of theorem 1.2 and hence, in this case, we
must stick to the case $m=1$, $\omega=(-1,1)$,.

Finally, we come to our last main result, which shows the asymptotic behaviour
of the first eigenvalue of eq. 1.3.

######  Theorem 1.5.

Let $0<s<1$, $1<p<\infty$, $\ell>0$ and
$\Omega_{\ell}=\ell\omega_{1}\times\omega$ in $\mathbb{R}^{n}$ with $1\leq
m<n$, where $\omega_{1},\ \omega$ are bounded open subsets of $\mathbb{R}^{m}$
and $\mathbb{R}^{n-m}$ respectively. We then have

$P_{n-m,s,p}^{2}(\omega)\leq P^{2}_{n,s,p}(\Omega_{\ell})\leq
P_{n-m,s,p}^{2}(\omega)+\frac{C_{1}}{\ell^{s}}+\frac{C_{2}}{\ell^{sp}},$

where $C_{1},C_{2}>0$ are constants independent of $\ell.$ Furthermore, if
$\Omega_{\infty}=\bigcup_{\ell>0}\Omega_{\ell}$

$\lim\limits_{\ell\to\infty}P^{2}_{n,s,p}(\Omega_{\ell})=P^{2}_{n-m,s,p}(\omega)=P^{2}_{n,s,p}(\Omega_{\infty}).$

This article is organized in the following way: In section 2 we recall some
results, already known in the literature. In section 3 we give proofs of
theorems 1.4, 1.2 and 1.5.

##  2\. Some known results and conventions

Here we briefly discuss the notations that we shall use throughout the paper.

  * •

$s$ will always be understood to be in $(0,1)$.

  * •

For any positive integer $n$ and a measurable set
$\Omega\subset\mathbb{R}^{n}$ we write $\mathcal{L}^{n}(\Omega)$ to denote the
Lebesgue measure of $\Omega$, or shortly $|\Omega|$ if $n$ is understood from
the context.

  * •

$B_{R}(x)$ denotes a ball of radius $R$ centered at $x.$ We shall also write
$B_{R}$ for $B_{R}(0)$.

  * •

$\mathbb{S}^{m-1}$ is the unit sphere in the Euclidean space $\mathbb{R}^{m}.$

  * •

$\mathcal{H}^{k}$ denotes the $k$-dimensional Hausdorff measure, so that

$\mathcal{H}^{n-1}\left(\mathbb{S}^{n-1}\right)=\frac{2\pi^{\frac{n}{2}}}{\Gamma\left(\frac{n}{2}\right)},\quad\text{where
$\Gamma$ is the standard gamma function}.$ (2.1)

  * •

The beta function, for $x,y>0$, is defined by

$B(x,y):=\int_{0}^{1}t^{x-1}(1-t)^{y-1}dt=B(x,y)=2\int_{0}^{\frac{\pi}{2}}(\sin\theta)^{2x-1}(\cos\theta)^{2y-1}d\theta=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}.$
(2.2)

We now state some definitions and results, already known in literature, which
we shall be using in the subsequent sections in this article. But before
defining these, we would like to recall a result which follows from [DPFBLR18,
Lemma 2.7].

######  Lemma 2.1.

Let $\Omega\subset\mathbb{R}^{n}$ be an open set, $p>1$. Then

$\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))\psi(x)}{|x-y|^{n+sp}}dxdy\\\
=\frac{1}{2}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))(\psi(x)-\psi(y))}{|x-y|^{n+sp}}dxdy,$

whenever the integral in the left hand side is finite or
$[u]_{s,p,\mathbb{R}^{n}},\ \|\psi\|_{L^{p}(\Omega)}<\infty$. Here the
integral in the LHS is to be understood in the principle value (P.V.) sense.

###### Proof.

Let $u,\psi\in W^{s,p}_{\Omega}(\mathbb{R}^{n})$. Set

$I:=\mbox{P.V.}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))\psi(x)}{|x-y|^{n+sp}}dxdy\\\
:=\lim\limits_{\epsilon\to
0}\iint_{|x-y|\geq\epsilon}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))\psi(x)}{|x-y|^{n+sp}}dxdy=\lim\limits_{\epsilon\to
0}I_{\epsilon}.$

If, $I_{\epsilon}$ is finite, we can write,
$I=\frac{1}{2}\lim\limits_{\epsilon\to 0}(I_{\epsilon}+I_{\epsilon})$, then we
make a change of variable, by interchanging $x$ and $y$, in the second
$I_{\epsilon}$ in the RHS and then we add the two terms in the RHS to conclude
the equality.

It remains to show that $[u]_{s,p,\mathbb{R}^{n}},\
\|\psi\|_{L^{p}(\Omega)}<\infty$ implies finiteness of $I_{\epsilon}$. The
following calculation is done in [BC18, lemma 2.3]. However, we include it
here for the sake of completeness. For a fixed $\epsilon>0$, we have

$\left|\iint_{|x-y|\geq\epsilon}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))\psi(x)}{|x-y|^{n+sp}}dxdy\right|\\\
\leq\iint_{|x-y|\geq\epsilon}\frac{|u(x)-u(y)|^{p-1}|\psi(x)|}{|x-y|^{n+sp}}dxdy\\\
=\iint_{|x-y|\geq\epsilon}\frac{|u(x)-u(y)|^{\frac{p}{p^{\prime}}}|\psi(x)|}{|x-y|^{\frac{n+sp}{p^{\prime}}}|x-y|^{\frac{n+sp}{p}}}dxdy\\\
\leq\left(\iint_{|x-y|\geq\epsilon}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}dxdy\right)^{\frac{1}{p^{\prime}}}\left(\iint_{|x-y|\geq\epsilon}\frac{|\psi(x)|^{p}}{|x-y|^{n+sp}}dxdy\right)^{\frac{1}{p}}\\\
\leq[u]_{s,p,\mathbb{R}^{n}}^{\frac{p}{p^{\prime}}}\left(\iint_{|x-y|\geq\epsilon}\frac{|\psi(x)|^{p}}{|x-y|^{n+sp}}dxdy\right)^{\frac{1}{p}}=C[u]_{s,p,\mathbb{R}^{n}}^{\frac{p}{p^{\prime}}}\|\psi\|_{L^{p}(\Omega)}\epsilon^{-s}<\infty.$

∎

Using this lemma, we define

######  Definition 2.2.

Let $\omega\subset\mathbb{R}^{n-m}$ be a bounded open set. A function $u\in
W^{s,p}_{\omega}(\mathbb{R}^{n-m})$ is said to be a weak solution of eq. 1.4
if $u$ satisfies

$\frac{C_{n-m,s,p}}{2}\int_{\mathbb{R}^{n-m}}\int_{\mathbb{R}^{n-m}}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))(\psi(x)-\psi(y))}{|x-y|^{n-m+sp}}dydx\\\
=C_{n-m,s,p}\int_{\mathbb{R}^{n-m}}\int_{\mathbb{R}^{n-m}}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))\psi(x)}{|x-y|^{n-m+sp}}dydx\\\
=P^{2}_{n-m,s,p}(\omega)\int_{\omega}|u(x)|^{p-2}u(x)\psi(x)dx,\text{ for all
}\psi\in W^{s,p}_{\omega}(\mathbb{R}^{n-m}).$

Any such $u$, not identically zero, is also called an eigenfunction of eq.
1.4, corresponding to the eigenvalue $P^{2}_{n-m,s,p}(\omega)$.

######  Lemma 2.3 (see [BP16]).

The constant $P^{2}_{n,s,p}(\omega)$ is the first eigenvalue of the problem
eq. 1.4, and the corresponding eigenfunction is strictly positive in the
domain. Moreover, the corresponding eigenspace is of dimension one.

######  Lemma 2.4 (see [LS10, Lemma 2.4] ).

Let $p>0,$ $0<s<1$ and $\Omega\subset\mathbb{R}^{n}$ be a measurable set. Then
for any $u\in C_{c}^{\infty}(\Omega)$

$\displaystyle
2\int_{\Omega}\int_{\Omega}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}dxdy\vskip
3.0pt plus 1.0pt minus 1.0pt$ $\displaystyle=$
$\displaystyle\int_{\mathbb{S}^{n-1}}d\mathcal{H}^{n-1}(w)\int_{\\{x:\;x\cdot
w=0\\}}d\mathcal{H}^{n-1}(x)\int_{\\{\ell:\,x+\ell
w\in\Omega\\}}\int_{\\{t:\,x+tw\in\Omega\\}}\frac{|u(x+\ell
w)-u(x+tw)|^{p}}{|\ell-t|^{1+sp}}dtd\ell.$

######  Lemma 2.5 (Discrete Picone inequality, [BF14]).

Let $p\in(1,\infty)$ and let $f,g:\mathbb{R}^{n}\to\mathbb{R}$ be two
measurable functions with $f\geq 0,\;g>0$, then $L(f,g)\geq 0\text{ in
}\mathbb{R}^{n}\times\mathbb{R}^{n},$ where

$L(f,g)(x,y)=|f(x)-f(y)|^{p}-|g(x)-g(y)|^{p-2}(g(x)-g(y))\left(\frac{f(x)^{p}}{g(x)^{p-1}}-\frac{f(y)^{p}}{g(y)^{p-1}}\right).$

The equality holds if and only if $f=\alpha g$ a.e. in $\mathbb{R}^{n}$ for
some constant $\alpha.$

The following result is well known in literature. It follows directly from the
definition of the Poincaré constant and of the Gagliardo seminorm.

######  Proposition 2.6.

Let $0<s<1$ and $p\in[1,\infty)$ we have

  1. (1)

(Domain monotonicity:) If
$\Omega_{1}\subseteq\Omega_{2}\subset\mathbb{R}^{n},$ then
$P^{2}_{n,s,p}(\Omega_{2})\leq P^{2}_{n,s,p}(\Omega_{1}).$

  2. (2)

(Dilation:) Let $\Omega\subset\mathbb{R}^{n}$ be an open set and let $u\in
W^{s,p}(t\Omega)$, for $t>0.$ We define $v_{t}(x)=u(tx)\in W^{s,p}(\Omega)$.
Then $[u]_{s,p,t\Omega}^{p}=t^{n-sp}[v_{t}]_{s,p,\Omega}^{p}$, and furthermore

$P^{1}_{n,s,p}(\Omega)=t^{n-sp}P^{1}_{n,s,p}(t\Omega),\text{ and
}P^{2}_{n,s,p}(\Omega)=t^{n-sp}P^{2}_{n,s,p}(\Omega).$

######  Remark 2.7.

To the best of our knowledge, the domain monotonicity property for
$P^{1}_{n,s,p}$ is not known in literature.

The proof of the following well-known result, in the case $sp<1$ can be found
in [Tri83, Theorem 3.4.3] for bounded $C^{\infty}$-domains. For bounded
Lipschitz domains, in the case $sp\leq 1$, this can be found in [Dyd04,
Section 2], [AW19, Theorem 2.1], [War15, Example 4.11]. For various related
results, we refer the reader to [DK21].

######  Lemma 2.8.

Let $p\in[1,\infty)$, and $\Omega$ be an open bounded set in $\mathbb{R}^{n}$
with Lipschitz boundary. Then $W^{s,p}_{0}(\Omega)=W^{s,p}(\Omega)$ if
$0<s\leq\frac{1}{p}$. In particular, in this case, we have
$P^{1}_{n,s,p}(\Omega)=0$.

######  Lemma 2.9.

Let $p\in[1,\infty)$, and $\Omega\subset\mathbb{R}^{n}$ be an open set. Then
$C_{c}^{1}(\Omega)\subset W^{1,p}_{0}(\Omega)\subset
W^{s,p}_{\Omega}(\mathbb{R}^{n})$.

###### Proof.

To show the first inclusion, let us take an arbitrary $v\in
C_{c}^{1}(\Omega)$. Then we apply [BMR20, Lemma 8] to get a bounded open set
$\Omega_{1}$ with smooth boundary such that
$\mbox{supp}(v)\subset\Omega_{1}\subset\Omega$. Clearly $v\in
W^{1,p}(\Omega_{1})$. Then we can say, from the well known trace theorem for
Sobolev spaces, that $v\in W^{1,p}_{0}(\Omega_{1})\subset
W^{1,p}_{0}(\Omega)$.

The last inclusion follows from [DNPV12, Proposition 2.2], which implies that
for any $u\in C_{c}^{\infty}(\Omega)$,

$\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}dxdy\leq
C(n,s,p)\left(\int_{\Omega}|u(x)|^{p}dx+\int_{\Omega}|\nabla
u(x)|^{p}dx\right)$

as for any $v\in W^{1,p}_{0}(\Omega)$, there exists a sequence of functions
$v_{n}\in C_{c}^{\infty}(\Omega)$, converging to $v$ in $W^{1,p}_{0}(\Omega)$.
The above inequality then suggests that the same sequence will converge to $v$
in $W^{s,p}_{\Omega}(\mathbb{R}^{n})$ as well. ∎

### Hyper-spherical Coordinates

Before moving further, let us recall the hyper-spherical coordinates and
derive an equality, which will be used in the forthcoming section.

Let

$A_{n-1}=(0,\pi)^{n-2}\times(0,2\pi)\subset\mathbb{R}^{n-1}.$

The hyper spherical coordinates
$H=(H_{1},\cdots,H_{n}):A_{n-1}\to\mathbb{S}^{n-1}$ are defined as follows:
for $k=1,\cdots,n$ and $\sigma=(\sigma_{1},\cdots,\sigma_{n-1})$

$H_{k}(\sigma)=\cos\sigma_{k}\prod_{l=0}^{k-1}\sin\sigma_{l}\quad\text{ with
the convention }\sigma_{0}=\frac{\pi}{2}\mbox{ and }\sigma_{n}=0.$

An elementary calculation shows $d_{i}(\sigma):=\left\langle\frac{\partial
H}{\partial\sigma_{i}},\frac{\partial
H}{\partial\sigma_{i}}\right\rangle=\prod_{l=0}^{i-1}\sin^{2}\sigma_{l}>0.$
One can easily verify that the metric tensor, in these coordinates, is
diagonal, that is $g_{ij}(\sigma)=\left\langle\frac{\partial
H}{\partial\sigma_{i}},\frac{\partial
H}{\partial\sigma_{j}}\right\rangle=\delta_{ij}d_{i}(\sigma),$ (Here
$\delta_{ij}$ denotes the usual ‘Kronecker delta’) and hence the surface
element $g_{n-1}$ is given by

$g_{n-1}(\sigma)=\sqrt{\det
g_{ij}(\sigma)}=\sqrt{\prod_{k=1}^{n-1}d_{k}(\sigma)}=\prod_{k=1}^{n-1}\prod_{l=0}^{k-1}\sin\sigma_{l}=\prod_{k=1}^{n-2}(\sin\sigma_{k})^{n-k-1}.$

##  3\. Proof of Main Results

######  Lemma 3.1.

Let $0<s<1$, $1\leq p<\infty$, and for any $m,n\in\mathbb{N}$ with $1\leq
m<n$. Let $C_{n,s,p}$ be the constant as in eq. 1.2. Then we have the
following:

(i) $C_{n,s,p}\Theta_{m,n,p}=C_{n-m,s,p},\quad\text{ where
}\quad\Theta_{m,n,p}=\mathcal{H}^{m-1}\left(\mathbb{S}^{m-1}\right)\int_{0}^{\infty}\frac{t^{m-1}}{(1+t^{2})^{\frac{n+sp}{2}}}\;dt$

(ii) If $a>0$ and $z\in\mathbb{R}^{m}$ then

$\displaystyle\int_{\mathbb{R}^{m}}\frac{dx}{\left(1+\frac{|x-z|^{2}}{a^{2}}\right)^{\frac{n+sp}{2}}}=a^{m}\Theta_{m,n,p}.$

###### Proof.

(i) Applying the change of variable $t=\tan\theta$ in the expression of
$\Theta_{m,n,p}$, followed by eqs. 2.1 and 2.2, we obtain

$\Theta_{m,n,p}=\mathcal{H}^{m-1}\left(\mathbb{S}^{m-1}\right)\int_{0}^{\frac{\pi}{2}}(\sin\theta)^{m-1}(\cos\theta)^{n-m+sp-1}\;d\theta\\\
=\frac{1}{2}B\bigg{(}\frac{m}{2},\frac{n-m+sp}{2}\bigg{)}\frac{2\pi^{\frac{m}{2}}}{\Gamma(\frac{m}{2})}=\frac{\pi^{\frac{m}{2}}\Gamma\left(\frac{n-m+sp}{2}\right)}{\Gamma\left(\frac{n+sp}{2}\right)}.$

From eq. 1.2 we get the desired result.

(ii) Taking the change of variable $y=\frac{x-z}{a}$, the identity follows
immediately. ∎

######  Proposition 3.2.

Let $m,n\in\mathbb{N}$ with $m<n$, $0<s<1$, $1\leq p<\infty$ and
$\Omega_{\infty}=\mathbb{R}^{m}\times\omega$, where
$\omega\subset\mathbb{R}^{n-m}$ is a bounded open set. Then we have

  1. (1)

$P^{1}_{n,s,p}(\Omega_{\infty})\leq P^{1}_{n-m,s,p}(\omega)$.

  2. (2)

$P^{2}_{n,s,p}(\Omega_{\infty})\leq P^{2}_{n-m,s,p}(\omega).$

###### Proof.

First, we prove (1). Note that if we can show, for any $W\in
C_{c}^{\infty}(\omega)$ and $\epsilon>0$, that there exists $u\in
C_{c}^{\infty}(\Omega_{\infty})$ such that

$\frac{[u]_{s,p,\Omega_{\infty}}^{p}}{\|u\|^{p}_{L^{p}(\Omega_{\infty})}}\leq\frac{[W]^{p}_{s,p,\omega}}{\|W\|^{p}_{L^{p}(\omega)}}+\epsilon,$

then we are done.

Therefore, we start by choosing, arbitrarily, $W\in C_{c}^{\infty}(\omega)$
and $v\in C_{c}^{\infty}(\mathbb{R}^{m})$ which satisfies
$\int_{\mathbb{R}^{m}}|v|^{p}=1$. We define, for $\ell>0$,
$v_{\ell}(x)=\ell^{-\frac{m}{p}}v\left(\frac{x}{\ell}\right).$ Clearly
$v_{\ell}\in C_{c}^{\infty}(\mathbb{R}^{m})$ and

$\int_{\mathbb{R}^{m}}|v_{\ell}|^{p}=1\quad\text{ for all }\ell>0.$ (3.1)

We denote the point $x\in\mathbb{R}^{n}$ by $x=(X_{1},X_{2})$, where
$X_{1}\in\mathbb{R}^{m}$ and $X_{2}\in\mathbb{R}^{n-m}$. Now we define

$u_{\ell}(X_{1},X_{2})=v_{\ell}(X_{1})W(X_{2}).$

Note that we can always assume $\|W\|_{L^{p}(\omega)}=1$ by normalizing $W$
appropriately. Using eq. 3.1 we get

$\|u_{\ell}\|_{L^{p}(\Omega_{\infty})}^{p}=\int_{\mathbb{R}^{m}}\int_{\omega}|v_{\ell}(X_{1})|^{p}|W(X_{2})|^{p}dX_{2}\,dX_{1}=\|W\|_{L^{p}(\omega)}^{p}=1\;\;\text{
for all }\ell>0.$

Therefore it only remains to show that, for sufficiently large $\ell$,

$[u_{\ell}]^{p}_{s,p,\Omega_{\infty}}\leq[W]^{p}_{s,p,\omega}+\epsilon(\ell),$

where $\lim\limits_{\ell\to\infty}\epsilon(\ell)=0$; again, this will follow
immediately, if we can show, after redefining $\epsilon(\ell)$ appropriately,

$[u_{\ell}]_{s,p,\Omega_{\infty}}\leq[W]_{s,p,\omega}+\epsilon(\ell).$

Using the triangle inequality of
$L^{p}(\Omega_{\infty}\times\Omega_{\infty})$-norm, we obtain

$\displaystyle[u_{\ell}]_{s,p,\Omega_{\infty}}$
$\displaystyle=\left(\frac{C_{n,s,p}}{2}\int_{\Omega_{\infty}}\int_{\Omega_{\infty}}\frac{|u_{\ell}(x)-u_{\ell}(y)|^{p}}{|x-y|^{n+sp}}dxdy\right)^{\frac{1}{p}}$
$\displaystyle=\left(\frac{C_{n,s,p}}{2}\int_{\Omega_{\infty}}\int_{\Omega_{\infty}}\frac{|v_{\ell}(X_{1})W(X_{2})-v_{\ell}(Y_{1})W(Y_{2})|^{p}}{|x-y|^{n+sp}}dxdy\right)^{\frac{1}{p}}$
$\displaystyle=\left(\frac{C_{n,s,p}}{2}\int_{\Omega_{\infty}}\int_{\Omega_{\infty}}\left|\frac{v_{\ell}(Y_{1})\left(W(X_{2})-W(Y_{2})\right)}{|x-y|^{\frac{n}{p}+s}}+\frac{W(X_{2})\left(v_{\ell}(X_{1})-v_{\ell}(Y_{1})\right)}{|x-y|^{\frac{n}{p}+s}}\right|^{p}dxdy\right)^{\frac{1}{p}}$
$\displaystyle\leq I_{1}+I_{2},$ (3.2)

where

$I_{1}=\left(\frac{C_{n,s,p}}{2}\int_{\Omega_{\infty}}\int_{\Omega_{\infty}}\frac{|v_{\ell}(Y_{1})|^{p}|W(X_{2})-W(Y_{2})|^{p}}{|x-y|^{n+sp}}dxdy\right)^{1/p}$

and

$I_{2}=\left(\frac{C_{n,s,p}}{2}\int_{\Omega_{\infty}}\int_{\Omega_{\infty}}\frac{|W(X_{2})|^{p}|v_{\ell}(X_{1})-v_{\ell}(Y_{1})|^{p}}{|x-y|^{n+sp}}dxdy\right)^{1/p}.$

We shall now estimate the integrals $I_{1}$ and $I_{2}$.

Estimate for $I_{1}$: For $X_{2}\neq Y_{2}$ and by (ii) of lemma 3.1, we get

$\displaystyle\int_{\mathbb{R}^{m}}\frac{dX_{1}}{\left(1+\frac{|X_{1}-Y_{1}|^{2}}{|X_{2}-Y_{2}|^{2}}\right)^{\frac{n+sp}{2}}}=|X_{2}-Y_{2}|^{m}\Theta_{m,n,p}\hskip
8.53581pt\textrm{for any}\ Y_{1}\in\mathbb{R}^{m}.$

Applying this identity to the definition of $I_{1}$, together with (i) of
lemma 3.1 and eq. 3.1, we get

$\displaystyle I_{1}^{p}=$
$\displaystyle\frac{C_{n,s,p}}{2}\int_{\Omega_{\infty}}\int_{\Omega_{\infty}}\frac{|{v_{\ell}}(Y_{1})(W(X_{2})-W(Y_{2}))|^{p}}{|X_{2}-Y_{2}|^{n+sp}\left(1+\frac{|X_{1}-Y_{1}|^{2}}{|X_{2}-Y_{2}|^{2}}\right)^{\frac{n+sp}{2}}}dxdy\vskip
3.0pt plus 1.0pt minus 1.0pt$ $\displaystyle=$
$\displaystyle\frac{C_{n,s,p}}{2}\int_{\omega}\int_{\omega}\frac{|W(X_{2})-W(Y_{2})|^{p}}{|X_{2}-Y_{2}|^{n+sp}}\int_{\mathbb{R}^{m}}\left(\int_{\mathbb{R}^{m}}\frac{dX_{1}}{\big{(}1+\frac{|X_{1}-Y_{1}|^{2}}{|X_{2}-Y_{2}|^{2}}\big{)}^{\frac{n+sp}{2}}}\right)|v_{\ell}(Y_{1})|^{p}dY_{1}dX_{2}dY_{2}\vskip
3.0pt plus 1.0pt minus 1.0pt$ $\displaystyle=$
$\displaystyle\frac{C_{n,s,p}}{2}\Theta_{m,n,p}\int_{\omega}\int_{\omega}\frac{\left|W(X_{2})-W(Y_{2})\right|^{p}}{\left|X_{2}-Y_{2}\right|^{n-m+sp}}dX_{2}dY_{2}\int_{\mathbb{R}^{m}}|v_{\ell}(Y_{1})|^{p}dY_{1}\vskip
3.0pt plus 1.0pt minus 1.0pt$ $\displaystyle=$
$\displaystyle\frac{C_{n-m,s,p}}{2}\int_{\omega}\int_{\omega}\frac{\left|W(X_{2})-W(Y_{2})\right|^{p}}{\left|X_{2}-Y_{2}\right|^{n-m+sp}}dX_{2}dY_{2}=[W]_{s,p,\omega}^{p}\,.$

Estimate for $I_{2}$: We can write $I_{2}$ as

$\displaystyle I_{2}^{p}$
$\displaystyle=\frac{C_{n,s,p}}{2}\int_{\Omega_{\infty}}\int_{\Omega_{\infty}}\frac{|({v_{\ell}}(X_{1})-{v_{\ell}}(Y_{1}))W(X_{2})|^{p}}{|X_{1}-Y_{1}|^{n+sp}\left(1+\frac{|X_{2}-Y_{2}|^{2}}{|X_{1}-Y_{1}|^{2}}\right)^{\frac{n+sp}{2}}}dxdy\vskip
3.0pt plus 1.0pt minus 1.0pt$
$\displaystyle=\frac{C_{n,s,p}}{2}\int_{\mathbb{R}^{m}}\int_{\mathbb{R}^{m}}\frac{|{v_{\ell}}(X_{1})-{v_{\ell}}(Y_{1})|^{p}}{|X_{1}-Y_{1}|^{n+sp}}\int_{\omega}\Bigg{(}\int_{\omega}\frac{dY_{2}}{\left(1+\frac{|X_{2}-Y_{2}|^{2}}{|X_{1}-Y_{1}|^{2}}\right)^{\frac{n+sp}{2}}}\Bigg{)}|W(X_{2})|^{p}\;dX_{2}dX_{1}dY_{1}\,.$

Using lemma 3.1 (ii) we get

$\displaystyle\int_{\omega}\frac{dY_{2}}{\left(1+\frac{|X_{2}-Y_{2}|^{2}}{|X_{1}-Y_{1}|^{2}}\right)^{\frac{n+sp}{2}}}\leq$
$\displaystyle\int_{\mathbb{R}^{n-m}}\frac{dY_{2}}{\left(1+\frac{|X_{2}-Y_{2}|^{2}}{|X_{1}-Y_{1}|^{2}}\right)^{\frac{n+sp}{2}}}=|X_{1}-Y_{1}|^{n-m}\Theta_{n-m,n,p}.$

Applying this to the definition of $I_{2}$ and using the fact that
$\|W\|_{L^{p}(\omega)}=1$, we obtain

$I_{2}^{p}\leq\frac{C_{n,s,p}\;\Theta_{n-m,n,p}}{2}\int_{\mathbb{R}^{m}}\int_{\mathbb{R}^{m}}\frac{\left|v_{\ell}(X_{1})-v_{\ell}(Y_{1})\right|^{p}}{|X_{1}-Y_{1}|^{m+sp}}dX_{1}dY_{1}=[v_{\ell}]_{s,p,\mathbb{R}^{m}}^{p}.$

By a change of variables in the definition of $v_{\ell}$, we get

$[v_{\ell}]_{s,p,\mathbb{R}^{m}}^{p}=\frac{\ell^{m-sp}}{\ell^{m}}[v]_{s,p,\mathbb{R}^{m}}^{p}=\frac{1}{\ell^{sp}}[v]_{s,p,\mathbb{R}^{m}}^{p}\quad\Rightarrow\quad
I_{2}\leq\frac{[v]_{s,p,\mathbb{R}^{m}}}{\ell^{s}}.$

Now plugging the above finer estimates of $I_{1}$ and $I_{2}$ into section 3,
we obtain

$[u_{\ell}]_{s,p,\Omega_{\infty}}\leq[W]_{s,p,\omega}+\frac{[v]_{s,p,\mathbb{R}^{m}}}{\ell^{s}}.$

This finishes the proof of (1). Proof of (2) is similar and hence omitted. ∎

In the next result we shall use the concept of weak formulation (see
definition 2.2).

######  Lemma 3.3.

Let $x=(X_{1},X_{2})\in\Omega_{\infty}$ and define $u^{*}(x):=W(X_{2})$, where
$W$ is a weak solution of eq. 1.4. Then

$C_{n,s,p}\int_{\mathbb{R}^{n-m}}\int_{\mathbb{R}^{n}}\frac{|u^{*}(x)-u^{*}(y)|^{p-2}(u^{*}(x)-u^{*}(y))\psi(X_{2})}{|x-y|^{n+sp}}dydX_{2}\\\
=P^{2}_{n-m,s,p}(\omega)\int_{\omega}|W(X_{2})|^{p-2}W(X_{2})\psi(X_{2})dX_{2},\text{
for all }\psi\in W^{s,p}_{\omega}(\mathbb{R}^{n-m}).$

###### Proof.

Let $\psi\in W^{s,p}_{\omega}(\mathbb{R}^{n-m})$. In the following
calculation, we use the fact that $W$ is a weak solution of eq. 1.4; also, we
use (i) of lemma 3.1 in the second equality, and (ii) of the same lemma, with
choices $a=|X_{2}-Y_{2}|$ and $z=X_{1}$, in the third equality. We, then,
have:

$P^{2}_{n-m,s,p}(\omega)\int_{\omega}|W(X_{2})|^{p-2}W(X_{2})\psi(X_{2})dX_{2}\\\
=C_{n-m,s,p}\int_{\mathbb{R}^{n-m}}\int_{\mathbb{R}^{n-m}}\frac{|W(X_{2})-W(Y_{2})|^{p-2}(W(X_{2})-W(Y_{2}))\psi(X_{2})}{|X_{2}-Y_{2}|^{n-m+sp}}dY_{2}dX_{2}\\\
={C_{n,s,p}}\int_{\mathbb{R}^{n-m}}\int_{\mathbb{R}^{n-m}}\frac{|W(X_{2})-W(Y_{2})|^{p-2}(W(X_{2})-W(Y_{2}))\psi(X_{2})}{|X_{2}-Y_{2}|^{n+sp}}\\\
\int_{\mathbb{R}^{m}}\frac{dY_{1}}{\left(1+\frac{|X_{1}-Y_{1}|^{2}}{|X_{2}-Y_{2}|^{2}}\right)^{\frac{n+sp}{2}}}dY_{2}dX_{2}\\\
=C_{n,s,p}\int_{\mathbb{R}^{n-m}}\int_{\mathbb{R}^{m}}\int_{\mathbb{R}^{n-m}}\frac{|W(X_{2})-W(Y_{2})|^{p-2}(W(X_{2})-W(Y_{2}))\psi(X_{2})}{\left(|X_{1}-Y_{1}|^{2}+|X_{2}-Y_{2}|^{2}\right)^{\frac{n+sp}{2}}}dY_{2}dY_{1}dX_{2}\\\
=C_{n,s,p}\int_{\mathbb{R}^{n-m}}\int_{\mathbb{R}^{n}}\frac{|u^{*}(x)-u^{*}(y)|^{p-2}(u^{*}(x)-u^{*}(y))\psi(X_{2})}{|x-y|^{n+sp}}dydX_{2}.$

Since $\psi$ is arbitrary, the lemma follows. ∎

###### Proof of theorem 1.2.

Suppose $W$ is the first eigenfunction corresponding to the first eigenvalue,
$P^{2}_{n-m,s,p}(\omega)$ of the problem eq. 1.4, which is strictly positive
in $\omega$ (by lemma 2.3). Fix any $v\in C_{c}^{\infty}(\Omega_{\infty})$
arbitrarily. Let $\\{\rho_{k}\\}$ be the standard mollifiers in
$\mathbb{R}^{n-m}$. Now Define
$\phi(X_{1},X_{2}):=\frac{|v|^{p}}{W(X_{2})^{p-1}}$, and
$\phi_{k}(X_{1},X_{2}):=\frac{|v|^{p}}{W_{k}(X_{2})^{p-1}}$, where
$W_{k}:=W*\rho_{k}$. At this point, we fix any $X_{1}\in\mathbb{R}^{m}$ so
that $v(X_{1},\cdot)\in C_{c}^{\infty}(\omega)$, and $\phi_{k}(X_{1},\cdot)\in
C_{c}^{1}(\omega)\subset W^{s,p}_{\omega}(\mathbb{R}^{n-m})$ (by lemma 2.9).
Then, as $W>0$ in $\omega$, $W_{k}$ are strictly positive and smooth in
$\omega$. Note that there exists $\alpha>0$ such that $W,W_{k}>\alpha$ in
$\mbox{Supp }v(X_{1},\cdot)$ for any $k$. Therefore, for any
$X_{2}\in\mbox{Supp }v(X_{1},\cdot)$,

$|\phi_{k}(X_{1},X_{2})-\phi_{k}(X_{1},Y_{2})|=\left|\frac{|v(X_{1},X_{2})|^{p}}{W_{k}(X_{2})^{p-1}}-\frac{|v(X_{1},Y_{2})|^{p}}{W_{k}(Y_{2})^{p-1}}\right|\\\
=\left|\frac{|v(X_{1},X_{2})|^{p}-|v(X_{1},Y_{2})|^{p}}{W_{k}^{p-1}(X_{2})}+\frac{|v(X_{1},Y_{2})|^{p}(W_{k}^{p-1}(Y_{2})-W_{k}^{p-1}(X_{2}))}{W_{k}^{p-1}(X_{2})W_{k}^{p-1}(Y_{2})}\right|\\\
\leq\alpha^{p-1}\left||v(X_{1},X_{2})|^{p}-|v(X_{1},Y_{2})|^{p}\right|+||v||_{\infty}^{p}\frac{|W_{k}^{p-1}(Y_{2})-W_{k}^{p-1}(X_{2})|}{W_{k}^{p-1}(X_{2})W_{k}^{p-1}(Y_{2})}\\\
\leq
p\alpha^{p-1}\left(|v(X_{1},X_{2})|^{p-1}+|v(X_{1},Y_{2})|^{p-1}\right)|v(X_{1},X_{2})-v(X_{1},Y_{2})|\\\
+(p-1)||v||_{\infty}^{p}\frac{(W_{k}^{p-2}(Y_{2})+W_{k}^{p-2}(X_{2}))}{W_{k}^{p-1}(X_{2})W_{k}^{p-1}(Y_{2})}|W_{k}(X_{2})-W_{k}(Y_{2})|\\\
\leq
C(p,\alpha,||v||_{\infty})\left(|v(X_{1},X_{2})-v(X_{1},Y_{2})|+|W_{k}(X_{2})-W_{k}(Y_{2})|\right).$

This shows that

$\int_{\mathbb{R}^{n-m}}\int_{\mathbb{R}^{n-m}}\frac{|\phi_{k}(X_{1},X_{2})-\phi_{k}(X_{1},Y_{2})|^{p}}{|X_{2}-Y_{2}|^{n-m+sp}}dX_{2}dY_{2}\\\
\leq\int_{\mathbb{R}^{n-m}}\int_{\mathbb{R}^{n-m}}C(p,W,v)\frac{|W_{k}(X_{2})-W_{k}(Y_{2})|^{p}}{|X_{2}-Y_{2}|^{n-m+sp}}dX_{2}dY_{2}+C(p,W,v)[v(X_{1},\cdot)]^{p}_{s,p,\mathbb{R}^{n-m}}.$

Now one can easily check that $W_{k}$ converges to $W$ in
$W^{s,p}_{\omega}(\mathbb{R}^{n-m})$ (see [FSV15, Lemma 11]) and also
pointwise. We can apply generalised dominated convergence theorem (see [RF10,
Theorem 19, Section 4.4]) to conclude that $\phi\in
W^{s,p}_{\omega}(\mathbb{R}^{n-m})$. Define $u(X_{1},X_{2}):=W(X_{2})$ and
apply discrete Picone inequality lemma 2.5 on $u$ and $|v|$ to obtain

$|u(X_{1},X_{2})-u(Y_{1},Y_{2})|^{p-2}(u(X_{1},X_{2})-u(Y_{1},Y_{2}))(\phi(X_{1},X_{2})-\phi(Y_{1},Y_{2}))\\\
\leq|v(X_{1},X_{2})-v(Y_{1},Y_{2})|^{p}.$

This gives

$\frac{C_{n,s,p}}{2}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|W(X_{2})-W(Y_{2})|^{p-2}(W(X_{2})-W(Y_{2}))(\phi(x)-\phi(y))}{|x-y|^{n+sp}}dxdy\\\
\leq\frac{C_{n,s,p}}{2}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|v(x)-v(y)|^{p}}{|x-y|^{n+sp}}dxdy.$
(3.3)

Now, observe the following calculation, where we have used Fubini’s theorem,
lemma 3.3 and positiveness of $W$:

$C_{n,s,p}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|W(X_{2})-W(Y_{2})|^{p-2}(W(X_{2})-W(Y_{2}))\phi(x)}{|x-y|^{n+sp}}dxdy\\\
=\int_{\mathbb{R}^{m}}C_{n,s,p}\int_{\mathbb{R}^{n-m}}\int_{\mathbb{R}^{n}}\frac{|W(X_{2})-W(Y_{2})|^{p-2}(W(X_{2})-W(Y_{2}))\phi(X_{1},X_{2})}{|x-y|^{n+sp}}dydX_{2}dX_{1}\\\
=P^{2}_{n-m,s,p}(\omega)\int_{\mathbb{R}^{m}}\int_{\omega}|W(X_{2})|^{p-2}W(X_{2})\phi(X_{1},X_{2})dX_{2}dX_{1}\\\
=P^{2}_{n-m,s,p}(\omega)\int_{\Omega_{\infty}}|v(x)|^{p}dx<\infty.$

This finiteness of the integrand allows us to rewrite it as:

$\frac{C_{n,s,p}}{2}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|W(X_{2})-W(Y_{2})|^{p-2}(W(X_{2})-W(Y_{2}))(\phi(x)-\phi(y)}{|x-y|^{n+sp}}dxdy\\\
=C_{n,s,p}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|W(X_{2})-W(Y_{2})|^{p-2}(W(X_{2})-W(Y_{2}))\phi(x)}{|x-y|^{n+sp}}dxdy\\\
=P^{2}_{n-m,s,p}(\omega)\int_{\Omega_{\infty}}|v(x)|^{p}dx,$

which, when combined with eq. 3.3, gives

$P^{2}_{n-m,s,p}(\omega)\int_{\Omega_{\infty}}|v(x)|^{p}dx\leq\frac{C_{n,s,p}}{2}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|v(x)-v(y)|^{p}}{|x-y|^{n+sp}}dxdy.$

As this is true for any $v\in C_{c}^{\infty}(\Omega_{\infty})$, we have
$P^{2}_{n-m,s,p}(\omega)\leq P^{2}_{n,s,p}(\Omega_{\infty})$, by density of
$C_{c}^{\infty}(\Omega_{\infty})$ in
$W^{s,p}_{\Omega_{\infty}}(\mathbb{R}^{n})$. The upper bound of
$P^{2}_{n,s,p}(\Omega_{\infty})$ follows from (2) of proposition 3.2.

Now for the last part of the theorem, suppose that there exist a function $u$
such that
$P^{2}_{n,s,p}(\Omega_{\infty})=\frac{[u]_{s,p.\mathbb{R}^{n}}^{p}}{\int_{\Omega_{\infty}}|u(x)|^{p}dx}$.
Then $u$ is a weak solution of the problem

$\begin{cases}(-\Delta_{n,p})^{s}u=P_{n,s,p}^{2}(\Omega_{\infty})|u|^{p-2}u\text{
in }\Omega_{\infty},\\\ u=0\text{ in
}\mathbb{R}^{n}\setminus\Omega_{\infty}.\end{cases}$ (3.4)

In other words, $u$ is an eigenfunction corresponding to the first eigenvalue
$P^{2}_{n,s,p}(\Omega_{\infty})$. Let $h\in\mathbb{R}^{m}$, define
$v_{h}(x)=u(X_{1}+h,X_{2})$. By change of variable, we also have $v_{h}$ is an
eigenfunction of eq. 3.4 associated to the eigenvalue
$P^{2}_{n,s,p}(\Omega_{\infty})$ for any $h$. Since
$P^{2}_{n,s,p}(\Omega_{\infty})$ is simple (see lemma 2.3),
$u=\alpha_{h}v_{h}$ for some constant $\alpha_{h}.$ Therefore, by a change of
variable, we have

$\int_{\Omega_{\infty}}|u|^{p}dx=\int_{\mathbb{R}^{m}}\int_{\omega}|u(X_{1},X_{2})|^{p}dX_{2}dX_{1}=|\alpha_{h}|^{p}\int_{\mathbb{R}^{m}}\int_{\omega}|v_{h}(X_{1},X_{2})|^{p}dX_{2}dX_{1}=|\alpha_{h}|^{p}\int_{\Omega_{\infty}}|u|^{p}dx$

Thus, we get $|\alpha_{h}|^{p}=1$ and this imply that $\alpha_{h}=1$, because
$u$ has constant sign in $\Omega_{\infty}.$ Therfore, we get
$u(X_{1},X_{2})=v_{h}(X_{1},X_{2})$ for any $h\in\mathbb{R}^{m}$. Hence $u$ is
independent of $X_{1}$ variable. In particular,
$||u||_{L^{p}(\Omega_{\infty})}$ is infinite, which gives a contradiction.
This completes the proof of theorem 1.2. ∎

######  Corollary 3.4.

Let $\\{\Omega_{\ell}\\}$ be an increasing sequence of bounded open sets in
$\mathbb{R}^{n}$ that is $\Omega_{\ell}\subseteq\Omega_{\ell_{1}}$ for any
$0<\ell<\ell_{1}$. If $\Omega=\bigcup_{\ell>0}\Omega_{\ell}$. Then we have

$P^{2}_{n,s,p}(\Omega)=\inf\limits_{\ell>0}P^{2}_{n,s,p}(\Omega_{\ell}).$

###### Proof.

By domain monotonicity property (2) of proposition 2.6, we have
$\inf\limits_{\ell>0}P^{2}_{n,s,p}(\Omega_{\ell})\geq P^{2}_{n,s,p}(\Omega)$.
So, to establish the result, we only need to show
$\inf\limits_{\ell>0}P^{2}_{n,s,p}(\Omega_{\ell})\leq P^{2}_{n,s,p}(\Omega)$.
Now, for any $v\in C_{c}^{\infty}(\Omega)$, there exists an $\ell>0$, big
enough, such that $\mbox{supp}(v)\subset\Omega_{\ell}$. Then we have
$\|v|_{\Omega_{\ell}}\|_{L^{p}(\Omega_{\ell})}=\|v\|_{L^{p}(\Omega)}$ and
$[v|_{\Omega_{\ell}}]_{s,p,\mathbb{R}^{n}}=[v]_{s,p,\mathbb{R}^{n}}$. So
$\inf\limits_{\ell>0}P^{2}_{n,s,p}(\Omega_{\ell})\leq
P^{2}_{n,s,p}(\Omega_{\ell})\leq\frac{[v]_{s,p,\mathbb{R}^{n}}^{p}}{\|v\|_{L^{p}(\Omega)}^{p}}$.
Since this holds for any $v\in C_{c}^{\infty}(\Omega)$, we conclude
$\inf\limits_{\ell>0}P^{2}_{n,s,p}(\Omega_{\ell})\leq P^{2}_{n,s,p}(\Omega)$.
∎

######  Lemma 3.5.

Let $\frac{1}{p}<s<1$, $\Omega\subset\mathbb{R}^{n}$ be a measurable set, and
$f:\mathbb{S}^{n-1}\to[0,\infty)$ be an $\mathcal{H}^{n-1}$-measurable
function satisfying

$P^{1}_{1,s,p}\left(\\{t\in\mathbb{R}:\ x+tw\in\Omega\\}\right)\geq f(w)$

for a.e. $w\in\mathbb{S}^{n-1}$ and a.e. $x\in\\{y\in\mathbb{R}^{n}:\,y\cdot
w=0\\}.$ Then

$P^{1}_{n,s,p}(\Omega)\geq\frac{C_{n,s,p}}{2C_{1,s,p}}\int_{\mathbb{S}^{n-1}}f(w)d\mathcal{H}^{n-1}(w).$

###### Proof.

Let us choose $w\in\mathbb{S}^{n-1}$ and $x\in
L_{w}:=\\{y\in\mathbb{R}^{n}:\,y\cdot w=0\\}$ arbitrarily. Denote
$\Omega_{w,x}:=\\{t\in\mathbb{R}:\ x+tw\ \in\Omega\\}.$ Then from the
hypotheses, we have

$\displaystyle\frac{C_{1,s,p}}{2}\int_{\\{\ell:\,x+\ell
w\in\Omega\\}}\int_{\\{t:\,x+tw\in\Omega\\}}\frac{|u(x+\ell
w)-u(x+tw)|^{p}}{|\ell-t|^{1+sp}}dtd\ell\geq$ $\displaystyle
P^{1}_{1,s,p}(\Omega_{w,x})\int_{\Omega_{w,x}}|u(x+tw)|^{p}dt\vskip 3.0pt plus
1.0pt minus 1.0pt$ $\displaystyle\geq$ $\displaystyle
f(w)\int_{\Omega_{w,x}}|u(x+tw)|^{p}dt.$

We apply Fubini’s theorem to get, for any $w\in\mathbb{S}^{n-1}$,

$\int_{L_{w}}d\mathcal{H}^{n-1}(x)\int_{\Omega_{x,w}}|u(x+tw)|^{p}dt=\int_{\Omega}|u|^{p},$

which, along with lemma 2.4, gives

$[u]_{s,p,\Omega}^{p}\geq\frac{C_{n,s,p}}{2C_{1,s,p}}\left(\int_{\mathbb{S}^{n-1}}f(w)d\mathcal{H}^{n-1}(w)\right)\int_{\Omega}|u|^{p}.$

This proves the lemma. ∎

Before proving theorem 1.4, observe that for any function $f$ depending only
on $\sigma_{1}$, where $\sigma=(\sigma_{1},\cdots,\sigma_{n-1})$, we have,
from the discussion on Hyper-spherical Coordinates that

$\displaystyle\int_{A_{n-1}}f(\sigma_{1})g_{n-1}(\sigma)d\sigma=$
$\displaystyle\int_{0}^{\pi}f(\sigma_{1})(\sin\sigma_{1})^{n-2}\left(\int_{Q_{n-2}}(\sin\sigma_{2})^{n-3}\cdots\sin\sigma_{n-2}d\sigma_{2}\cdots
d\sigma_{n-2}\right)d\sigma_{1}\vskip 3.0pt plus 1.0pt minus 1.0pt$
$\displaystyle=$
$\displaystyle\int_{0}^{\pi}f(\sigma_{1})(\sin\sigma_{1})^{n-2}\left(\int_{Q_{n-2}}g_{n-2}(\theta)d\theta\right)d\sigma_{1}\vskip
3.0pt plus 1.0pt minus 1.0pt$ $\displaystyle=$
$\displaystyle\mathcal{H}^{n-2}(\mathbb{S}^{n-2})\int_{0}^{\pi}f(\sigma_{1})(\sin\sigma_{1})^{n-2}d\sigma_{1}.$

In particular, using eqs. 2.1 and 2.2, for $f(\sigma)=|\cos\sigma_{1}|^{sp}$
we obtain

$\begin{split}\int_{A_{n-1}}|\cos\sigma_{1}|^{sp}g_{n-1}(\sigma)d\sigma=&2\mathcal{H}^{n-2}(\mathbb{S}^{n-2})\int_{0}^{\frac{\pi}{2}}(\cos\sigma_{1})^{sp}(\sin\sigma_{1})^{n-2}d\sigma_{1}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
=&\frac{2\pi^{\frac{n-1}{2}}}{\Gamma\left(\frac{n-1}{2}\right)}B\left(\frac{n-1}{2},\frac{sp+1}{2}\right).\end{split}$
(3.5)

Now we are ready to prove theorem 1.4.

###### Proof of theorem 1.4.

Part (1): Assume $s\in(0,\frac{1}{p}]$. Note that the case $p=1$ is covered
here, with $sp<1$. We apply proposition 3.2 with $m=n-1$ and
$\omega=(-1,1)\subset\mathbb{R}$ to deduce $P^{1}_{n,s,p}(\Omega_{\infty})\leq
P^{1}_{1,s,p}((-1,1)).$ Now applying lemma 2.8 on $P^{1}_{1,s,p}((-1,1))=0$ we
get the result.

Part (2): Let us assume $s\in(\frac{1}{p},1)$. We know, from proposition 3.2,
that $P_{n,s,p}^{1}(\Omega_{\infty})\leq P_{1,s,p}^{1}((-1,1))$. So, it is
enough to prove that

$P_{n,s,p}^{1}(\Omega_{\infty})\geq P_{1,s,p}^{1}((-1,1)).$ (3.6)

We shall show this using lemma 3.5. Choose
$w=(w_{1},\cdots,w_{n})\in\mathbb{S}^{n-1}$ and $x\in\mathbb{R}^{n}$ such that
$w_{1}\neq 0$ and $x.w=0$. Notice that
$\mathcal{L}^{1}(\\{t\in\mathbb{R}:\,x+tw\in\Omega_{\infty}\\})$, i.e. the
length of the intersection $\Omega_{\infty}\cap\\{x+tw:\,t\in\mathbb{R}\\}$,
is independent of $x\in\omega^{\perp}$. So we have

$\mathcal{L}^{1}(\\{t\in\mathbb{R}:\,x+tw\in\Omega_{\infty}\\})=\mathcal{H}^{1}\left(\Omega_{\infty}\cap\\{x+tw:\,t\in\mathbb{R}\\}\right)\\\
=\mathcal{H}^{1}\left(\Omega_{\infty}\cap\\{(-1,0,\ldots,0)+tw:\,t\in\mathbb{R}\\}\right)=|t_{0}(w)|,$

where $-1+t_{0}(w)w_{1}=1$ i.e. $t_{0}(w)=\frac{2}{w_{1}}$. From (ii) of
proposition 2.6 we see that

$P_{1,s,p}^{1}\left(\\{t\in\mathbb{R}:\,x+tw\in\Omega_{\infty}\\}\right)=\left(\frac{|w_{1}|}{2}\right)^{sp}P_{1,s,p}^{1}((0,1))=|w_{1}|^{sp}P_{1,s,p}^{1}((-1,1)).$

The above equality enables us to apply lemma 3.5, with the choice
$f(w)=P_{1,s,p}^{1}((-1,1))|w_{1}|^{sp}$. We get

$P_{n,s,p}^{1}(\Omega_{\infty})\geq
P_{1,s,p}^{1}((-1,1))\frac{C_{n,s,p}}{2C_{1,s,p}}\int_{\mathbb{S}^{n-1}}|w_{1}|^{sp}d\mathcal{H}^{n-1}.$

Again, using Hyper-spherical Coordinates, in the RHS of the above inequality,
we get

$P_{n,s,p}^{1}(\Omega_{\infty})\geq
P_{1,s,p}^{1}((-1,1))\frac{C_{n,s,p}}{2C_{1,s,p}}\int_{A_{n-1}}|\cos\sigma_{1}|^{sp}g_{n-1}(\sigma)d\sigma.$

Now, eq. 3.5 gives

$P_{n,s,p}^{1}(\Omega_{\infty})\geq
P_{1,s,p}^{1}((-1,1))\frac{C_{n,s,p}}{2C_{1,s,p}}\frac{2\pi^{\frac{n-1}{2}}}{\Gamma\left(\frac{n-1}{2}\right)}B\left(\frac{n-1}{2},\frac{sp+1}{2}\right).$

Again, using eqs. 1.2 and 2.2, we find that

$\frac{C_{n,s,p}}{C_{1,s,p}}\frac{\pi^{\frac{n-1}{2}}}{\Gamma\left(\frac{n-1}{2}\right)}B\left(\frac{n-1}{2},\frac{sp+1}{2}\right)=1,$

consequently $P_{n,s,p}^{1}(\Omega_{\infty})\geq P_{1,s,p}^{1}((-1,1))$. This
concludes the proof of eq. 3.6 and hence the theorem follows. ∎

###### Proof of theorem 1.5.

The domain monotonicity property ((i) of proposition 2.6) and theorem 1.2,
implies $P^{2}_{n-m,s,p}(\omega)\leq P^{2}_{n,s,p}(\Omega_{\ell})$. For the
reverse inequality, following the same proof as in (1) of proposition 3.2,
where the domain of integration $\Omega_{\infty}$ is replaced by
$\Omega_{\ell}$, we obtain

$P^{2}_{n,s,p}(\Omega_{\ell})\leq\left(P^{2}_{n-m,s,p}(\omega)^{\frac{1}{p}}+\frac{[v]_{s,p,\mathbb{R}^{m}}}{\ell^{s}}\right)^{p}\\\
\leq
P^{2}_{n-m,s,p}(\omega)+p2^{p-1}\left(\frac{P^{2}_{n-m,s,p}(\omega)^{\frac{p-1}{p}}[v]_{s,p,\mathbb{R}^{m}}}{\ell^{s}}+\frac{[v]_{s,p,\mathbb{R}^{m}}^{p}}{\ell^{sp}}\right)=P^{2}_{n-m,s,p}(\omega)+\frac{C_{1}}{\ell^{s}}+\frac{C_{2}}{\ell^{sp}},$

where we used the following elementary inequality: $(a+b)^{q}\leq
a^{q}+q2^{q-1}(a^{q-1}b+b^{q})$ for $a,b\geq 0$ and $q\geq 1$. Combining these
two estimates of $P^{2}_{n,s,p}(\Omega_{\ell})$, the first part of the theorem
follows. Now letting $\ell\to\infty$ and applying theorem 1.2 we conclude the
last equality. This finishes the proof of theorem 1.5.

∎

Acknowledgment: The authors would like to thanks Prof. Prosenjit Roy, Prof.
Gyula Csató and Dr. Indranil Chowdhury for fruitful discussions on this
subject.

## References

  * [AF03] Robert A. Adams and John J. F. Fournier.  Sobolev spaces, volume 140 of Pure and Applied Mathematics (Amsterdam).  Elsevier/Academic Press, Amsterdam, second edition, 2003.
  * [AFM20] Vincenzo Ambrosio, Lorenzo Freddi, and Roberta Musina.  Asymptotic analysis of the Dirichlet fractional Laplacian in domains becoming unbounded.  J. Math. Anal. Appl., 485(2):123845, 17, 2020.
  * [AW19] Harbir Antil and Mahamadi Warma.  Optimal control of the coefficient for the regional fractional $p$-Laplace equation: approximation and convergence.  Math. Control Relat. Fields, 9(1):1–38, 2019.
  * [BC18] Lorenzo Brasco and Eleonora Cinti.  On fractional Hardy inequalities in convex sets.  Discrete Contin. Dyn. Syst., 38(8):4019–4040, 2018.
  * [BF14] Lorenzo Brasco and Giovanni Franzina.  Convexity properties of Dirichlet integrals and Picone-type inequalities.  Kodai Math. J., 37(3):769–799, 2014.
  * [BLP14] Lorenzo Brasco, Erik Lindgren, and Enea Parini.  The fractional Cheeger problem.  Interfaces Free Bound., 16(3):419–458, 2014.
  * [BMR20] Kaushik Bal, Kaushik Mohanta, and Prosenjit Roy.  Bourgain-Brezis-Mironescu domains.  Nonlinear Anal., 199:111928, 10, 2020.
  * [BMRS20] Kaushik Bal, Kaushik Mohanta, Prosenjit Roy, and Firoj Sk.  Hardy and Poincaré inequalities in fractional Orlicz-Sobolev space.  arXiv preprint arXiv:2009.07035, 2020.
  * [BP16] Lorenzo Brasco and Enea Parini.  The second eigenvalue of the fractional $p$-Laplacian.  Adv. Calc. Var., 9(4):323–355, 2016.
  * [BS19] Lorenzo Brasco and Ariel Salort.  A note on homogeneous Sobolev spaces of fractional order.  Ann. Mat. Pura Appl. (4), 198(4):1295–1330, 2019.
  * [CCRS21] Indranil Chowdhury, Gyula Csató, Prosenjit Roy, and Firoj Sk.  Study of fractional Poincaré inequalities on unbounded domains.  Discrete Contin. Dyn. Syst., 41(6):2993–3020, 2021.
  * [CR17] Indranil Chowdhury and Prosenjit Roy.  On the asymptotic analysis of problems involving fractional Laplacian in cylindrical domains tending to infinity.  Commun. Contemp. Math., 19(5):1650035, 21, 2017.
  * [CRS13] Michel Chipot, Prosenjit Roy, and Itai Shafrir.  Asymptotics of eigenstates of elliptic problems with mixed boundary data on domains tending to infinity.  Asymptot. Anal., 85(3-4):199–227, 2013.
  * [DK21] Bartłomiej Dyda and Michał Kijaczko.  On density of compactly supported smooth functions in fractional Sobolev spaces.  arXiv preprint arXiv:2104.08953, 2021.
  * [DNPV12] Eleonora Di Nezza, Giampiero Palatucci, and Enrico Valdinoci.  Hitchhiker’s guide to the fractional Sobolev spaces.  Bull. Sci. Math., 136(5):521–573, 2012.
  * [DPFBLR18] Leandro Del Pezzo, Julián Fernández Bonder, and Luis López Ríos.  An optimization problem for the first eigenvalue of the $p$-fractional Laplacian.  Mathematische Nachrichten, 291(4):632–651, 2018.
  * [Dyd04] Bartłomiej Dyda.  A fractional order Hardy inequality.  Illinois J. Math., 48(2):575–588, 2004.
  * [ERS21] Luca Esposito, Prosenjit Roy, and Firoj Sk.  On the asymptotic behavior of the eigenvalues of nonlinear elliptic problems in domains becoming unbounded.  Asymptot. Anal., 123(1-2):79–94, 2021.
  * [FP14] Giovanni Franzina and Giampiero Palatucci.  Fractional $p$-eigenvalues.  Riv. Math. Univ. Parma (N.S.), 5(2):373–386, 2014.
  * [FSV15] Alessio Fiscella, Raffaella Servadei, and Enrico Valdinoci.  Density properties for fractional Sobolev spaces.  Ann. Acad. Sci. Fenn. Math., 40(1):235–253, 2015.
  * [LL14] Erik Lindgren and Peter Lindqvist.  Fractional eigenvalues.  Calc. Var. Partial Differential Equations, 49(1-2):795–826, 2014\.
  * [LS10] Michael Loss and Craig Sloane.  Hardy inequalities for fractional integrals on general domains.  J. Funct. Anal., 259(6):1369–1379, 2010.
  * [RF10] Halsey Lawrence Royden and Patrick Fitzpatrick.  Real analysis.  New Jersey: Printice-Hall, fourth edition, 2010.
  * [Tri83] Hans Triebel.  Theory of function spaces, volume 78 of Monographs in Mathematics.  Birkhäuser Verlag, Basel, 1983.
  * [War15] Mahamadi Warma.  The fractional relative capacity and the fractional Laplacian with Neumann and Robin boundary conditions on open sets.  Potential Anal., 42(2):499–547, 2015.
  * [Yer14] Karen Yeressian.  Asymptotic behavior of elliptic nonlocal equations set in cylinders.  Asymptot. Anal., 89(1-2):21–35, 2014.