Fractional (s,p)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\varvec{s}},{\varvec{p}})$$\end{document}-Robin–Venttsel’ problems on extension domains

We study a nonlocal Robin–Venttsel’-type problem for the regional fractional p-Laplacian in an extension domain Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document} with boundary a d-set. We prove existence and uniqueness of a strong solution via a semigroup approach. Markovianity and ultracontractivity properties are proved. We then consider the elliptic problem. We prove existence, uniqueness and global boundedness of the weak solution.


Introduction
Aim of this paper is to study a parabolic problem for the regional fractional p-Laplacian with nonlocal Robin-Venttsel' boundary conditions in extension domains.
Nowadays the literature on fractional operators is huge, due to the fact that they describe mathematically many physical phenomena exhibiting deviations from standard diffusion, the so-called anomalous diffusion. This is an important topic not only in physics, but also in finance and probability [1,29,42,44].
Several models appear in the literature to describe such diffusion, e.g. the fractional Brownian motion, the continuous time random walk, the Lévy flight as well as random walk models based on evolution equations of single and distributed fractional order in time and/or space [21,26,41,44,46].
In the present paper, we consider the following evolution problem for the regional fractional p-Laplacian with nonlocal dynamical Robin-Venttsel' boundary conditions. suitable notion of p-fractional normal derivative on irregular sets, via a generalized fractional Green formula, see Theorem 2.2.
We then consider the fractional energy functional Φ p,s defined in (3.2), which is proper, convex and weakly lower semicontinuous, and the corresponding associated subdifferential A. In Theorem 3.3 we prove, via a semigroup approach, existence and uniqueness of a strong solution for a suitable abstract Cauchy problem (P ) for the operator A. Then, via Theorem 3.6, we prove that problem (P ) is the strong formulation of the abstract problem (P ). In Theorem 3.10 we prove that the associated (nonlinear) semigroup is orderpreserving and non-expansive on L ∞ , and in Theorem 4.7 we prove that it is ultracontractive.
We then consider the elliptic problem. After proving existence and uniqueness of a weak solution in suitable functional spaces in Theorem 5.1, we prove its global boundedness in Theorem 5.4 via Lemma 5.2.
The plan of the paper is the following. In Sect. 1 we introduce the extension domain Ω and we recall some preliminary results on fractional Sobolev spaces, embeddings and traces.
In Sect. 2 we introduce the regional fractional p-Laplacian and the notion of weak p-fractional normal derivative by proving a generalized p-fractional Green formula.
In Sect. 3 we introduce the energy functional Φ p,s which is proper, convex and weakly lower semicontinuous and the associated subdifferential A which is the generator of a nonlinear C 0 semigroup. We prove existence and uniqueness of a strong solution for the corresponding abstract Cauchy problem and we give a strong interpretation. Moreover, we prove that the semigroup is Markovian.
In Sect. 4 we prove that the associated semigroup is ultracontractive. The proof, as usual, relies on a fractional logarithmic-Sobolev inequality adapted to the present case.
In Sect. 5 we consider the elliptic problem. We prove existence and uniqueness of the weak solution and its global boundedness.
it becomes a Banach space. Moreover, we denote by |u| W s,p (G) the seminorm associated to u W s,p (G) and we define, for every u, v ∈ W s,p (G), In the following we will denote by |A| the Lebesgue measure of a subset A ⊂ R N . For f ∈ W s,p (G), we define the trace operator γ 0 as at every point x ∈ G where the limit exists. The above limit exists at quasi every x ∈ G with respect to the (s, p)-capacity (see Definition 2.2.4 and Theorem 6.2.1 page 159 in [2]). From now one we denote the trace operator simply by f | G ; sometimes we will omit the trace symbol and the interpretation will be left to the context. For 1 ≤ q, r ≤ ∞, we introduce the space If q = r < ∞, we denote the above space simply by X q (Ω, ∂Ω) and we endow it with the following norm: If q = r = ∞, we endow X ∞ (Ω, ∂Ω) with the following norm: In the following, for a function f with well-defined trace f | ∂Ω on ∂Ω, we simply denote f = (f, f | ∂Ω ).
We say that F is an ( , δ) domain if, whenever x, y ∈ F with |x − y| < δ, there exists a rectifiable arc γ ∈ F joining x to y such that In this paper, we consider two particular classes of ( , δ) domains Ω ⊂ R N . More precisely, Ω can be a ( , δ) domain having as boundary either a d-set or an arbitrary closed set in the sense of [31]. For the sake of simplicity, from now on we restrict ourselves to the case in which ∂Ω is a d-set (see [32]).
The measure μ is called d-measure.
We suppose that Ω can be approximated by a sequence {Ω n } of domains such that for every n ∈ N: The reader is referred to [15,16] for examples of such domains. We recall the definition of Besov space specialized to our case. For generalities on Besov spaces, we refer to [32].
is the space of functions for which the following norm is finite: Let p be the Hölder conjugate exponent of p. In the following, we will denote the dual of the Besov space B p,p α (F) with (B p,p α (F)) ; we point out that this space coincides with the space B p ,p −α (F) (see [33]). From now on, let We point out that p * ≥p ≥ p.
We define the average of u on Ω and on ∂Ω respectively as Fractional (s, p)-Robin-Venttsel'. . . Page 7 of 33 31 Using Theorems 1.7 and 1.8 and Hölder inequality, one can easily prove that (1.10) We point out that (1.9) and (1.10) imply that, for every 1 < q ≤ p * , it holds (1.11)

The regional fractional p-Laplacian and the Green formula
We recall the definition of the regional fractional p-Laplacian. We refer to [51] and the references listed in. Let s ∈ (0, 1) and p > 1. For G ⊆ R N , we define the space (2.1) provided that the limit exists, for every function u ∈ L p−1 s (G). The positive constant C N,p,s is defined as follows: where Γ is the Euler function. We now introduce the notion of p-fractional normal derivative on (ε, δ) domains having as boundary a d-set and satisfying hypotheses (H) in Sect. 1.2 via a p-fractional Green formula by suitably generalizing the results in [16]. We recall its proof for the reader's convenience. For the case p = 2, we refer to [27,28] for the smooth case and to [15] for irregular sets.
We define the space which is a Banach space equipped with the norm We first define the p-fractional normal derivative on Lipschitz domains.
for every v ∈ W s,p (T ). In this case, g is uniquely determined and we call We point out that, when s → 1 − in (2.2), we recover the quasi-linear Green formula for Lipschitz domains [8].

The following generalized Green formula holds for every
From Hölder inequality, we get We now prove that the operator l(u) is independent from the choice of v and it is an element of (B p,p α (∂Ω)) . From Proposition 1.4, for every v ∈ B p,p α (∂Ω) there exists a functionw : The thesis follows from (2.4) and (4.21). We now recall that Ω is approximated by a sequence of Lipschitz domains Ω n , for n ∈ N, satisfying conditions (H) in Sect. 1.2. From (2.2) we have that From the dominated convergence theorem, we have . Hence, we define the p-fractional normal derivative on Ω as Remark 2.3. We note that p (1−s) = 2−β, where β = ps−1 p−1 +1, thus recovering the usual notation for the p-fractional normal derivative in (2.3).

The energy functional
From now on, we suppose that p ≥ 2, sp < N and that b ∈ C(Ω) is strictly positive and continuous on Ω. We denote by H := X 2 (Ω, ∂Ω) the Lebesgue space defined in Sect. 1.1.

Abstract Cauchy problem
Let T be a fixed positive number. We consider the abstract Cauchy problem where A is the subdifferential of Φ p,s andF = (f, g) and u 0 are given data.
According to [3, Section 2.1, chapter II], we give the following definition.  From Theorem 1 and Remark 2 in [7] (see also [3]) we have the following result. We denote by T p,s (t) the nonlinear semigroup generated by −∂Φ p,s . From Proposition 3.2, page 176 in [45], the following result follows.

The strong problem
We give a characterization of ∂Φ p,s in order to prove that the strong solution of the abstract Cauchy problem solves problem (P ).
This means that We choose ψ = u + tv, with v ∈ W s,p (Ω) and 0 < t ≤ 1 in (3.3), thus obtaining (3.4) We first take v ∈ D(Ω) in (3.4) and, by passing to the limit for t → 0 + , we get If we take −v in (3.4) we obtain the opposite inequality, thus getting the equality Since v ∈ D(Ω) and p ≤ 2, it turns out that in particular f ∈ L p (Ω). Hence, the p-fractional Green formula for irregular domains given by Theorem 2.2 yields that (and in particular in L 2 (Ω)). We go back to (3.4). Dividing by t > 0 and passing to the limit for t → 0 + , we get As before, by taking −v we obtain the opposite inequality, hence we get the equality. Then, from Theorem 2.2 and (3.5) we get Hence (3.6) holds in (B p,p α (∂Ω)) and we get the thesis. We now prove the converse. Let then u ∈ W s,p (Ω) be the weak solution of problem (P ). We have to prove that Φ p, and taking into account that u is the weak solution of (P ), by using as test functions v and u respectively, we get thus concluding the proof.
From Theorem 3.6, we deduce that the unique strong solution u of the abstract Cauchy problem (P ) solves the following Venttsel'-type problem(P ) on Ω for a.e. t ∈ (0, T ] in the following weak sense: where we recall that α = s − N −d p .

Well-posedness of the (homogeneous) heat equation
In this subsection we prove that the homogeneous heat equation is well-posed. This will be achieved by investigating the order-preserving and Markovian properties for the semigroup T p,s (t) generated by −A = −∂Φ p,s for every p ≥ 2.
For the sake of completeness, we recall the following definitions. We refer to [11] for details. Definition 3.7. Let X be a locally compact metric space andμ be a Radon measure on X. Let {T (t)} t≥0 be a strongly continuous semigroup on L 2 (X,μ). The semigroup is order-preserving if, for every u, v ∈ L 2 (X,μ) such that u ≤ v, The semigroup is non-expansive on L q (X,μ) if for every t ≥ 0 The semigroup is Markovian if it is order-preserving and non-expansive on L ∞ (X,μ).
We give two equivalent conditions for proving order-preserving and Markovian properties.

Theorem 3.10. The semigroup {T p,s (t)} t≥0 generated by −A is Markovian on
Proof. We will apply Propositions 3.8 and 3.9.
We begin by proving the order-preserving property. We set, for u, v ∈ W s,p (Ω), Since W s,p (Ω) is a lattice, we have that both g(u, v) and h(u, v) belong to W s,p (Ω). Proceeding as in [47,Theorem 3.1.4], we prove that Moreover, from the convexity of the functional and proceeding as in [48, Theorem 3.4] we have that  In order to complete the proof, we apply Proposition 3.9. In the notation of Proposition 3.9, given u, v ∈ W s,p (Ω) andα > 0, we set We point out that gα(u, v) ∈ W s,p (Ω

Ultracontractivity of semigroups
We now focus on proving the ultracontractivity of the semigroup T p,s (t). We first prove a logarithmic Sobolev inequality adapted to our case. Then there exists a positive constantC =C(N, s, p, d, Ω) such that, for every ε > 0,

2)
where u p log u := (u p log u, u| p ∂Ω log u| ∂Ω ) andū Ω andū ∂Ω are defined in (1.8). Proof. Following [6, Proposition 2.1], we apply Jensen's inequality with q = p − p = p(d−N +sp) N −sp and we obtain Moreover, from the properties of the logarithmic function, for every ε > 0 we have that log u p p ≤ ε u p p − log ε. From (1.11) with q =p, we estimate u p p in (4.4). Hence, there exists a positive constantC such that thus concluding the proof.
In order to prove the ultracontractivity of T p,s (t), we now prove some preliminary lemmas. We adapt to the fractional framework the results of [34,Section 3.2], see also [49,51,52].
We first recall some known numerical inequalities. For more details we refer to [5].
We remark that (4.6) implies where c * p > 0 is the constant given in Proposition 4.2 and b 0 = min Ω b. Proof. The proof can be obtained by suitably adapting the proof of Lemma 3.4 in [34].
We remark that, as a consequence of Lemma 4.3, we have that G r (t) := where C r,p := (r − 1) The next two lemmas follow by adapting to the fractional setting Lemmas 3.5 and 3.6 in [34] (see also [47]).  Proof. From the chain rule, we have that Then, from Lemma 4.3 the following holds: Recalling the definition of Λ, using Lemma 4.4 and estimating the boundary term with zero, we get where fulfills the hypotheses of Proposition 4.1. Thus we have that, for every ε > 0, (4.13) We now point out that Moreover, we have that an analogous equality holds for |F ∂Ω | p . Hence, from (4.13), (4.14) and (4.15) we deduce and, defining we get the thesis. where Proof. We first take u 0 , v 0 ∈ X ∞ (Ω, ∂Ω) and we use the same assumptions and notations of Lemma 4.6. In particular, we consider an increasing differentiable function r : [0, ∞) → [2, ∞) and we define A(t) and B(t) as in (4.18) and (4.19) respectively.
We set then, from (4.17), y(t) satisfies the following ordinary differential inequality: We now consider the following ODE: The unique solution x(t) of (4.29) can be written in the following way: hence, the solution y(t) of the ordinary differential inequality (4.28) is such that y(t) ≤ x(t) for every t ∈ [0, ∞). We now fix t > 0, for any given q ≥ 2 and for τ ∈ [0, t) we set The function r(·) satisfies the hypotheses of Lemma 4.5, i.e. it is increasing and differentiable on [0, t) and r(τ ) ≥ 2 for every τ ∈ [0, t). Using (4.31), we obtain that Our aim is now to write x(t) in a more explicit way. From standard calculations, we have that (4.32) Moreover, again from standard cumbersome calculations, we can prove that whereČ is a suitable positive constant depending on N , s, p, Ω, d and q and I (1) , I (2) and I (3) are integral terms which do not depend on t and can be explicitly computed as in [10, proof of Lemma 3.9]. From (4.32) and (4.33) it follows that where C 2 = We remark that also in the linear case, i.e. p = 2, the semigroup T 2,s (t) is ultracontractive. The proof follows by adapting the techniques of [25, Theorem 2.16].

The elliptic problem
In this section we investigate the elliptic Venttsel' problem, under the same assumptions and notations of the previous sections. In particular, we prove a priori estimates for its (unique) weak solution.
Let (f, g) ∈ X q,r (Ω, ∂Ω). The elliptic Venttsel' problem is formally given by We observe that, from Theorems 1.7 and 1.8, the space W s,p (Ω) is continuously embedded in X q,r (Ω, ∂Ω) for every q ∈ [1, p * ] and r ∈ [1,p ]; hence, there exists a positive constant C such that, for every u ∈ W s,p (Ω), u q,r ≤ C u W s,p (Ω) . (5.1) We first aim to prove the existence and uniqueness of a weak solution of the elliptic problem (P e ).
We say that u ∈ W s,p (Ω) is a weak solution of problem (P e ) if