On two-dimensional nonlocal Venttsel' problems in piecewise smooth domains

We establish the regularity results for solutions of nonlocal Venttsel' problems in polygonal and piecewise smooth two-dimensional domains.


Introduction
In this paper we investigate an elliptic nonlocal Venttsel' problem for the Laplace operator in a bounded polygonal domain Ω ⊂ R 2 .
Lately Venttsel' problems in irregular domains (for example having prefractal or fractal boundary) have been widely investigated, see e.g. [12] and [10] and the references listed in.
In [12] the reader can also find motivations for the study of such problems.
Our aim in this paper is to study the regularity in weighted Sobolev spaces of the weak solution of a nonlocal Venttsel' problem in a polygonal domain. These results will be crucial to obtain optimal a priori error estimates for the numerical approximation of the problem at hand; to this regard, for the local case, see [5] and [6].
We first point out that a general nonlocal term appears also in the pioneering original paper by Venttsel' [19]. Here we consider a nonlocal term which can be regarded as a version of the fractional Laplace operator (−∆) s , for 0 < s < 1, on the boundary. The presence of this term could, in principle, deteriorate the regularity of the solution on the boundary. We prove that this is not the case, and that the weak solution of the nonlocal Venttsel' problem belongs to H 2 (∂Ω), i.e. it has the same regularity as in the local case (see [5]).
It is well known that solutions of boundary value problems in piecewise smooth domains usually belong to weighted Sobolev spaces. In our case, the interplay between the boundary equation and the equation in the domain essentially influences the range of weight exponents, see (2.2).
We remark that the techniques used in the local case to prove the regularity on the boundary are very different from the ones used in this paper.
The obtained results are a starting point in order to investigate the regularity of the solution of nonlocal Venttsel' problems in the case of domains with fractal boundary (for example of Koch-type domains).
The paper is organized as follows. In Section 1 we define the domain and the functional spaces which will appear in this paper and we state the problem. In Section 2 we prove a key a priori estimate for the solution. In Section 3 we give an existence and uniqueness result for the weak and strong solutions of the nonlocal Venttsel' problem.

Statement of the problem
Let Ω ⊂ R 2 be a domain with polygonal boundary ∂Ω. Namely, we suppose that ∂Ω is made by a finite number of segments, which form a finite number N of angles α j , for j = 1, . . . , N, and let us denote with α the opening of the largest angle in ∂Ω.
In the following we denote by L 2 (Ω) the Lebesgue space with respect to the Lebesgue measure dx on Ω, and by L 2 (∂Ω) the Lebesgue space on the boundary with respect to the arc length dℓ. By H s (Ω), for s ∈ N, we denote the standard Sobolev spaces. By C(∂Ω) we denote the set of continuous functions on ∂Ω.
By H s (∂Ω), for 0 < s < 1, we denote the Sobolev space on ∂Ω defined by local Lipschitz charts as in [16]. For s ≥ 1, we define the Sobolev space H s (∂Ω) by using the characterization given by Brezzi-Gilardi in [4]: where M denotes a side of ∂Ω and We denote the trace of u on ∂Ω with γ 0 u. Sometimes we will use the same symbol to denote u and its trace γ 0 u. The interpretation will be left to the context.
We now recall the Friedrichs inequality, see [13, page 24] for more details.
Let r = r(x) be the distance from the set of vertices. For γ ∈ R, and s = 1, 2, . . . , we denote by H s γ (Ω) the Kondratev (or weighted Sobolev) space of functions for which the norm is finite, see [9]. For s = 0, this space evidently coincides with the weighted Lebesgue space L 2 γ (Ω). We also define, for s > 0 integer, the space H We define the composite spaces We consider the problem formally stated as where f and g are given functions, ∆ ℓ = ∂ 2 ∂ℓ 2 , ν the unit vector of exterior normal, b ∈ L ∞ (∂Ω) and we set θ s : H s (∂Ω) → H −s (∂Ω) as follows: for every u, v ∈ H s (∂Ω) where ·, · denotes the duality pairing between H −s (∂Ω) and H s (∂Ω). We remark that the nonlocal term θ s (·) can be regarded as an analogue of the fractional Laplace operator (−∆) s on the boundary.
We now define the bilinear form as follows: We consider the weak formulation of problem (1.2)-(1. 3): In what follows we denote by C all positive constants. The dependence of constants on some parameters is given in parentheses. We do not indicate the dependence of C on the geometry of Ω.

A priori estimates
. Suppose that s < 3/4. Then there exists a positive constant C = C(σ) such that (recall that α is the opening of the largest angle in ∂Ω).
Proof. We use the so-called Munchhausen trick. We consider the right-hand side in (1.3) as known functions. Then we easily have that First we estimate θ s (u) 2 L 2 (∂Ω) . Since u ∈ H 2 (∂Ω), it is sufficient to consider the local behavior of u near the vertices. Without loss of generality, we can assume that the vertex is located at the origin. We introduce a smooth cutoff function η and rectify ∂Ω near the origin. It is easy to see that uη| ∂Ω becomes a function on R which is the sum of a smooth function and a term c|t|η(t) (hereη is a one-dimensional cutoff function near the origin).
By choosing ε sufficiently small we obtain .
Since D 2 U vanishes in neighborhoods of vertices, without loss of generality we can assume that for every γ ∈ R If we consider the function v = u − U, from Hardy inequality applied on each segment of ∂Ω (see [8]) we obtain that v ∈ H 2 γ=0 (∂Ω). By rescaling we deduce v ∈ H (here we used the last restriction in (2.2)). From Theorem 3.1, Chapter 2 in [15] (with l = 0) it follows that v ∈ H 2 σ (Ω) if |σ − 1| < π/α (we recall that α is the opening of the largest angle in ∂Ω). With regard to (2.5) and (2.6), this implies (to estimate the first term, we also used that σ ≤ 1 in (2.2)).

Solvability of the Venttsel' problem
We begin the existence and uniqueness of the weak solution.

By the Poincaré inequality, E[u]
generates an equivalent norm on the subspace of functions in V 1 (Ω, ∂Ω) orthogonal to constants. Since the term ∂Ω bu 2 dℓ does not degenerate on constants, the statement follows.
The following existence and uniqueness result holds. where C depends only on the coercivity constant of E.
We finally prove the desired regularity for the weak solution of the nonlocal Venttsel' problem.

2)
where C depends on σ and the coercivity constant of E.
Proof. We introduce the set of operators L µ : V 2 σ (Ω, ∂Ω) → L 2 σ (Ω) × L 2 (∂Ω) We claim that the operator L 0 is invertible. Indeed, it corresponds to the boundary value problem −∆u = f in Ω, −∆ ℓ u + bu = g on ∂Ω. Here is compact. Since Ker(L 1 ) is trivial by Corollary 3.2, the operator L 1 is also invertible, and the proof is complete.
If Ω is a convex polygon, then α < π. So we can put σ = 0 and obtain the following result.
where C depends on the coercivity constant of E.
Remark 3.6. Without any sign condition on the coefficient b, the problem (1.2)-(1.3) is not necessarily solvable, but it has the Fredholm property.
Remark 3.7. All our results easily hold for an arbitrary piecewise smooth domain Ω ⊂ R 2 without cusps.