Singular boundary behaviour and large solutions for fractional elliptic equations

Abstract

We show that the boundary behaviour of solutions to nonlocal fractional equations posed in bounded domains strongly differs from the one of solutions to elliptic problems modelled upon the Laplace-Poisson equation with zero boundary data. In this classical case it is known that, at least in a suitable weak sense, solutions of non-homogeneous Dirichlet problem are unique and tend to zero at the boundary. Limits of these solutions then produce solutions of some non-homogeneous Dirichlet problem as the interior data concentrate suitably to the boundary. Here, we show that such results are false for equations driven by a wide class of nonlocal fractional operators, extending previous findings for some models of the fractional Laplacian operator. Actually, different blow-up phenomena may occur at the boundary of the domain. We describe such explosive behaviours and obtain precise quantitative estimates depending on simple parameters of the nonlocal operators. Our unifying technique is based on a careful study of the inverse operator in terms of the corresponding Green function. Contents

Combined effects in mixed local-nonlocal stationary problems

“…• Optimal boundary behavior: To prove this, we study the action of the Green's operator on the inverse of the distance function perturbed with logarithmic nonlinearity. Using the lower and upper bound estimates of the Green's kernel [26] and borrowing some techniques from [2], we show…”

Singular solutions for fractional parabolic boundary value problems

“…However, for the Censored (or Regional) Fractional Laplacian (CFL), 2s− γ − 1 = 0 and this is a Dirichlet type condition. As mentioned in [4], no examples are known to satisfy 2s−γ −1 > 0. In this last setting, Eu = 0 seems to be a redundant condition u = 0 in ∂Ω, which calls into question if the Green operator G, introduced in (2.1), comes from a reasonable direct operator L. Nevertheless, as in [4] we will include a mathematical framework for these problems.…”

“…A well-posed problem follows in that case. Two questions immediately arise from the elliptic studies, in particular [4]: whether this singular behaviour is preserved in the parabolic problem, and what is the corresponding theory. We will try to give a satisfactory answer to those questions in this paper.…”

Heat kernel estimates for subordinate Markov processes and their applications

“…(2) For all (t, x, y) ∈ [1, ∞) × D × D, q(t, x, y) ≃ e −tλ β 1 δ D (x) α/2 δ D (y) α/2 , where λ 1 is the smallest eigenvalue of the Dirichlet (fractional) Laplacian (−∆) α/2 D . From Theorem 1.1 (1), one can see that, for x, y away from the boundary (in the sense that δ ∧ (x, y) ≥ |x − y| ∨ t 1/(αβ) ), and for all β ∈ (0, 1), it holds that q(t, x, y) ≃ t −d/(αβ) ∧ t |x − y| d+αβ .…”

“…1,1 open set in R d and α ∈ (0, 2). Let Y D be the part process in D of an isotropic α-stable process.…”