Related papers: The Wiener criterion for nonlocal Dirichlet proble…
In this paper we study nonlocal nonlinear equations of fractional $(s,p)$-Laplacian type on $\mathbf{R}^n$. We show that the irregular boundary points for the Dirichlet problem can be divided into two disjoint classes: semiregular and…
Solutions to nonlinear integro-differential systems are regular outside a negligible closed subset whose Hausdorff dimension can be explicitly bounded from above. This subset can be characterized using quantitative, universal energy…
We study a Dirichlet problem in the entire space for some nonlocal degenerate elliptic operators with internal nonlinearities. With very mild assumptions on the boundary datum, we prove existence and uniqueness of the solution in the…
This paper establishes a Wiener criterion at $\infty$ to characterise the unique solvability of the Dirichlet problem for degenerate elliptic equations with power-like weights in arbitrary open sets. In the measure-theoretical context, the…
We study boundary value problems at infinity for the graph $p$-Laplacian on infinite, connected, locally finite weighted graphs. Our main result is a Wiener criterion for $p$-massiveness. Assuming volume doubling and a weak…
Using the Caffarelli--Silvestre extension, we show for a general open set $\Om\subset\R^n$ that a boundary point $x_0$ is regular for the fractional Laplace equation $(-\Delta)^su=0$, $0<s<1$, if and only if $(x_0,0)$ is regular for the…
In this paper we are concerned with hypoelliptic diffusion operators $\mathcal{H}$. Our main aim is to show, with an axiomatic approach, that a Wiener-type test of $\mathcal{H}$-regularity of boundary points can be derived starting from the…
In this note we set up the elliptic and the parabolic Dirichlet problem for linear nonlocal operators. As opposed to the classical case of second order differential operators, here the "boundary data" are prescribed on the complement of a…
This paper is devoted to prove the existence of positive solutions of a second order differential equation with a nonhomogeneous Dirichlet conditions given by a parameter dependence integral. The studied problem is a nonlocal perturbation…
In this paper we are interested in integro-differential elliptic and parabolic equations involving nonlocal operators with order less than one, and a gradient term whose coercivity growth makes it the leading term in the equation. We obtain…
We study semilinear problems in bounded $C^{1,1}$ domains for non-local operators with a boundary condition. The operators cover and extend the case of the spectral fractional Laplacian. We also study harmonic functions with respect to the…
We develop an existence, regularity and potential theory for nonlinear integrodifferential equations involving measure data. The nonlocal elliptic operators considered are possibly degenerate and cover the case of the fractional…
We study a Dirichlet boundary value problem associated to an anisotropic differential operator on a smooth bounded of $\Bbb R^N$. Our main result establishes the existence of at least two different non-negative solutions, provided a certain…
We deal with a wide class of nonlinear nonlocal equations led by integro-differential operators of order $(s,p)$, with summability exponent $p \in (1,\infty)$ and differentiability exponent $s\in (0,1)$, whose prototype is the fractional…
We present an existence and uniqueness result for weak solutions of Dirichlet boundary value problems governed by a nonlocal operator in divergence form and in the presence of a datum which is assumed to belong only to $L^1(\Omega)$ and to…
In this paper we survey some results on the Dirichlet problem \[\left\{ \begin{array}{rcll} L u &=&f&\textrm{in }\Omega \\ u&=&g&\textrm{in }\mathbb R^n\backslash\Omega \end{array}\right.\] for nonlocal operators of the form…
This paper investigates the Dirichlet problem for a non-divergence form elliptic operator $L$ in a bounded domain of $\mathbb{R}^d$. Under certain conditions on the coefficients of $L$, we first establish the existence of a unique Green's…
The paper deals with second order parabolic equations on bounded domains with Dirichlet conditions in arbitrary Euclidean spaces. Their interest comes from being models for describing reaction-diffusion processes in several frameworks. A…
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.…
A Dirichlet-type problem is studied for an equation of even order with variable coefficients. A criterion for the uniqueness of a solution is given. The solution is built in the form of a Fourier series. When justifying the convergence of…