Related papers: Wiener type regularity for non-linear integro-diff…
A Wiener-type condition for the continuity at the boundary points of Q-minima, is established, in terms of the divergence of a suitable Wiener integral.
We prove a nonlocal, nonlinear commutator estimate concerning the transfer of derivatives onto testfunctions. For the fractional $p$-Laplace operator it implies that solutions to certain degenerate nonlocal equations are higher…
We study robust regularity estimates for a class of nonlinear integro-differential operators with anisotropic and singular kernels. In this paper, we prove a Sobolev-type inequality, a weak Harnack inequality, and a local H\"older estimate.
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…
We consider a class of nonlinear integro-differential equations whose leading operator is obtained as a superposition of $(-\Delta_{p})^{s}$ and $(-\Delta_{p})^{t}$, where $0<s<t<1<p<\infty$, weighted via two possibly degenerate…
We give necessary and sufficient conditions for the existence of weak solutions to the model equation $$-\Delta_p u=\sigma \, u^q \quad \text{on} \, \, \, \R^n,$$ in the case $0<q<p-1$, where $\sigma\ge 0$ is an arbitrary locally integrable…
In this paper we study regularity of partial differential equations with polynomial coefficients in non isotropic Beurling spaces of ultradifferentiable functions of global type. We study the action of transformations of Gabor and Wigner…
In this paper we prove a Wiener-type characterization of boundary regularity, in the spirit of a classical result by Landis, for a class of evolutive H\"ormander operators. We actually show the validity of our criterion for a larger class…
We survey some recent regularity results for fractional p-Laplacian elliptic equations, especially focusing on pure and weighted boundary H\"older continuity of the solutions of related Dirichlet problems. Then, we present some applications…
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 set up a general framework tailor-made to solve complement value problems governed by symmetric nonlinear integrodifferential $p$-L\'evy operators. A prototypical example of integrodifferential $p$-L\'evy operators is the well-known…
This work investigates the Sobolev regularity of solutions to perturbed fractional 1-Laplace equations. Under the assumption that weak solutions are locally bounded, we establish that the regularity properties are analogous to those…
Following ideas of Caffarelli and Silvestre in~\cite{CS}, and using recent progress in hyperbolic fillings, we define fractional $p$-Laplacians $(-\Delta_p)^\theta$ with $0<\theta<1$ on any compact, doubling metric measure space…
In this paper, we consider a class of nonlinear fractional differential equations involving Hilfer derivative with boundary conditions. First, we obtain an equivalent integral for the given boundary value problem in weighted space of…
In this paper, we prove Wiener's criterion for parabolic equations with singular and degenerate coefficients. To be precise, we study the problem of the regularity of boundary points for the Dirichlet problem for degenerate parabolic…
Perron's method and Wiener's criterion have entirely solved the Dirichlet problem for the Laplace equation. Since then, this approach has attracted the attention of many mathematicians for applying these ideas in the more general equations.…
A sharp pointwise differential inequality for vectorial second-order partial differential operators, with Uhlenbeck structure, is offered. As a consequence, optimal second-order regularity properties of solutions to nonlinear elliptic…
In this article, we investigate the existence, uniqueness, nonexistence, and regularity of weak solutions to the nonlinear fractional elliptic problem of type $(P)$ (see below) involving singular nonlinearity and singular weights in smooth…
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…
Given a bounded finely open set $V$ and a function $f$ on the fine boundary of $V$, we introduce four types of upper Perron solutions to the nonlinear Dirichlet problem for $p$-energy minimizers, $1<p<\infty$, with $f$ as boundary data.…