Related papers: Wiener's criterion for degenerate parabolic equati…
We study the boundary regularity of solutions to the porous medium equation $u_t = \Delta u^m$ in the degenerate range $m>1$. In particular, we show that in cylinders the Dirichlet problem with positive continuous boundary data on the…
We study the boundary regularity properties and derive a priori pointwise supremum estimates of weak solutions and their derivatives in terms of suitable weighted $L^2$-norms for a class of degenerate parabolic equations that satisfy…
Wiener's criterion for the regularity of a boundary point with respect to the Dirichlet problem for the Laplace equation has been extended to various classes of elliptic and parabolic partial differential equations. They include linear…
We study existence and uniqueness of solutions to a class of nonlinear degenerate parabolic equations, in bounded domains. We show that there exists a unique solution which satisfies possibly inhomogeneous Dirichlet boundary conditions. To…
In this note we prove a Wiener criterion of regularity of boundary points for the Dirichlet problem related to $X$-elliptic operators in divergence form enjoying the doubling condition and the Poincar\'e inequality. As a step towards this…
As an application of the theory of linear parabolic differential equations on noncompact Riemannian manifolds, developed in earlier papers, we prove a maximal regularity theorem for nonuniformly parabolic boundary value problems in…
In this work there is established an optimal existence and regularity theory for second order linear parabolic differential equations on a large class of noncompact Riemannian manifolds. Then it is shown that it provides a general unifying…
In this paper, we study parabolic equations in divergence form with coefficients that are singular degenerate as some Muckenhoupt weight functions in one spatial variable. Under certain conditions, weighted reverse H\"{o}lder's inequalities…
We study the boundary behavior of solutions to the Dirichlet problems for integro-differential operators with order of differentiability $s \in (0, 1)$ and summability $p>1$. We establish a nonlocal counterpart of the Wiener criterion,…
In this paper we obtain sharp Petrovskii criteria for the p-parabolic equation, both in the degenerate case p>2 and the singular case 1<p<2. We also give an example of an irregular boundary point at which there is a barrier, thus showing…
In this study, we firstly establish the well-posedness of a degenerate parabolic equation under Dirichlet boundary conditions. Following this, we introduce a shape design problem, which acts as a framework for approximating the degenerate…
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 study the Cauchy-Dirichlet problem associated to a phase transition modeled upon the degenerate two-phase Stefan problem. We prove that weak solutions are continuous up to the parabolic boundary and quantify the continuity by deriving a…
We study the boundary continuity of solutions to fully nonlinear elliptic equations. We first define a capacity for operators in non-divergence form and derive several capacitary estimates. Secondly, we formulate the Wiener criterion, which…
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 consider the Dirichlet problem for a class of elliptic and parabolic equations in the upper-half space $\mathbb{R}^d_+$, where the coefficients are the product of $x_d^\alpha, \alpha \in (-\infty, 1),$ and a bounded uniformly elliptic…
In this work we deal with degenerate parabolic equations with three lines of degeneration. Using "a-b-c" method we prove the uniqueness theorems defining conditions to parameters. We show nontrivial solutions for considered problems, when…
In this paper we prove Holder regularity of the gradient for solutions of Dirichlet problem associate to degenerate elliptic equations, extending the recent result of Imbert and Silvestre. Indeed we obtain regularity up to the boundary and…
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.…
In the framework of Potential Theory we prove existence or the Perron-Weiner-Brelot-Bauer solution to the Dirichlet problem related to a family of totally degenerate, in the sense of Bony, differential operators. We also state and prove a…