Related papers: $p(x)$-Stability of the Dirichlet problem for Pois…

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 Dirichlet problem for p-harmonic functions on metric spaces with respect to arbitrary compactifications. A particular focus is on the Perron method, and as a new approach to the invariance problem we introduce Sobolev-Perron…

For the $p$-Laplace Dirichlet problem (where $\varphi (t)=t|t|^{p-2}$, $p>1$) \[ \varphi(u'(x))'+ f(u(x))=0 \;\;\;\; \mbox{for $-1<x<1$}, \;\; u(-1)=u(1)=0 \] assume that $f'(u)>(p-1)\frac{f(u)}{u}>0$ for $u>\gamma>0$, while $\int_u^\gamma…

We study the regularity up to the boundary of solutions to the Dirichlet problem for the fractional Laplacian. We prove that if $u$ is a solution of $(-\Delta)^s u = g$ in $\Omega$, $u \equiv 0$ in $\R^n\setminus\Omega$, for some…

We consider the elliptic equation $-\Delta u = u^q|\nabla u|^p$ in $\mathbb R^n$ for any $p\ge 2$ and $q>0$. We prove a Liouville-type theorem, which asserts that any positive bounded solution is constant. The proof technique is based on…

The Dirichlet boundary value problem for the Stokes operator with $L^p$ data in any dimension on domains with conical singularity (not necessary a Lipschitz graph) is considered. We establish the solvability of the problem for all $p\in…

We study the convergence of the Method of Reflections for the Dirichlet problem of the Poisson and the Stokes equations in perforated domains which consist in the exterior of balls. We prove that the method converges if the balls are…

On a smooth domain $\Omega\subset\subset\mathbb C^n$, we consider the Dirichlet problem for the complex Monge-Amp\`ere equation $((dd^cu)^n=fdV,\,u|_{b\Omega}\equiv\phi)$. We state the H\"older regularity of the solution $u$ when the…

We provide an answer to a question raised by Levine and Weinberger in their $1986$ paper concerning the difference between Dirichlet and Neumann eigenvalues of the Laplacian on bounded domains in $\mathbb{R}^{n}$. More precisely, we show…

Consider a complete $d$-dimensional Riemannian manifold $(\mathcal M,g)$, a point $p\in\mathcal M$ and a nonlinearity $f(q,u)$ with $f(p,0)>0$. We prove that for any odd integer $N\ge3$, there exists a sequence of smooth domains…

We present a nonvariational setting for the Neumann problem for the Poisson equation for solutions that are H\"{o}lder continuous and that may have infinite Dirichlet integral. We introduce a distributional normal derivative on the boundary…

We establish new quantitative estimates for localized finite differences of solutions to the Poisson problem for the fractional Laplace operator with homogeneous Dirichlet conditions of solid type settled in bounded domains satisfying the…

In this paper we study numerical approximations of the evolution problem for the nonlocal $p$-Laplacian with homogeneous Neumann boundary conditions. First, we derive a bound on the distance between two continuous-in-time trajectories…

We prove the homogenization of fully nonlinear parabolic equations with periodic oscillating Dirichlet boundary conditions on certain general prescribed space-time domains. It was proved in [9,10] that for elliptic equations, the…

In this paper we study the following singular perturbation problem for the $p_\varepsilon(x)$-Laplacian: \[ \Delta_{p_\varepsilon(x)}u^\varepsilon:=\mbox{div}(|\nabla u^\varepsilon(x)|^{p_\varepsilon(x)-2}\nabla…

This paper is concerned with the homogenization of Dirichlet problem of elliptic systems in a bounded, smooth domain of finite type. Both the coefficients of the elliptic operator and the Dirichlet boundary data are assumed to be periodic…

In this paper, we develop a framework for the discretization of a mixed formulation of quasi-reversibility solutions to ill-posed problems with respect to Poisson's equations. By carefully choosing test and trial spaces a formulation that…

We consider a Dirichlet problem driven by the anisotropic $(p,q)$-Laplacian and a reaction with gradient dependence (convection). The presence of the gradient in the source term excludes from consideration a variational approach in dealing…

We establish existence and uniqueness results for nonlinear elliptic Dirichlet boundary value problems on n-dimensional time scale domains. Time scales provide a unified framework that encompasses continuous, discrete, and hybrid settings.…

In this paper we deal with the well-posedness of Dirichlet problems associated to nonlocal Hamilton-Jacobi parabolic equations in a bounded, smooth domain $\Omega$, in the case when the classical boundary condition may be lost. We address…