Related papers: A $C^1$ regularity result for the inhomogeneous no…

We prove regularity results such as interior Lipschitz regularity and boundary continuity for the Cauchy-Dirichlet problem associated to a class of parabolic equations inspired by the evolutionary $p$-Laplacian, but extending it at a wide…

This paper is devoted to analyse the Dirichlet problem for a nonlinear elliptic equation involving the $1$--Laplacian and a total variation term, that is, the inhomogeneous case of the equation arising in the level set formulation of the…

We study finite element approximations of the nonhomogeneous Dirichlet problem for the fractional Laplacian. Our approach is based on weak imposition of the Dirichlet condition and incorporating a nonlocal analogous of the normal derivative…

We study the equation $u_{11}u_{22} = 1$ in $\mathbb{R}^2$. Our results include an interior $C^2$ estimate, classical solvability of the Dirichlet problem, and the existence of non-quadratic entire solutions. We also construct global…

We study the Dirichlet problem of the following discrete infinity Laplace equation on unbounded subgraphs \begin{equation*} \Delta_{\infty}u(x):=\inf_{y\sim x}u(y)+\sup_{y\sim x}u(y)-2u(x)=f(x). \end{equation*} For the homogeneous case…

We provide a sharp $C^{1,\alpha}$ estimate up to the boundary for a viscosity solution of a degenerate fully nonlinear elliptic equation with the oblique boundary condition on a $C^1$ domain. To this end, we first obtain a uniform boundary…

In this paper we prove that any solution of the $m$-polyharmonic Poisson equation in a Reifenberg-flat domain with homogeneous Dirichlet boundary condition, is $\mathscr{C}^{m-1,\alpha}$ regular up to the boundary. To achieve this result we…

In this paper, we prove the boundary pointwise $C^{0}$-regularity of weak solutions for Dirichlet problem of elliptic equations in divergence form with distributional coefficients, where the boundary value equals to zero. This is a…

In this paper we investigate the problem of identifying the source term in an elliptic system from a single noisy measurement couple of the Neumann and Dirichlet data. A variational method of Tikhonov-type regularization with specific…

This paper is devoted to the study of the Dirichlet problem associated with the Dunkl Laplacian $\Delta_k$. We establish, under some condition on a bounded domain $D$ of $\R^d$, the existence of a unique continuous function $h$ on $\R^d$…

We investigate existence and uniqueness of solutions to the filtration equation with an inhomogeneous density in ${\mathbb R}^N$, approaching at infinity a given continuous datum of Dirichlet type.

It is shown that solutions of the Neumann problem for the Poisson equation in an arbitrary convex $n$-dimensional domain are uniformly Lipschitz. Applications of this result to some aspects of regularity of solutions to the Neumann problem…

For a one-phase free boundary problem involving a fractional Laplacian, we prove that "flat free boundaries" are $C^{1,\alpha}$. We recover the regularity results of Caffarelli for viscosity solutions of the classical Bernoulli-type free…

In a previous paper we considered a class of infinitely degenerate quasilinear equations and derived a priori bounds for high order derivatives of solutions in terms of the Lipschitz norm. We now show that it is possible to obtain bounds…

In this paper, we consider the pointwise boundary Lipschitz regularity of solutions for the semilinear elliptic equations in divergence form mainly under some weaker assumptions on nonhomogeneous term and the boundary. If the domain…

The Monge-Amp\`ere type equations over bounded convex domains arise in a host of geometric applications. In this paper, we focus on the Dirichlet problem for a class of Monge-Amp\`ere type equations, which can be degenerate or singular near…

We prove the solvability of the Dirichlet problem for the variable exponent $p$-Laplacian with boundary data in $W^{1,p(x)}(\Omega)$ on a bounded, smooth domain $\Omega \subset {\mathbb R}^n$. Our main focus will be on an a.e. finite…

We consider the Dirichlet problem for the Beltrami equation in some simply connected domain. We consider the class of all homeomorphic solutions of such a problem with a normalization condition and set-theoretic constraints on their complex…

We study the exterior Dirichlet problem for the homogeneous $k$-Hessian equation. The prescribed asymptotic behavior at infinity of the solution is zero if $k<\frac{n}{2}$, it is $\log|x|+O(1)$ if $k=\frac{n}{2}$ and it is…

In this paper, we prove two unique continuation results for second order elliptic equations with Robin boundary conditions on $C^{1,1}$ domains. The first one is a sharp vanishing order estimate of Robin problems with Lipschitz coefficients…