Related papers: Regularity for a Local-Nonlocal Transmission Probl…
We develop the regularity theory of viscosity solutions to transmission problems for fully nonlinear second order uniformly elliptic equations. Our results give a complete theory of existence, uniqueness, comparison principle, and…
A classical pseudodifferential operator $P$ on $R^n$ satisfies the $\mu$-transmission condition relative to a smooth open subset $\Omega $, when the symbol terms have a certain twisted parity on the normal to $\partial\Omega $. As shown…
We study a free transmission problem driven by degenerate fully nonlinear operators. Our first result concerns the existence of solutions to the associated Dirichlet problem. By framing the equation in the context of viscosity inequalities,…
We derive the solvability and regularity of the Dirichlet problem for fully non-linear elliptic equations possibly with degenerate right-hand side on Hermitian manifolds, through establishing a quantitative version of boundary estimate…
In this note we study the boundary regularity of solutions to nonlocal Dirichlet problems of the form $Lu=0$ in $\Omega$, $u=g$ in $\mathbb R^N\setminus\Omega$, in non-smooth domains $\Omega$. When $g$ is smooth enough, then it is easy to…
In this paper we investigate the regularity and solvability of solutions to Dirichlet problem for fully non-linear elliptic equations with gradient terms on Hermitian manifolds, which include among others the Monge-Amp\`ere equation for…
We consider elliptic transmission problems in several space dimensions near an interface which is $C^{1,1}$ diffeomorphic to an axisymmetric reference-interface with a singular point of cusp type. We establish the regularity of the gradient…
We consider a transmission problem consisting of a semilinear parabolic equation in a general non-smooth setting with emphasis on rough interfaces which bear a fractal-like geometry and nonlinear dynamic (possibly, nonlocal)\ boundary…
We study the higher H\"older regularity of local weak solutions to a class of nonlinear nonlocal elliptic equations with kernels that satisfy a mild continuity assumption. An interesting feature of our main result is that the obtained…
We prove optimal Lipschitz regularity for weak solutions of the measure-valued $p$-Poisson equation $-\Delta_p u = Q \; \mathcal{H}^{n-1} \llcorner \Gamma$. Here $p \in (1,2)$, $\Gamma$ is a compact and connected $C^2$-hypersurface without…
In this paper we obtain interior regularity estimates for viscosity solutions of nonlocal Dirichlet problems that degenerate when the gradient of the solution vanishes. Interior H\"older estimates are obtained when the order of the…
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…
In this work, we present a systematic approach to investigate the existence, multiplicity, and local gradient regularity of solutions for nonlocal quasilinear equations with local gradient degeneracy. Our method involves an interactive…
This paper is concerned with the regularity theory of a transmission problem arising in composite materials. We give a new self-contained proof for the $C^{k,\alpha}$ estimates on both sides of the interface under the minimal assumptions on…
We study the elliptic equation with a line Dirac delta function as the source term subject to the Dirichlet boundary condition in a two-dimensional domain. Such a line Dirac measure causes different types of solution singularities in the…
We establish that solving an optimal transportation problem in which the source and target densities are defined on manifolds with different dimensions, is equivalent to solving a new nonlocal analog of the Monge-Amp\`ere equation,…
For $2a$-order strongly elliptic operators $P$ generalizing $(-\Delta )^a$, $0<a<1$, the treatment of the homogeneous Dirichlet problem on a bounded open set $\Omega \subset R^n$ by pseudodifferential methods, has been extended in a recent…
In this work, we study one-dimensional nonlocal elliptic transmission problems with piecewise constant coefficients that may change sign across an interface. In the local setting, we recall the T-coercive structure of the problem and…
We consider elliptic equations of order $2m$ in a bounded domain $Q\subset\mathbb R^n$ with nonlocal boundary-value conditions connecting the values of a solution and its derivatives on $(n-1)$-dimensional smooth manifolds $\Gamma_i$ with…
We provide a rather complete description of the sharp regularity theory for a family of heterogeneous, two-phase variational free boundary problems, $\mathcal{J}_\gamma \to $ min, ruled by nonlinear, $p$-degenerate elliptic operators.…