Related papers: Regularity of Infinity for Elliptic Equations with…
Under structural conditions which are almost optimal, we derive a quantitative version of boundary estimate then prove existence of solutions to Dirichlet problem for a class of fully nonlinear elliptic equations on Hermitian manifolds.
In this paper, we study the nonhomogeneous Dirichlet problem concerning general semilinear elliptic equations in divergence form. We establish that the boundary Lipschitz regularity of solutions under some more weaker conditions on the…
We prove sharp boundary regularity of solutions to nonlocal elliptic equations arising from operators comparable to the fractional Laplacian over Reifenberg flat sets and with null exterior condition. More precisely, if the operator has…
As explained in detail in the prologue to this manuscript, boundedness of weak solutions for general classes of elliptic equations in divergence form is a classic tool for achieving higher regularity. We propose here some global boundedness…
We provide regularity results at the boundary for continuous viscosity solutions to nonconvex fully nonlinear uniformly elliptic equations and inequalities in Euclidian domains. We show that (i) any solution of two sided inequalities with…
We study connections between the problem of the existence of positive solutions for certain nonlinear equations and weighted norm inequalities. In particular, we obtain explicit criteria for the solvability of the Dirichlet problem…
We study boundary value problems at infinity for the graph $p$-Laplacian on infinite, connected, locally finite weighted graphs. Our main result is a Wiener criterion for $p$-massiveness. Assuming volume doubling and a weak…
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,…
We consider weak solutions $u \in u_0 + W^{1,2}_0(\Omega,R^N) \cap L^{\infty}(\Omega,R^N)$ of second order nonlinear elliptic systems of the type $- div a (\cdot, u, Du) = b(\cdot,u,Du)$ in $\Omega$ with an inhomogeneity satisfying a…
We develop a higher regularity theory for general quasilinear elliptic equations and systems in divergence form with random coefficients. The main result is a large-scale $L^\infty$-type estimate for the gradient of a solution. The estimate…
In this paper, we study the 3D Lam\'e system and establish its weighted positive definiteness for a certain range of elastic constants. By modifying the general theory developed in Maz'ya (2002), we then show, under the assumption of…
We prove continuity for bounded weak solutions of a nonlinear nonlocal parabolic type equation associated to a Dirichlet form with a rough kernel. The equation is allowed to be singular at the level zero, and solutions may change sign. If…
We study uniform Lipschitz regularity estimates for elliptic systems in divergence form with continuous coefficients, based on rapidly oscillating periodic coefficients derived from homogenization theory. We extend a result by Avellaneda…
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…
The Dirichlet problem in arbitrary domains for a wide class of anisotropic elliptic equations of the second order with variable exponent nonlinearities and the right-hand side as a measure is considered. The existence of an entropy solution…
We study the Dirichlet problem for a second-order elliptic operator $L^*$ in double divergence form, also known as the stationary Fokker-Planck-Kolmogorov equation. Assuming that the leading coefficients have Dini mean oscillation, we…
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…
In this paper, we extend the uniform regularity estimates obtained by M. Avellanda and F. Lin in the paper of Compactness methods in the theory of homogenization (Comm. Pure Appl. Math. 40(1987), no.6, 803-847) to the more general second…
We define the notion of an exceptional manifold to be a flat Riemannian manifold with boundary which supports a positive harmonic function satisfying simultaneously a zero Dirichlet condition and a constant (nonzero) Neumann condtion at the…
We study the relationship between the Regularity and Dirichlet boundary value problems for parabolic equations of the form $Lu=\text{div}(A \nabla u)-u_t=0$ in Lip$(1,1/2)$ time-varying cylinders, where the coefficient matrix $A = \left[…