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In this manuscript we present an approach to analyze the discontinuous Galerkin solution for general quasilinear elliptic problems. This approach is sufficiently general to extend most of the well-known discretization schemes, including…
Consider positive solutions to second order elliptic equations with measurable coefficients in a bounded domain, which vanish on a portion of the boundary. We give simple necessary and sufficient geometric conditions on the domain, which…
A flag domain in $\mathbb{R}^{3}$ is a subset of $\mathbb{R}^{3}$ of the form $\{(x,y,t) : y < A(x)\}$, where $A \colon \mathbb{R} \to \mathbb{R}$ is a Lipschitz function. We solve the Dirichlet and Neumann problems for the sub-elliptic…
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…
We construct the fundamental solution or Green function for a divergence form elliptic system in two dimensions with bounded and measurable coefficients. We consider the elliptic system in a Lipschitz domain with mixed boundary conditions.…
The squared Laplace operator acting on symmetric rank-two tensor fields is studied on a (flat) Riemannian manifold with smooth boundary. Symmetry of this fourth-order elliptic operator is obtained provided that such tensor fields and their…
In terms of layer potential methods, this paper is devoted to study the $L^2$ boundary value problems for nonhomogeneous elliptic operators with rapidly oscillating coefficients in a periodic setting. Under a low regularity assumption on…
For a Jordan domain with sufficiently smooth boundaries, the solution to the Dirichlet problem for second order skew-symmetric strongly elliptic system with constant coefficients and regular enough boundary data is constructed in the form…
We consider the Dirichlet problem for solutions to general second-order homogeneous elliptic equations with constant complex coefficients. We prove that any Jordan domain with $C^{1,\alpha}$-smooth boundary, $0<\alpha<1$, is not regular…
The diffuse domain method for partial differential equations on complicated geometries recently received strong attention in particular from practitioners, but many fundamental issues in the analysis are still widely open. In this paper we…
This paper is concerned with the convergence of the solution of general elliptic boundary value problems in cylindrical domain, when some directions of the domain go to infinity.
We show that, under general conditions, the operator $\bigl (-\nabla \cdot \mu \nabla +1\bigr)^{1/2}$ with mixed boundary conditions provides a topological isomorphism between $W^{1,p}_D(\Omega)$ and $L^p(\Omega)$, for $p \in {]1,2[}$ if…
We establish Dahlberg's perturbation theorem for non-divergence form operators L = A\nabla^2. If L_0 and L_1 are two operators on a Lipschitz domain such that the L^p Dirichlet problem for the operator L_0 is solvable for some p in…
We consider the mixed problem for the Laplace operator in a class of Lipschitz graph domains in two dimensions with Lipschitz constant at most 1. The boundary of the domain is decomposed into two disjoint sets D and N. We suppose the…
In this paper, we use a probabilistic approach to show that there exists a unique, bounded continuous solution to the Dirichlet boundary value problem for a general class of second order non-symmetric elliptic operators $L$ with singular…
This paper investigates the Dirichlet problem for a non-divergence form elliptic operator $L$ in a bounded domain of $\mathbb{R}^d$. Under certain conditions on the coefficients of $L$, we first establish the existence of a unique Green's…
We derive a priori second order estimates for solutions of a class of fully nonlinear elliptic equations on Riemannian manifolds under some very general structure conditions. We treat both equations on closed manifolds, and the Dirichlet…
In this paper we are concerned with the initial boundary value problems of linear and semi-linear parabolic equations with mixed boundary conditions on non-cylindrical domains in spatial-temporal space. We obtain the existence of a weak…
We study generalized Robin boundary conditions, Robin-to-Dirichlet maps, and Krein-type resolvent formulas for Schr\"odinger operators on bounded Lipschitz domains in $\bbR^n$, $n\ge 2$. We also discuss the case of bounded…
We study the solvability of boundary-value problems for differential-operator equations of the second order in L p (0, 1; X), with 1 < p < +$\infty$, X being a UMD complex Banach space. The originality of this work lies in the fact that we…