Related papers: The Dirichlet problem for double divergence form e…

We prove regularity estimates for weak solutions to the Dirichlet problem for a divergence form elliptic operator. We give $L^p$ estimates for the second derivative for $p<2$. Our work generalizes results due to Miranda [28].

In this paper we present and analyse a discontinuous Galerkin finite element method (DGFEM) for the approximation of solutions to elliptic partial differential equations in nondivergence form, with oblique boundary conditions, on curved…

In this work we study regularity properties of solutions to fractional elliptic problems with mixed Dirichlet-Neumann boundary data when dealing with the Spectral Fractional Laplacian.

We consider planar solutions to certain quasilinear elliptic equations subject to the Dirichlet boundary conditions; the boundary data is assumed to have finite number of relative maximum and minimum values. We are interested in certain…

We solve the Dirichlet problem for fully nonlinear elliptic equations on Riemannian manifolds under essentially optimal structure conditions, especially with no restrictions to the curvature of the underlying manifold and the second…

We report on some recent existence and uniqueness results for elliptic equations subject to Dirichlet boundary condition and involving a singular nonlinearity. We take into account the following types of problems: (i) singular problems with…

We consider the Dirichlet problem for a class of elliptic and parabolic equations in the upper-half space $\mathbb{R}^d_+$, where the coefficients are the product of $x_d^\alpha, \alpha \in (-\infty, 1),$ and a bounded uniformly elliptic…

The present paper establishes a certain duality between the Dirichlet and Regularity problems for elliptic operators with $t$-independent complex bounded measurable coefficients ($t$ being the transversal direction to the boundary). To be…

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 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…

We study Dirichlet problems for fractional Laplace equations of the form $(-\Delta)^{\frac{\alpha}{2}} u = f(x,u)$ in $\mathbb{R}^{n}$ for $0<\alpha<n$ where the nonlinearity $f(x,u) = \sum_{i=1}^{M} \sigma_{i} u^{q_i} + \omega$ involves…

We establish $L^p$ solvability of the Dirichlet problem, for some finite $p$, in a 1-sided chord-arc domain $\Omega$ (i.e., a uniform domain with Ahlfors-David regular boundary), for elliptic equations of the form \[ Lu=-\text{div}(A\nabla…

This paper deals with the Landesman-Lazer type problem of elliptic equations associated with homogeneous Dirichlet boundary conditions. By using some dynamical arguments we derive some new results on bifurcation from infinity and…

The present paper pioneers the study of the Dirichlet problem with $L^q$ boundary data for second order operators with complex coefficients in domains with lower dimensional boundaries, e.g., in $\Omega := \mathbb R^n \setminus \mathbb R^d$…

We consider the Dirichlet problem for second-order linear elliptic equations in divergence form \begin{equation*} -\mathrm{div }(A\nabla u)+\mathbf{b} \cdot \nabla u+\lambda u=f+\mathrm{div } \mathbf{F}\quad \text{in }…

In this paper, we prove that there exists a unique solution to the Dirichlet boundary value problem for a general class of semilinear second order elliptic partial differential equations. Our approach is probabilistic. The theory of…

We study the homogenization of the equation $-A(\frac{\cdot}{\varepsilon}):D^2 u_{\varepsilon} = f$ posed in a bounded convex domain $\Omega\subset \mathbb{R}^n$ subject to a Dirichlet boundary condition and the numerical approximation of…

We prove $L_p$ estimates of solutions to a conormal derivative problem for divergence form complex-valued higher-order elliptic systems on a half space and on a Reifenberg flat domain. The leading coefficients are assumed to be merely…

We consider the Dirichlet problem for a class of quasilinear elliptic systems in domain with irregular boundary. The principal part satisfies componentwise coercivity condition and the nonlinear terms are Carath\'eodory maps having Morrey…

We study spectral stability estimates of elliptic operators in divergence form $-\textrm{div} [A(w) \nabla g(w)]$ with the Dirichlet boundary condition in non-Lipschitz domains $\widetilde{\Omega} \subset \mathbb C$. The suggested method is…