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We study elliptic and parabolic problems governed by singular elliptic operators \begin{equation*} \mathcal L =\sum_{i,j=1}^{N+1}q_{ij}D_{ij}+\frac c y D_y \end{equation*} in the half-space $\mathbb{R}^{N+1}_+=\{(x,y): x \in \mathbb{R}^N,…

We study an elliptic differential operator A on a manifold with conic points. Assuming A to be defined on the smooth functions supported away from the singularities, we first address the question of possible closed extensions of A to L^p…

We consider the elliptic estimates for Dirichlet-Neumann operator related to the water-wave problem on a two-dimensional corner domain in this paper. Due to the singularity of the boundary, there will be singular parts in the solution of…

The main goal of this paper is to address global hypoellipticity issues for the following class of operators: $L = D_t + C(t,x,D_x)$, where $(t,x) \in \mathbb{T} \times M$, $\mathbb{T}$ is the one-dimensional torus, $M$ is a closed manifold…

We study the closed extensions (realizations) of differential operators subject to homogeneous boundary conditions on weighted L_p-Sobolev spaces over a manifold with boundary and conical singularities. Under natural ellipticity conditions…

In the first part of the article, we give necessary and sufficient conditions for the solvability of a class of nonlinear elliptic boundary value problems with nonlinear boundary conditions involving the q-Laplace-Beltrami operator. In the…

Let $M$ be a smooth manifold with boundary $\partial M$ and bounded geometry, $\partial_D M \subset \partial M$ be an open and closed subset, $P$ be a second order differential operator on $M$, and $b$ be a first order differential operator…

We consider four-dimensional Euclidean gravity in a finite cavity. Dirichlet conditions do not yield a well-posed elliptic system, and Anderson has suggested boundary conditions that do. Here we point out that there exists a one-parameter…

We investigate the spectral properties of a differential elliptic operator on $H^1(\bar{\Omega}\cup \Sigma)$, where $\Omega$ is a smooth domain surrounded by a layer $\Sigma$. The thickness of the layer is given by $\varepsilon h$, where…

In this paper, we study fully nonlinear second-order elliptic and parabolic equations with Neumann boundary conditions on compact Riemannian manifolds with smooth boundary. We derive oscillation bounds for admissible solutions with Neumann…

We consider second-order partial differential operators $H$ in divergence form on $\Ri^d$ with a positive-semidefinite, symmetric, matrix $C$ of real $L_\infty$-coefficients and establish that $H$ is strongly elliptic if and only if the…

We consider a strongly nonlinear elliptic problem with the homogeneous Dirichlet boundary condition. The growth and the coercivity of the elliptic operator is assumed to be indicated by an inhomogeneous anisotropic $\mathcal{N}$-function.…

Let $M$ be a Riemannian manifold with a smooth boundary. The main question we address in this article is: "When is the Laplace-Beltrami operator $\Delta\colon H^{k+1}(M)\cap H^1_0(M) \to H^{k-1}(M)$, $k\in \mathbb{N}_0$, invertible?" We…

In a class of inner product H\"ormander spaces, we investigate a general elliptic problem for which the maximum of orders of boundary conditions is grater than or equal to the order of elliptic equation. The order of regularity for these…

In this short note, we consider the Dirichlet problem associated to an even order elliptic system with antisymmetric first order potential. Given any continuous boundary data, we show that weak solutions are continuous up to boundary.

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…

We consider a one-parameter family of degenerately elliptic operators $\cal{L}_\gamma$ on the closed disk $\mathbb{D}$, of Keldysh (or Kimura) type, which appears in prior work [Mishra et al., Inverse Problems (2022)] by the authors and…

We study functional determinants for Dirac operators on manifolds with boundary and discuss the ellipticity of boundary problems by using the Calder\'on projector. We give, for local boundary conditions, an explicit formula relating these…

We prove sharp lower bound estimates for the first nonzero eigenvalue of the non-linear elliptic diffusion operator $L_p$ on a smooth metric measure space, without boundary or with a convex boundary and Neumann boundary condition,…

Consider a second order, strongly elliptic negative semidefinite differential operator $L$ (maybe a system) on a compact Riemannian manifold $\overline{M}$ with smooth boundary, where the domain of $L$ is defined by a coercive boundary…