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Let ${cal A}(x;D_x)$ be a second-order linear differential operator in divergence form. We prove that the operator ${l}I- {cal A}(x;D_x)$, where $lincsp$ and $I$ stands for the identity operator, is closed and injective when ${rm Re}l$ is large enough and the domain of ${cal A}(x;D_x)$ consists of a special class of weighted Sobolev function spaces related to conical open bounded sets of $rsp^n$, $n ge 1$.
We study the adjoint of the double layer potential associated with the Laplacian (the adjoint of the Neumann-Poincare operator), as a map on the boundary surface $Gamma$ of a domain in $mathbb{R}^3$ with conical points. The spectrum of this operator
In this paper, we derive Carleman estimates for the fractional relativistic operator. Firstly, we consider changing-sign solutions to the heat equation for such operators. We prove monotonicity inequalities and convexity of certain energy functionals
For a domain $Omega subset mathbb{R}^n$ and a small number $frak{T} > 0$, let [ mathcal{E}_0(Omega) = lambda_1(Omega) + {frak{T}} {text{tor}}(Omega) = inf_{u, w in H^1_0(Omega)setminus {0}} frac{int | abla u|^2}{int u^2} + {frak{T}} int frac{1}{2
We study the oblique derivative problem for uniformly elliptic equations on cone domains. Under the assumption of axi-symmetry of the solution, we find sufficient conditions on the angle of the oblique vector for Holder regularity of the gradient to
In this paper we classify all positive extremal functions to a sharp weighted Sobolev inequality on the upper half space, which involves divergent operators with degeneracy on the boundary. As an application of the results, we can derive a sharp Sobo