Related papers: Unique Continuation for Sublinear Elliptic Equatio…
We prove strong unique continuation property for the differential inequality $|(\partial_t +\Delta)u(x,t)|\le V(x,t)|u(x,t)|$ with $V$ contained in weak spaces. In particular, we establish the strong unique continuation property for $V\in…
In this article we give an extention of the L^2-theory of anisotropic singular perturbations for elliptic problems. We study a linear and some nonlinear problems involving L^p data (1<p<2). Convergences in pseudo Sobolev spaces are proved…
In the present work, we consider weakly-singular integral equations arising from linear second-order strongly-elliptic PDE systems with constant coefficients, including, e.g., linear elasticity. We introduce a general framework for optimal…
In this paper we study the game $p-$Laplacian on a tree, that is, $$ u(x)=\frac{\alpha}2\left\{\max_{y\in \S(x)}u(y) + \min_{y\in \S(x)}u(y)\right\} + \frac{\beta}{m}\sum_{y\in \S(x)} u(y), $$ here $x$ is a vertex of the tree and $S(x)$ is…
We consider positive singular solutions to semilinear elliptic problems with possibly singular nonlinearity. We deduce symmetry and monotonicity properties of the solutions via the moving plane procedure.
In this paper, we first establish a weak unique continuation property for time-fractional diffusion-advection equations. The proof is mainly based on the Laplace transform and the unique continuation properties for elliptic and parabolic…
In this paper, we prove the strong unique continuation property at the origin for solutions of the following scaling critical parabolic differential inequality \[ |\operatorname{div} (A(x,t) \nabla u) - u_t| \leq \frac{M}{|x|^{2}} |u|,\ \ \…
This paper is concerned with developing accurate and efficient discontinuous Galerkin methods for fully nonlinear second order elliptic and parabolic partial differential equations (PDEs) in the case of one spatial dimension. The primary…
The paper studies continutity of Moser nonlinearity in two dimensions with respect to weak convergence. Unlike the critical nonlinearity in the Sobolev inequality, which lacks weak continuity at any point, Moser functional fails to be…
In this paper, we study the observability and controllability of wave equations coupled by first or zero order terms on a compact manifold. We adopt the approach in Dehman-Lebeau's paper \cite{DehmanLebeau09} to prove that: the weak…
In this article we find the equivalent conditions to assure the existence and uniqueness of positive solutions to semilinear elliptic equations wih double power nonlinearities. As a bonus, we give a simpler proof of our former result that…
In this paper, we study the stabilizer-free weak Galerkin methods on polytopal meshes for a class of second order elliptic boundary value problems of divergence form and with gradient nonlinearity in the principal coefficient. With certain…
In this paper we develop a monotonicity formula for elliptic systems with Neumann boundary coupling, proving unique continuation and classification of blow-up profiles. As an application, we obtain strong unique continuation for some fourth…
We consider initial boundary value problems with the homogeneous Neumann boundary condition. Given an initial value, we establish the uniqueness in determining a spatially varying coefficient of zeroth-order term by a single measurement of…
We obtain a unique continuation result for fractional Schr\"odinger operators with potential in Morrey spaces. This is based on Carleman inequalities for fractional Laplacians.
We propose in this paper to study the solutions of some nonlinear elliptic equations with singular potential.
A second-order regularity theory is developed for solutions to a class of quasilinear elliptic equations in divergence form, including the $p$-Laplace equation, with merely square-integrable right-hand side. Our results amount to the…
We consider the Dirichlet and Neumann problems for second-order linear elliptic equations: \[ -\triangle u +\mathrm{div}(u\mathbf{b}) =f \quad\text{ and }\quad -\triangle v -\mathbf{b} \cdot \nabla v =g \] in a bounded Lipschitz domain…
This paper aims to establish counterparts of fundamental regularity statements for solutions to elliptic equations in the setting of low-dimensional structures such as, for instance, glued manifolds or CW-complexes. The main result proves…
{\bf Abstract} \,\,We prove exponential decay of the critical and subcritical semilinear inhomogeneous and anisotropic elastic wave equation with locally distributed damping on bounded domain. One novelty compared to previous results, is to…