相关论文: Widder's representation theorem for symmetric loca…
We study the $n$-dimensional Schr\"odinger equation with singular potential $V_\lambda(x)=\lambda |x|^{-2}$. Its solution space is studied as a global representation of $\widetilde{SL(2,\R)}\times O(n)$. A special subspace of solutions for…
In this paper we first prove a general representation theorem for generators of backward stochastic differential equations (BSDEs for short) by utilizing a localization method involved with stopping time tools and approximation techniques,…
In this paper we study the linear Weingarten equation defined by the fully non-linear PDE $$a\, \mbox{div}\frac{Du}{\sqrt{1+|Du|^2}}+b\, \frac{\mbox{det}D^2u}{(1+|Du|^2)^2}=\phi\left(\frac{1}{\sqrt{1+|Du|^2}}\right)$$ in a domain…
This paper is devoted to the homogenization of the heat conduction equation, with a homogeneous Dirichlet boundary condition, having a periodically oscillating thermal conductivity and a vanishing volumetric heat capacity. A homogenization…
In this thesis we describe a type of metric space called an Euclidean polyhedral complex. We define a Dirichlet form on it; this is used to give a corresponding heat kernel. We provide a uniform small time Poincare inequality for complexes…
In this paper, we prove the diffusion phenomenon for the linear wave equation with space-dependent damping. We prove that the asymptotic profile of the solution is given by a solution of the corresponding heat equation in the $L^2$-sense.
In this paper authors consider representations of graphs in Hilbert spaces applying a restriction of local scalarity on them. It enables to obtain a theory, similar to the classical theory of representations of graphs in vector spaces. In…
A sharp double-sided Harnack bound is derived for positive solutions of a fractional order heat equation.
We consider a nonlinear parametric Dirichlet problem driven by the $p$-Laplace differential operator and a reaction which has the competing effects of a parametric singular term and of a Carath\'eodory perturbation which is ($p-1$)-linear…
This paper concerns with the heat equation in the half-space $\mathbb{R}_{+}^{n}$ with nonlinearity and singular potential on the boundary $\partial\mathbb{R}_{+}^{n}$. We develop a well-posedness theory (without using Kato and Hardy…
In this article, we prove that the least energy nodal solutions to Lane-Emden equation $-{\Delta}u = |u|^{p-2}u$ with zero Dirichlet boundary conditions on a square are odd with respect to one diagonal and even with respect to the other one…
The weak well-posedness results of the strongly damped linear wave equation and of the non linear Westervelt equation with homogeneous Dirichlet boundary conditions are proved on arbitrary three dimensional domains or any two dimensional…
The aim of this work is to study comparability of nonlocal Dirichlet forms. We provide sufficient conditions on the kernel for local and global comparability. As an application we prove a-priori estimates in H\"{o}lder spaces for solutions…
We give a new proof for the local existence of a smooth isometric embedding of a smooth $3$-dimensional Riemannian manifold with nonzero Riemannian curvature tensor into $6$-dimensional Euclidean space. Our proof avoids the sophisticated…
Liouville theorems for scaling invariant nonlinear parabolic equations and systems (saying that the equation or system does not possess positive entire solutions) guarantee optimal universal estimates of solutions of related initial and…
We study qualitative and quantitative properties of local weak solutions of the fast $p$-Laplacian equation, $\partial_t u=\Delta_{p}u$, with $1<p<2$. Our main results are quantitative positivity and boundedness estimates for locally…
We obtain Lipschitz regularity results for a fairly general class of nonlinear first-order PDEs. These equations arise from the inner variation of certain energy integrals. Even in the simplest model case of the Dirichlet energy the…
Stationary solutions asymptoting to nonlinear plane waves of the nonlinear Schr\"odinger equation with a PT-symmetric, complex linear potential are characterized. The potential includes both a spatially varying gain-loss profile and a…
In this paper, we introduce the unconditional uniqueness of solutions in Herz spaces for the Hardy--H\'enon parabolic equation, which is a semilinear heat equation with a power-type weight in the nonlinear term $|x|^\gamma|u|^{\alpha-1}u$.…
We focus on three different convexity principles for local and nonlocal variational integrals. We prove various generalizations of them, as well as their equivalences. Some applications to nonlinear eigenvalue problems and Hardy-type…