In this paper we provide a new (probabilistic) proof of a classical result in partial differential equations, viz. if φ is a tempered distribution, then the solution of the heat equation for the Laplacian, with initial condition φ , is given by the convolution of φ with the heat kernel (Gaussian density). Our results also extend the probabilistic representation of solutions of the heat equation...

The symmetry operators for the Laplacian in flat space were recently described and here we consider the same question for the square of the Laplacian. Again, there is a close connection with conformal geometry. There are three main steps in our construction. The first is to show that the symbol of a symmetry is constrained by an overdetermined partial differential equation. The second is to sho...

We study a Dirichlet optimal control problem for a quasi-linear monotone elliptic equation, the so-called weighted p-Laplace problem. The coefficient of the p-Laplacian, the weight u, we take as a control in BV (Ω) ∩ L∞(Ω). In this article, we use box-type constraints for the control such that there is a strictly positive lower and some upper bound. In order to handle the inherent degeneracy of...

This paper presents an image interpolation model with nonlocal p-Laplacian regularization. The nonlocal p-Laplacian regularization overcomes the drawback of the partial differential equation PDE proposed by Belahmidi and Guichard 2004 that image density diffuses in the directions pointed by local gradient. The grey values of images diffuse along image feature direction not gradient direction un...

This paper is a continuation of [MNS], where an analytic smoothing effect was proved for long-range type perturbations of the Laplacian H0 on R . In this paper, we consider short-range type perturbations H of the Laplacian on R, and we characterize the analytic wave front set of the solution to the Schrödinger equation: ef , in terms of that of the free solution: e0f , for t < 0 in the forward ...

Abstract. Two optimal monotone integral principles (equivalently for the Laplacian, two sharp iso-weighted-volume inequalities) are established through extending the first and second integral bounds of H. Weinberger for the Green functions (i.e., fundamental solutions) of uniformly elliptic equations in terms of the layer-cake formula, a one-dimensional monotone integral principle, and the isop...

We investigates the global regularity issue concerning a model equation proposed by Hou and Lei [3] to understand the stabilizing effects of the nonlinear terms in the 3D axisymmetric Navier-Stokes and Euler equations. Two major results are obtained. The first one establishes the global regularity of a generalized version of their model with a fractional Laplacian when the fractional power sati...

We study the global structure of the set of radial solutions of a nonlinear Dirichlet eigenvalue problem involving the p-Laplacian with p > 2, in the unit ball of RN , N > 1. We show that all non-trivial radial solutions lie on smooth curves of respectively positive and negative solutions, bifurcating from the first eigenvalue of a weighted p-linear problem. Our approach involves a local bifurc...

We study a linear fractional Fokker–Planck equation that models non-local diffusion in the presence of a potential field. The non-locality is due to the appearance of the 'fractional Laplacian' in the corresponding PDE, in place of the classical Laplacian which distinguishes the case of regular diffusion. We prove existence of weak solutions by combining a splitting technique together with a Wa...

Let G = (V, E) be a locally finite graph, Ω ⊂ V be a bounded open domain, ∆ be the usual graph Laplacian, and λ1(Ω) be the first eigenvalue of −∆ with respect to Dirichlet boundary condition. Using the mountain pass theorem of Ambrosette and Rabinowitz, We prove that if α < λ1(Ω), then for any p > 2, there exists a positive solution to  −∆u − αu = |u| p−2u in Ω◦, u = 0 on ∂Ω, where Ω◦ and...