Related papers: Notes on the Infinity-Laplace Equation
This paper deals with the \emph{integral} version of the Dirichlet homogeneous fractional Laplace equation. For this problem weighted and fractional Sobolev a priori estimates are provided in terms of the H\"older regularity of the data. By…
A new notion of displacement convexity on a matrix level is developed for density flows arising from mean-field games, compressible Euler equations, entropic interpolation, and semi-classical limits of non-linear Schr\"odinger equations.…
We study a counterpart of the classical Poisson integral for a family of weighted Laplace differential equations in Euclidean half space, solutions of which are known as generalized axially symmetric potentials. These potentials appear…
The aim of this work is to derive new explicit solutions to the $\infty$-Laplace equation, the fundamental PDE arising in Calculus of Variations in the space $L^\infty$. These solutions obey certain symmetry conditions and are derived in…
In this paper, we study the Dirichlet problem for Laplace's equation in an open disk. The uniqueness of solutions is ensured by the well-known weak maximum principle. We introduce a novel approach to demonstrate the existence of a solution…
The Laplace approximation is an old, but frequently used method to approximate integrals for Bayesian calculations. In this paper we develop an extension of the Laplace approximation, by applying it iteratively to the residual, i.e., the…
We give a unified statement and proof of a class of wellknown mean value inequalities for nonnegative functions with a nonlinear bound on the Laplacian. We generalize these to domains with boundary, requiring a (possibly nonlinear) bound on…
The time-fractional Fokker-Planck equation is a key model for characterizing anomalous diffusion, stochastic transport, and non-equilibrium statistical mechanics with applications in finance, chaotic dynamics, optical physics, and…
Laplace-type results characterize the limit of sequence of measures $(\pi_\varepsilon)_{\varepsilon >0}$ with density w.r.t the Lebesgue measure $(\mathrm{d} \pi_\varepsilon / \mathrm{d} \mathrm{Leb})(x) \propto \exp[-U(x)/\varepsilon]$…
We consider weighted graphs with an infinite set of vertices. We show that boundedness of all functions of finite energy can be seen as a notion of `relative compactness' for such graphs and study sufficient and necessary conditions for…
The purpose of this paper is to determine the main properties of Laplace contour integrals $$\Lambda(z)=\frac1{2\pi i}\int_\CC\phi_L(t)e^{-zt}\,dt,$$ that solve linear differential equations…
The narrow escape problem consists of deriving the asymptotic expansion of the solution of a drift-diffusion equation with the Dirichlet boundary condition on a small absorbing part of the boundary and the Neumann boundary condition on the…
We apply the method of inverse iteration to the Laplace eigenvalue problem with Robin and mixed Dirichlet-Neumann boundary conditions, respectively. For each problem, we prove convergence of the iterates to a non-trivial principal…
The purpose of this paper is to prove a uniform convergence rate of the solutions of the $p$-Laplace equation $\Delta_p u = 0$ with Dirichlet boundary conditions to the solution of the infinity-Laplace equation $\Delta_\infty u = 0$ as…
This paper investigates the Dirichlet problem for a non-divergence form elliptic operator $L$ in a bounded domain of $\mathbb{R}^d$. Under certain conditions on the coefficients of $L$, we first establish the existence of a unique Green's…
We study the Dirichlet problem of the following discrete infinity Laplace equation on a subgraph with finite width $$\Delta_{\infty} u(x) = \inf_{y \sim x}u(y)+\sup_{y \sim x}u(y)-2u(x) = f(x).$$ We say that a subgraph has finite width if…
The notion of Laplace invariants is transferred to the lattices and discrete equations which are difference analogs of hyperbolic PDE's with two independent variables. The sequence of Laplace invariants satisfy the discrete analog of…
In the present paper we consider the problem of Laplace deconvolution with noisy discrete non-equally spaced observations on a finite time interval. We propose a new method for Laplace deconvolution which is based on expansions of the…
We represent in this note the solutions of the electronic Schr\"odinger equation as traces of higher-dimensional functions. This allows to decouple the electron-electron interaction potential but comes at the price of a degenerate elliptic…
In this paper, under suitable settings, we can obtain the existence and uniqueness of solutions to a class of Hessian quotient equations with Dirichlet boundary condition in Lorentz-Minkowski space $\mathbb{R}^{n+1}_{1}$, which can be seen…