The Laplace Equation in Space

A space mapping approach for the p-Laplace equation

The Laplace Equation

Definition 1. Among the most important and ubiquitous of all partial differential equations is Laplace’s Equation: ∆u = 0, where the Laplacian operator ∆ acts on the function u : U → R (U is open in R) by taking the sum of the unmixed partial derivatives. For example: n = 1: ∆u = ∂ 2u ∂x2 = u = 0 In this simple case, the solution u = ax + b is found by integrating twice. n = 2: ∆u = ∂ 2u ∂x1 + ...

Inhomogeneous infinity Laplace equation ✩

We present the theory of the viscosity solutions of the inhomogeneous infinity Laplace equation ∂xi u∂xj u∂ 2 xixj u = f in domains in Rn. We show existence and uniqueness of a viscosity solution of the Dirichlet problem under the intrinsic condition f does not change its sign. We also discover a characteristic property, which we call the comparison with standard functions property, of the visc...

Shape sensitivity analysis of the Neumann problem of the Laplace equation in the Half-Space

Shape optimization for the Neumann problem of the Laplace equation is important for application and also from the numerical point of view. Mathematical analysis of such problem in the half space is not available. In this paper we prove the shape differentiability of solutions in appropriate weighted Sobolev spaces which describe the behavior of solutions at infinity. We will consider two differ...

Shape differentiability of the Neumann problem of the Laplace equation in the half - space

Shape optimization for the Neumann problem of the Laplace equation is important for application and also from the numerical point of view. Mathematical analysis of such problem in the half space is not available. In this paper we prove the shape differentiability of solutions in appropriate weighted Sobolev spaces which describe the behavior of solutions at infinity. We will consider two differ...