Perturbations of the Dirichlet Problem and Error Bounds

2014/01/07 by Martin, Charles Z.
#31A25 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1401.1519

Abstract

The Dirichlet problem on a bounded planar domain is more readily understood and solved for the Laplace operator than it is for a Schrodinger operator. When the potential function is small, we might hope to approximate the solution to the Schrodinger equation with the solution to the Laplace equation. In this vein we develop a series expansion for the solution and give explicit bounds on the error terms when truncating the series. We also examine a handful of examples and derive similar results for the Green function and Dirichlet-to-Neumann map.

Related

• Dirichlet Problems in Perforated Domains
• Increasing stability for determining the potential in the Schrödinger equation with attenuation from the Dirichlet-to-Neumann map
• Boundary quasi-orthogonality and sharp inclusion bounds for large Dirichlet eigenvalues
• Singular asymptotic expansions for Dirichlet eigenvalues and eigenfunctions of the Laplacian on thin planar domains
• Note on the Dirichlet Approximation Theorem