On an Approximate Solution of the Dirichlet Problem for the Generalized Laplacian

✍ Scribed by Nikolai N. Tarkhanov

• Publisher: John Wiley and Sons
• Year: 2006
• Tongue: English
• Weight: 750 KB
• Volume: 169
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.19941690122

✦ Synopsis

Abstract

For an arbitrary differential operator P of order p on an open set X ⊂ R^n^, the Laplacian is defined by Δ = P*P. It is an elliptic differential operator of order 2p provided the symbol mapping of P is injective. Let O be a relatively compact domain in X with smooth boundary, and B~j~(j = 0…,p — 1) be a Dirichlet system of order p − 1 on ∂O. By {C~j~} we denote the Dirichlet system on ∂O adjoint for {B~j~} with respect to the Green formula for P. The Hardy space H^2^(O) is defined to consist of all the solutions f of Δ__f__ = 0 in O of finite order of growth near the boundary such that the weak boundary values of the expression {B~j~f} and {C~j~(Pf)} belong to the Lebesgue space L^2^(∂O). Then the Dirichlet problem consists of finding a solution f ϵ H^2^(O) with prescribed data {B~j~f} on ∂O. We develop the classical Fischer‐Riesz equations method to derive a solvability condition of the Dirichlet problem as well as an approximate formula for solutions.