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On an Approximate Solution of the Dirichlet Problem for the Generalized Laplacian
Author(s) -
Tarkhanov Nikolai N.
Publication year - 1994
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19941690122
Subject(s) - mathematics , dirichlet problem , boundary (topology) , open set , dirichlet distribution , order (exchange) , operator (biology) , laplace operator , mathematical analysis , differential operator , combinatorics , space (punctuation) , boundary value problem , pure mathematics , biochemistry , chemistry , philosophy , finance , repressor , transcription factor , economics , gene , linguistics
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.