On an Approximate Solution of the Dirichlet Problem for the Generalized Laplacian
Николай Тарханов · Mathematische Nachrichten · 1994
Abstract For an arbitrary differential operatorPof orderpon an open setX⊂Rn, the Laplacian is defined by Δ =P*P. It is an elliptic differential operator of order2pprovided the symbol mapping ofPis injective. LetObe a relatively compact domain inXwith smooth boundary, andBj(j= 0…,p— 1) be a Dirichlet system of orderp− 1 on ∂O. By {Cj} we denote the Dirichlet system on ∂Oadjoint for {Bj} with respect to the Green formula forP.The Hardy spaceH2(O) is defined to consist of all the solutionsfof Δf= 0 inOof finite order of growth near the boundary such that the weak boundary values of the expression {Bjf} and {Cj(Pf)} belong to the Lebesgue spaceL2(∂O). Then the Dirichlet problem consists of finding a solutionfϵH2(O) with prescribed data {Bjf} 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.