Тип публикации: статья из журнала
Год издания: 2019
Ключевые слова: elliptic operators, Green's formulas,weighted Sobolev spaces, ill-posed problems, the Cauchy problem
Аннотация: We consider the ill-posed Cauchy problem in a bounded domain D of R-n for an elliptic differential operator A (x, δ) A (x,) with data on a relatively open subset S of the boundary â â¡ δ D. We do it in weighted Sobolev spaces H s, γ (Δy) {H-s,γ}(D) containing the elements with prescribed smoothness s {s\in\mathbb{N}} and growth neaПоказать полностьюr S S in δ D, controlled by a real number γ. More precisely, using proper (left) fundamental solutions of ¢ (x, -) A, we obtain a Green-type integral formula for functions from H s, γ (δY) {H-s,\gamma D. Then a Neumann-type series, constructed with the use of iterations of the (bounded) integral operators applied to the data, gives a solution to the Cauchy problem in H s, γ D (δY) {H-s,γ D whenever this solution exists.
Журнал: Journal of Inverse and Ill-Posed Problems
ISSN журнала: 09280219
Издатель: Brill Academic Publishers