The Dirichlet Problem for the Perturbed Elliptic Equation

Ulyana Yarka, Solomiia Fedushko and Peter Veselý
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Ulyana Yarka: Social Communication and Information Activities Department, Lviv Polytechnic National University, 79000 Lviv, Ukraine
Solomiia Fedushko: Social Communication and Information Activities Department, Lviv Polytechnic National University, 79000 Lviv, Ukraine
Peter Veselý: Faculty of Management, Comenius University in Bratislava, 81499 Bratislava, Slovakia

Mathematics, 2020, vol. 8, issue 12, 1-14

Abstract: In this paper, the authors consider the construction of one class of perturbed problems to the Dirichlet problem for the elliptic equation. The operators of both problems are isospectral, which makes it possible to construct solutions to the perturbed problem using the Fourier method. This article focuses on the Dirichlet problem for the elliptic equation perturbed by the selected variable. We established the spectral properties of the perturbed operator. In this work, we found the eigenvalues and eigenfunctions of the perturbed task operator. Further, we proved the completeness, minimal spanning system, and Riesz basis system of eigenfunctions of the perturbed operator. Finally, we proved the theorem on the existence and uniqueness of the solution to the boundary value problem for a perturbed elliptic equation.

Keywords: differential equations; perturbed problem; isospectrality; Dirichlet problem; Riesz basis; spectral properties (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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