A posteriori verification of the positivity of solutions to elliptic boundary value problems

Kazuaki Tanaka () and Taisei Asai
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Kazuaki Tanaka: Waseda University
Taisei Asai: Waseda University

Partial Differential Equations and Applications, 2022, vol. 3, issue 1, 1-25

Abstract: Abstract The purpose of this paper is to develop a unified a posteriori method for verifying the positivity of solutions of elliptic boundary value problems by assuming neither $$H^2$$ H 2 -regularity nor $$ L^{\infty } $$ L ∞ -error estimation, but only $$ H^1_0 $$ H 0 1 -error estimation. In (J Comput Appl Math 370:112647, 2020), we proposed two approaches to verify the positivity of solutions of several semilinear elliptic boundary value problems. However, some cases require $$ L^{\infty } $$ L ∞ -error estimation and, therefore, narrow applicability. In this paper, we extend one of the approaches and combine it with a priori error bounds for Laplacian eigenvalues to obtain a unified method that has wide application. We describe how to evaluate some constants required to verify the positivity of desired solutions. We apply our method to several problems, including those to which the previous method is not applicable.

Keywords: Computer-assisted proofs; Elliptic boundary value problems; Error bounds; Numerical verification; Positive solutions; 35J25; 35J61; 65N15 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s42985-021-00143-2

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