A numerical proof algorithm for the non-existence of solutions to elliptic boundary value problems

Abstract In 1988, M.T. Nakao developed an algorithm that was based on the fixed-point theorem on Sobolev spaces for the numerical proof of the existence of solutions to elliptic boundary value problems on a bounded domain with a Lipschitz boundary (Nakao (1988) [9] ). Thereafter, many researchers reported that the numerical existence proof algorithm to elliptic boundary value problems is actually significant and sufficiently useful. However, the numerical proof of the non-existence of solutions to the problem has hitherto not been considered due to several challenges. The purpose of this paper is to solve these difficulties and to propose an algorithm for the numerical proof of the non-existence of solutions in a closed ball B ¯ H 0 1 ( u ˆ , ρ ) = { u ∈ H 0 1 ( Ω ) | ‖ u − u ˆ ‖ H 0 1 ≤ ρ } to elliptic boundary value problems. We demonstrate some numerical examples that confirm the usefulness of the proposed algorithm.

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