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Numerical existence proofs and explicit bounds for solutions of nonlinear elliptic boundary value problems

Numerische Existenzbeweise und explizite Schranken für Lösungen nichtlinearer elliptischer Randwertprobleme

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Abstract

For elliptic boundary value problems of the form −ΔU+F(x, U, U x )=0 on Ω,B[U]=0 on ϖΩ, with a nonlinearityF growing at most quadratically with respect to the gradientU x and with a mixed-type linear boundary opeatorB, a numerical method is presented which can be used to prove the existence of a solution within a “close”H 1,4(Ω)-neighborhood of some approximate solution ω∈H 2(Ω) satisfying the boundary condition, provided that the defect-norm ∥−Δω +F(·, ω, ω x )∥2 is sufficiently small and, moreover, the linearization of the given problem at ω leads to an invertible operatorL. The main tools are explicit Sobolev imbeddings and eigenvalue bounds forL or forL*L. All kinds of monotonicity or inverse-positivity assumptions are avoided.

Zusammenfassung

Gegeben sei ein elliptisches Randwertproblem der Form −ΔU+F(x, U, U x )=0 auf Ω,B[U]=0 auf ϖΩ, mit einer NichtlinearitätF, die einer quadratischen Wachstumsbedingung bezüglich des GradientenU x genügt, und mit einem linearen RandoperatorB von gemischtem Typ. Es wird eine numerische Methode vorgestellt, mit deren Hilfe sich die Existenz einer Lösung innerhalb einer “kleinen”H 1,4(Ω)-Umgebung einer Näherungslösung ω∈H 2(Ω), die die Randbedingung erfüllt, nachweisen läßt, sofern die Defektnorm ∥−Δω +F(·, ω, ω x )∥2 hinreichend klein ist und ferner die Linearisierung des gegebenen Problems in ω auf einen invertierbaren OperatorL führt. Die wesentlichen Hilfsmittel sind explizite Sobolevsche Einbettungen und Eigenwertschranken fürL oderL*L. Jegliche Monotonie- und Inverspositivitätsbedingungen werden vermieden.

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Plum, M. Numerical existence proofs and explicit bounds for solutions of nonlinear elliptic boundary value problems. Computing 49, 25–44 (1992). https://doi.org/10.1007/BF02238648

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