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Subsolutions of Elliptic Operators in Divergence Form and Application to Two-Phase Free Boundary Problems

Abstract

Let be a divergence form operator with Lipschitz continuous coefficients in a domain, and let be a continuous weak solution of in. In this paper, we show that if satisfies a suitable differential inequality, then is a subsolution of away from its zero set. We apply this result to prove regularity of Lipschitz free boundaries in two-phase problems.

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Correspondence to Fausto Ferrari.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Ferrari, F., Salsa, S. Subsolutions of Elliptic Operators in Divergence Form and Application to Two-Phase Free Boundary Problems. Bound Value Probl 2007, 057049 (2006). https://doi.org/10.1155/2007/57049

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Keywords

  • Differential Equation
  • Partial Differential Equation
  • Ordinary Differential Equation
  • Weak Solution
  • Functional Equation