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Boundary estimates for solutions of two-phase obstacle problems

Abstract

We establish the optimal regularity (of class W 2 ) of a solution to the two-phase obstacle problem

$$\Delta u = \lambda ^ + \chi _{\{ u > 0\} } - \lambda ^ - \chi _{\{ u < 0\} } in \Omega , \lambda ^ \pm \geqslant 0, \lambda ^ + + \lambda ^ - > 0,$$

with a nonhomogeneous Dirichlet condition in a bounded domain Ω ⊂ ℝn with smooth boundary ∂Ω. Bibliography: 10 titles.

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Translated from Problemy Matematicheskogo Analiza, No. 34, 2006, pp. 3–11.

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Apushkinskaya, D.E., Uraltseva, N.N. Boundary estimates for solutions of two-phase obstacle problems. J Math Sci 142, 1723–1732 (2007). https://doi.org/10.1007/s10958-007-0083-8

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Keywords

  • Free Boundary
  • Obstacle Problem
  • Boundary Estimate
  • Homogenous Dirichlet Boundary Condition
  • Optimal Regularity