On the two-phase membrane problem with coefficients below the Lipschitz threshold |
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Authors: | Erik Lindgren Henrik Shahgholian Anders Edquist |
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Institution: | Department of Mathematics, Royal Institute of Technology, 100 44 Stockholm, Sweden |
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Abstract: | We study the regularity of the two-phase membrane problem, with coefficients below the Lipschitz threshold. For the Lipschitz coefficient case one can apply a monotonicity formula to prove the C1,1-regularity of the solution and that the free boundary is, near the so-called branching points, the union of two C1-graphs. In our case, the same monotonicity formula does not apply in the same way. In the absence of a monotonicity formula, we use a specific scaling argument combined with the classification of certain global solutions to obtain C1,1-estimates. Then we exploit some stability properties with respect to the coefficients to prove that the free boundary is the union of two Reifenberg vanishing sets near so-called branching points. |
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Keywords: | 35J70 35J60 35J85 |
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