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Improvement of flatness for vector valued free boundary problems

1 Department of Mathematics, Barnard College, Columbia University, New York, NY 10027, USA
2 Dipartimento di Matematica, Alma Mater Studiorum Universita di Bologna, Piazzà di Porta San Donato 5, 40126 Bologna, Italy

This contribution is part of the Special Issue: Contemporary PDEs between theory and modeling—Dedicated to Sandro Salsa, on the occasion of his 70th birthday
Guest Editor: Gianmaria Verzini
Link: https://www.aimspress.com/newsinfo/1429.html

Special Issues: Contemporary PDEs between theory and modeling—Dedicated to Sandro Salsa, on the occasion of his 70th birthday

For a vectorial Bernoulli-type free boundary problem, with no sign assumption on the components, we prove that flatness of the free boundary implies C1,α regularity, as well-known in the scalar case [1, 4]. While in [15] the same result is obtained for minimizing solutions by using a reduction to the scalar problem, and the NTA structure of the regular part of the free boundary, our result uses directly a viscosity approach on the vectorial problem, in the spirit of [8]. We plan to use the approach developed here in vectorial free boundary problems involving a fractional Laplacian, as those treated in the scalar case in [10, 11].
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