Abstract
For a vectorial Bernoulli-type free boundary problem, with no sign assumption on the components, we prove that flatness of the free boundary implies <i>C</i><sup>1, <i>α</i></sup> regularity, as well-known in the scalar case <sup>[<span class="xref"><a href="#b1" ref-type="bibr">1</a></span>,<span class="xref"><a href="#b4" ref-type="bibr">4</a></span>]</sup>. While in <sup>[<span class="xref"><a href="#b15" ref-type="bibr">15</a></span>]</sup> 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 <sup>[<span class="xref"><a href="#b8" ref-type="bibr">8</a></span>]</sup>. 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 <sup>[<span class="xref"><a href="#b10" ref-type="bibr">10</a></span>,<span class="xref"><a href="#b11" ref-type="bibr">11</a></span>]</sup>.
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