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We study the higher regularity of solutions and free boundaries in the Alt-Phillips problem u=u^-1, with (0, 1). Our main results imply that, once free boundaries are C^1, , then they are C^. In addition u/d^2{2-} and u^2-{2} are C^ too. In order to achieve this, we need to establish fine regularity estimates for solutions of linear equations with boundary-singular Hardy potentials - v = v/d² in, where d is the distance to the boundary and 14. Interestingly, we need to include even the critical constant =14, which corresponds to =23.
Restrepo et al. (Mon,) studied this question.
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