In the unit ball B(0, 1), let u and Ω (a domain in R N ) solve the following overdetermined problem:where χ Ω denotes the characteristic function, and the equation is satisfied in the sense of distributions.If the complement of Ω does not develop cusp singularities at the origin then we prove ∂Ω is analytic in some small neighborhood of the origin. The result can be modified to yield for more general divergence form operators. As an application of this, then, we obtain the regularity of the boundary of a domain without the Pompeiu property, provided its complement has no cusp singularities.

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We study the regularity of the free boundary in a Stefan-type problem \[ Δ u − ∂ t u = χ Ω in  D ⊂ R n × R , u = | ∇ u | = 0 on  D ∖ Ω \Delta u - \partial _t u = \chi _\Omega \quad \text {in $D\subset \mathbb {R}^n\times \mathbb {R}$}, \qquad u = |\nabla u| = 0 \quad \text {on $D\setminus \Omega $} \] with no sign assumptions on u u and the time derivative ∂ t u \partial _t u .

In this paper we prove the optimal C 1,1 (B 1 2 )-regularity for a general obstacle type problemunder the assumption that f * N is C 1,1 (B 1 ), where N is the Newtonian potential. This is the weakest assumption for which one can hope to get C 1,1regularity. As a by-product of the C 1,1 -regularity we are able to prove that, under a standard thickness assumption on the zero set close to a free boundary point x 0 , the free boundary is locally a C 1 -graph close to x 0 , provided f is Dini. This completely settles the question of the optimal regularity of this problem, that has been under much attention during the last two decades.

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