Regularity of a free boundary with application to the Pompeiu problem

By Luis Caffarelli, Lavi Karp, H. Shahgholian. 2000

137 citations recorded by the source corpus.

In the unit ball B(0; 1), let u and › (a domain in N ) solve the following overdetermined problem: ¢u = ´› in B(0; 1); 02 @› ;u =jruj =0 inB(0; 1)n ›; where ´› denotes the characteristic function, and the equation is satisfled in the sense of distributions. If the complement of › does not develop cusp singularities at the origin then we prove @› is analytic in some small neighborhood of the origin. The result can be modifled 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.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/0a55d19568eece942458a84043134916016f4b0d

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Citations in the corpus

On the continuity of the time derivative of the solution to the parabolic obstacle problem with variable coefficients (2006, methodology): “This proof follows the rst part of the proof of Lemma 4.2 in [6] which was adapted from [ 8 ].”
A General Class of Free Boundary Problems for Fully Nonlinear Elliptic Equations (2012, background): “Under assumptions (H0)-(H2) above, strong Ln solutions are also viscosity solutions [5], and hence regularity results for parabolic fully nonlinear equations [12, 13] show that u ∈ W 2,p x (Qρ)∩ W 1,p t (Qρ) for all ρ ∈ (0, 1) and p < ∞.”
A variational problem for the spatial segregation of reaction-diffusion systems (2003, methodology): “A key tool in studying the regularity of both the function U and the free boundary is a suitable version of the celebrated monotonicity theorem, see [2, 5].”
C1, 1 Regularity in semilinear elliptic problems (2003, methodology): “In a recent work (see [2]) the author, L.”