Boundary regularity and compactness for overdetermined problems

Blank, Ivan; Shahgholian, Henrik

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 2 (2003) no. 4, pp. 787-802.

Let $D$ be either the unit ball $B_1\left(0\right)$ or the half ball $B_1^{+}\left(0\right),$ let $f$ be a strictly positive and continuous function, and let $u$ and $\Omega \subset D$ solve the following overdetermined problem:

$$\Delta u\left(x\right)={\chi }_{{}_{\Omega }}\left(x\right)f\left(x\right)\text{in}D,0\in \partial \Omega ,u=\left|\nabla u\right|=0\text{in}{\Omega }^c,$$

where ${\chi }_{{}_{\Omega }}$ denotes the characteristic function of $\Omega ,$ ${\Omega }^c$ denotes the set $D\setminus \Omega ,$ and the equation is satisfied in the sense of distributions. When $D=B_1^{+}\left(0\right),$ then we impose in addition that

$$u\left(x\right)\equiv 0\text{on}\left\{(x^{\text{'}},x_n)\right|x_n=0\}.$$

We show that a fairly mild thickness assumption on ${\Omega }^c$ will ensure enough compactness on $u$ to give us “blow-up” limits, and we show how this compactness leads to regularity of $\partial \Omega .$ In the case where $f$ is positive and Lipschitz, the methods developed in Caffarelli, Karp, and Shahgholian (2000) lead to regularity of $\partial \Omega$ under a weaker thickness assumption

Classification : 35R35

[B] I. Blank, Sharp results for the regularity and stability of the free boundary in the obstacle problem, Indiana Univ. Math. J. 50 (2001), 1077-1112. | MR | Zbl

[C1] L. A. Caffarelli, The regularity of free boundaries in higher dimensions, Acta Math. 139 (1977), 155-184. | MR | Zbl

[C2] L. A. Caffarelli, Compactness methods in free boundary problems, Comm. Partial Differential Equations 5 (1980), 427-448. | MR | Zbl

[C3] L. A. Caffarelli, The obstacle problem revisited, J. Fourier Anal. Appl. 4 (1998), 383-402. | MR | Zbl

[CKS] L. A. Caffarelli - L. Karp - H. Shahgholian, Regularity of a free boundary with application to the Pompeiu problem, Ann. of Math. 151 (2000), 269-292. | MR | Zbl

[F] A. Friedman, “Variational Principles and Free Boundary Problems”, Wiley, 1982. | MR | Zbl

[GT] D. Gilbarg - N. S. Trudinger, “Elliptic Partial Differential Equations of Second Order”, 2nd ed., Springer-Verlag, 1983. | MR | Zbl

[I] V. Isakov, “Inverse Source Problems”, AMS Math. Surveys and Monographs 34, Providence, Rhode Island, 1990. | MR | Zbl

[KN] D. Kinderlehrer - L. Nirenberg, Regularity in free boundary value problems, Ann. Scuola Norm. Sup. Pisa 4 (1977), 373-391. | Numdam | MR | Zbl

[KS] L. Karp - H. Shahgholian, On the optimal growth of functions with bounded Laplacian, Electron. J. Differential Equations 2000 (2000), 1-9. | MR | Zbl

[KT] C.E. Kenig - T. Toro, Free boundary regularity for harmonic measures and Poisson Kernels, Ann. of Math. 150 (1999), 369-454. | MR | Zbl

[M] A. S. Margulis, Potential theory for $L^p$-densities and its applications to inverse problems of gravimetry, Theory and Practice of Gravitational and Magnetic Fields Interpretation in USSR, Naukova Dumka Press, Kiev, 1983, 188-197 (Russian).

[R] E. R. Reifenberg, Solution of the Plateau Problem for $m$-dimensional surfaces of varying topological type, Acta Math. 104 (1960), 1-92. | MR | Zbl

[Sc] D. G. Schaeffer, Some examples of singularities in a free boundary, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 4 (1977), 133-144. | Numdam | MR | Zbl

[St] V. N. Strakhov, The inverse logarithmic potential problem for contact surface, Physics of the Solid Earth 10 (1974), 104-114 [translated from Russian].

[SU] H. Shahgholian - N. Uraltseva, Regularity properties of a free boundary near contact points with the fixed boundary, Duke Univ. Math. J. 116 (2003), 1-34. | MR | Zbl

[T] T. Toro, Doubling and flatness: geometry of measures, Notices Amer. Math. Soc. 44 (1997), 1087-1094. | MR | Zbl