Boundary regularity and compactness for overdetermined problems
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 2 (2003) no. 4, p. 787-802

Let D be either the unit ball B 1 (0) or the half ball B 1 + (0), let f be a strictly positive and continuous function, and let u and ΩD solve the following overdetermined problem: Δu(x)=χ Ω (x)f(x)inD,0Ω,u=|u|=0inΩ c , where χ Ω denotes the characteristic function of Ω, Ω c denotes the set DΩ, and the equation is satisfied in the sense of distributions. When D=B 1 + (0), then we impose in addition that u(x)0on{(x ' ,x n )|x n =0}. We show that a fairly mild thickness assumption on Ω c will ensure enough compactness on u to give us “blow-up” limits, and we show how this compactness leads to regularity of Ω. In the case where f is positive and Lipschitz, the methods developed in Caffarelli, Karp, and Shahgholian (2000) lead to regularity of Ω under a weaker thickness assumption

@article{ASNSP_2003_5_2_4_787_0,
     author = {Blank, Ivan and Shahgholian, Henrik},
     title = {Boundary regularity and compactness for overdetermined problems},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola normale superiore},
     volume = {Ser. 5, 2},
     number = {4},
     year = {2003},
     pages = {787-802},
     zbl = {1170.35484},
     mrnumber = {2040643},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2003_5_2_4_787_0}
}
Blank, Ivan; Shahgholian, Henrik. Boundary regularity and compactness for overdetermined problems. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 2 (2003) no. 4, pp. 787-802. http://www.numdam.org/item/ASNSP_2003_5_2_4_787_0/

[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 1871348 | Zbl 1032.35170

[C1] L. A. Caffarelli, The regularity of free boundaries in higher dimensions, Acta Math. 139 (1977), 155-184. | MR 454350 | Zbl 0386.35046

[C2] L. A. Caffarelli, Compactness methods in free boundary problems, Comm. Partial Differential Equations 5 (1980), 427-448. | MR 567780 | Zbl 0437.35070

[C3] L. A. Caffarelli, The obstacle problem revisited, J. Fourier Anal. Appl. 4 (1998), 383-402. | MR 1658612 | Zbl 0928.49030

[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 1745013 | Zbl 0960.35112

[F] A. Friedman, “Variational Principles and Free Boundary Problems”, Wiley, 1982. | MR 679313 | Zbl 0564.49002

[GT] D. Gilbarg - N. S. Trudinger, “Elliptic Partial Differential Equations of Second Order”, 2nd ed., Springer-Verlag, 1983. | MR 737190 | Zbl 0562.35001

[I] V. Isakov, “Inverse Source Problems”, AMS Math. Surveys and Monographs 34, Providence, Rhode Island, 1990. | MR 1071181 | Zbl 0721.31002

[KN] D. Kinderlehrer - L. Nirenberg, Regularity in free boundary value problems, Ann. Scuola Norm. Sup. Pisa 4 (1977), 373-391. | Numdam | MR 440187 | Zbl 0352.35023

[KS] L. Karp - H. Shahgholian, On the optimal growth of functions with bounded Laplacian, Electron. J. Differential Equations 2000 (2000), 1-9. | MR 1735060 | Zbl 0937.35029

[KT] C.E. Kenig - T. Toro, Free boundary regularity for harmonic measures and Poisson Kernels, Ann. of Math. 150 (1999), 369-454. | MR 1726699 | Zbl 0946.31001

[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 114145 | Zbl 0099.08503

[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 516201 | Zbl 0354.35033

[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 1950478 | Zbl 1050.35157

[T] T. Toro, Doubling and flatness: geometry of measures, Notices Amer. Math. Soc. 44 (1997), 1087-1094. | MR 1470167 | Zbl 0909.31006