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.

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

Classification: 35R35
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     title = {Boundary regularity and compactness for overdetermined problems},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {787--802},
     publisher = {Scuola normale superiore},
     volume = {Ser. 5, 2},
     number = {4},
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     mrnumber = {2040643},
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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://archive.numdam.org/item/ASNSP_2003_5_2_4_787_0/

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