Uniqueness for an overdetermined

boundary value problem for the p-Laplacian

Authors:
Farid Bahrami and Henrik Shahgholian

Journal:
Proc. Amer. Math. Soc. **126** (1998), 745-750

MSC (1991):
Primary 31B20, 35J05, 35R35

MathSciNet review:
1422844

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Abstract | References | Similar Articles | Additional Information

Abstract: For set , and let be a measure with compact support. Suppose, for , there are functions and (bounded) domains , both containing the support of with the property that in (weakly) and in the complement of . If in addition is convex, then and .

**[gu]**Björn Gustafsson,*On quadrature domains and an inverse problem in potential theory*, J. Analyse Math.**55**(1990), 172–216. MR**1094715**, 10.1007/BF02789201**[g-s]**Björn Gustafsson and Henrik Shahgholian,*Existence and geometric properties of solutions of a free boundary problem in potential theory*, J. Reine Angew. Math.**473**(1996), 137–179. MR**1390686****[h-k-m]**Juha Heinonen, Tero Kilpeläinen, and Olli Martio,*Nonlinear potential theory of degenerate elliptic equations*, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1993. Oxford Science Publications. MR**1207810****[i]**Victor Isakov,*Inverse source problems*, Mathematical Surveys and Monographs, vol. 34, American Mathematical Society, Providence, RI, 1990. MR**1071181****[k-s]**L. Karp, H. Shahgholian, Regularity of free boundaries, To appear.**[l]**John L. Lewis,*Regularity of the derivatives of solutions to certain degenerate elliptic equations*, Indiana Univ. Math. J.**32**(1983), no. 6, 849–858. MR**721568**, 10.1512/iumj.1983.32.32058**[n]**P. S. Novikov, Sur le probléme inverse du potentiel, Dokl. Akad. Nauk SSSR, vol 18, 1938, 165-168.**[sa]**Makoto Sakai,*Quadrature domains*, Lecture Notes in Mathematics, vol. 934, Springer-Verlag, Berlin-New York, 1982. MR**663007****[shah]**Henrik Shahgholian,*Convexity and uniqueness in an inverse problem of potential theory*, Proc. Amer. Math. Soc.**116**(1992), no. 4, 1097–1100. MR**1137234**, 10.1090/S0002-9939-1992-1137234-2**[sh]**Harold S. Shapiro,*The Schwarz function and its generalization to higher dimensions*, University of Arkansas Lecture Notes in the Mathematical Sciences, 9, John Wiley & Sons, Inc., New York, 1992. A Wiley-Interscience Publication. MR**1160990****[t]**Peter Tolksdorf,*On the Dirichlet problem for quasilinear equations in domains with conical boundary points*, Comm. Partial Differential Equations**8**(1983), no. 7, 773–817. MR**700735**, 10.1080/03605308308820285**[z]**Lawrence Zalcman,*Some inverse problems of potential theory*, Integral geometry (Brunswick, Maine, 1984) Contemp. Math., vol. 63, Amer. Math. Soc., Providence, RI, 1987, pp. 337–350. MR**876329**, 10.1090/conm/063/876329

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Additional Information

**Farid Bahrami**

Affiliation:
Department of Mathematics, University of Tehran, P.O. Box 13145-1873, Tehran, Iran

**Henrik Shahgholian**

Affiliation:
Department of Mathematics, The Royal Institute of Technology, 100 44 Stockholm, Sweden

Email:
henriks@math.kth.se

DOI:
https://doi.org/10.1090/S0002-9939-98-04087-8

Keywords:
Inverse domain problem,
p-Laplacian,
uniqueness

Received by editor(s):
April 3, 1996

Received by editor(s) in revised form:
August 28, 1996

Communicated by:
J. Marshall Ash

Article copyright:
© Copyright 1998
American Mathematical Society