Duality theorems in some overdetermined boundary value problems. (English) Zbl 0698.35051

In 1971 J. Serrin [Arch. Ration. Mech. Anal. 43, 304-318 (1971; Zbl 0222.31007)] showed, that if u solves the n-dimensional problem \(\Delta u=-1\) in \(\Omega\), \(u=0\) on \(\partial \Omega\), \(\partial u/\partial \nu =-c\), c constant on \(\partial \Omega\), \(u\in C^ 2(\Omega)\cap C^ 1({\bar \Omega})\), then the domain \(\Omega\) must be an n-ball. This result was proved in an alternative manner by H. F. Weinberger [Arch. Ration. Mech. Anal. 43, 319-320 (1971; Zbl 0222.31008)]. Using an equivalent integral formulation (this is the background for the word “dual” in the title of the paper) and Green’s functions the authors prove such theorems for various classes of also higher order boundary value problems.
Reviewer: A.G pfert


35J40 Boundary value problems for higher-order elliptic equations
35J25 Boundary value problems for second-order elliptic equations
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