## 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

### MSC:

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

### Citations:

Zbl 0222.31007; Zbl 0222.31008
Full Text:

### References:

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