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On overdetermined boundary value problems for the biharmonic operator. (English) Zbl 0810.35020

Summary: We consider several overdetermined boundary value problems for the biharmonic operator and deduce that if a solution exists, then the domain must be a ball in \(N\)-space. When the auxiliary conditions are prescribed on the boundary, we reduce the problem to an equivalent second order overdetermined problem and use Rellich and Green identities and/or maximum principle arguments. When one or more auxiliary condition is specified on an interior surface, we use a particular solution of an associated problem to obtain the result.

MSC:

35J40 Boundary value problems for higher-order elliptic equations
31B30 Biharmonic and polyharmonic equations and functions in higher dimensions
58J70 Invariance and symmetry properties for PDEs on manifolds
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