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On the stability of the Serrin problem. (English) Zbl 1173.35019

Summary: We investigate stability issues concerning the radial symmetry of solutions to Serrin’s overdetermined problems. In particular, we show that, if \(u\) is a solution to \(\Delta u=n\) in a smooth domain \(\Omega \subset \mathbb R^n, u=0\) on \(\partial \Omega \) and \(|Du|\) is “close” to 1 on \(\partial \Omega \), then \(\Omega \) is “close” to the union of a certain number of disjoint unitary balls.

MSC:

35B35 Stability in context of PDEs
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35N05 Overdetermined systems of PDEs with constant coefficients
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