This analysis reveals geometric inequalities resulting from a serrin-type problem in riemannian manifolds, suggesting domains must be euclidean balls.
In this paper, we investigate an overdetermined boundary value problem of divergence type on bounded domains in Riemannian manifolds with non-negative Ricci curvature. Using integral identities and the P-function method, we derive geometric inequalities and rigidity results. Under natural conditions on the nonlinearity, we prove that equality implies the domain is isometric to a Euclidean ball, thereby extending classical symmetry results to the Riemannian setting.
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Batista et al. (2025) studied this question.
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