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Register a new userSerrins type overdetermined problems in convex cones
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We consider overdetermined problems of Serrins type in convex cones for (possibly) degenerate operators in the Euclidean space as well as for a suitable generalization to space forms. We prove rigidity results by showing that the existence of a solution implies that the domain is a spherical sector.
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In this paper we consider Serrins overdetermined problems in warped product manifolds and we prove Serrins type rigidity results by using the P-function approach introduced by Weinberger.
In this paper, we prove the existence of nontrivial unbounded domains $Omegasubsetmathbb{R}^{n+1},ngeq1$, bifurcating from the straight cylinder $Btimesmathbb{R}$ (where $B$ is the unit ball of $mathbb{R}^n$), such that the overdetermined elliptic problem begin{equation*} begin{cases} Delta u +f(u)=0 &mbox{in $Omega$, } u=0 &mbox{on $partialOmega$, } partial_{ u} u=mbox{constant} &mbox{on $partialOmega$, } end{cases} end{equation*} has a positive bounded solution. We will prove such result for a very general class of functions $f: [0, +infty) to mathbb{R}$. Roughly speaking, we only ask that the Dirichlet problem in $B$ admits a nondegenerate solution. The proof uses a local bifurcation argument.
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In this paper, we study a partially overdetermined mixed boundary value problem in a half ball. We prove that a domain in which this partially overdetermined problem admits a solution if and only if the domain is a spherical cap intersecting $ss^{n-1}$ orthogonally. As an application, we show a stationary point for a partially torsional rigidity under a volume constraint must be a spherical cap.
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