Improvements on overdetermined problems associated to the $ p $-Laplacian

<abstract><p>This work presents some improvements on related papers that investigate certain overdetermined problems associated to elliptic quasilinear operators. Our model operator is the $ p $-Laplacian. Under suitable structural conditions, and assuming that a solution exists, we show that the domain of the problem is a ball centered at the origin. Furthermore we discuss a convenient form of comparison principle for this kind of problems.</p></abstract>

Shape Perturbation of Grushin Eigenvalues

We consider the spectral problem for the Grushin Laplacian subject to homogeneous Dirichlet boundary conditions on a bounded open subset of $${\mathbb {R}}^N$$ R N . We prove that the symmetric functions of the eigenvalues depend real analytically upon domain perturbations and we prove an Hadamard-type formula for their shape differential. In the case of perturbations depending on a single scalar parameter, we prove a Rellich–Nagy-type theorem which describes the bifurcation phenomenon of multiple eigenvalues. As corollaries, we characterize the critical shapes under isovolumetric and isoperimetric perturbations in terms of overdetermined problems and we deduce a new proof of the Rellich–Pohozaev identity for the Grushin eigenvalues.

Serrin’s type problems in warped product manifolds

In this paper, we consider Serrin’s overdetermined problems in warped product manifolds and we prove Serrin’s type rigidity results by using the [Formula: see text]-function approach introduced by Weinberger.

On the stability of k-Hessian overdetermined and partially overdetermined problems in planar domain

In this paper, we study the stability of k-Hessian overdetermined problems under small perturbations in {R^{2}}. Additionally, we give a proof for k-Hessian partially overdetermined problems and the related stability problems in {R^{2}}. The proof mainly replies to choosing a suitable auxiliary f.