2020/05/03 by Lorenzo Cavallina, Cavallina, Lorenzo · 1 citation
Computer Science · Mathematics · #34K18 #35J15 #35N25 #35Q93 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Contact Mechanics and Variational Inequalities #FOS: Mathematics #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2005.01012
openalex publication_date 2020/05/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider an overdetermined problem for a two phase elliptic operator in divergence form with piecewise constant coefficients. We look for domains such that the solution u of a Dirichlet boundary value problem also satisfies the additional property that its normal derivative ∂n u is a multiple of the radius of curvature at each point on the boundary. When the coefficients satisfy some "non-criticality" condition, we construct nontrivial solutions to this overdetermined problem employing a perturbation argument relying on shape derivatives and the implicit function theorem. Moreover, in the critical case, we employ the use of the Crandall-Rabinowitz theorem to show the existence of a branch of symmetry breaking solutions bifurcating from trivial ones. Finally, some remarks on the one phase case and a similar overdetermined problem of Serrin type are given.