A free boundary problem for systems (The symmetric regime) - Layan El Hajj (23275180) (2025)
In this short note, we study a system of PDEs with free boundaries inside the unit ball. In particular, we prove that solutions to our problem exist and, furthermore, that any solution must be symmetric.The core difficulty arises from the non-variational nature of the system, coupled with a highly singular right-hand side term. These characteristics preclude the application of standard techniques. However, the inherent symmetry of the problem, derived from the geometry, allows us to circumvent these challenges. Although we treat the Laplacian case in more detail, the approach is easily generalized to more general operators, -component case, and -Laplacian operator as stated in the paper. Other Information Published in: Communications on Pure and Applied Analysis License: https://creativecommons.org/licenses/by/4.0/ See article on publisher's website: https://dx.doi.org/10.3934/cpaa.2025036