Sharp existence conditions and geometric inheritance for overdetermined free boundary problems of Laplacian and bi-Laplacian type

Abstract

This paper provides necessary and sufficient conditions for the existence of free boundaries in overdetermined problems for the Laplacian, and sufficient conditions for the bi-Laplacian, when the overdetermined boundary condition is non-constant. Using classical integral inequalities (Cauchy-Schwarz, Hölder, Hardy, eigenvalue bounds, Pohozaev and Reilly identities), we derive existence results for a broad class of free boundary problems arising in potential theory, plate theory, electromagnetism, and shape optimization. A regularity result for minimizers in the $C$-GNP class is established using the thickness function and the Wiener criterion, based on the geometric description of cusp points given in \cite{Barkatou2002}. We provide a new, self-contained geometric result: for almost every $t$, the level sets of the solution inherit the $C$-GNP property. This inheritance theorem justifies the variational framework and guarantees that the entire foliation generated by the state function remains within the admissible class. New results include refined estimates via interpolation inequalities, stability under perturbations, and connections with isoperimetric inequalities. The physical interpretation of the bi-Laplacian problem $\mathcal{B}(f,g)$ in the Kirchhoff-Love theory of thin plates is emphasized.