The multi-layer free boundary problem for the \(p\)-Laplacian in convex domains
Let \(K_1, K_{m+2}\) (with \(m\) a positive integer) be given convex domains in \(\mathbb{R}^N (N\geq 2)\) such that \(K_1 \subset \subset K_{m+2}\). The problem here studied consists in looking for convex domains \(K_1 \subset \subset K_2 \subset \subset \dots \subset \subset K_{m+1} \subset \subset K_{m+2}\) and for functions \(u_i, i=1,\dots, m\) such that \[ \begin{alignedat}{2} \Delta_p u_i&=0 &&\quad\text{in } K_{i+1}=K_i,\\ u&=\lambda_i &&\quad\text{on }\partial K_i,\\ u&=\lambda_{i+1} &&\quad\text{on }\partial K_{i+1}, \end{alignedat} \] where \(\Delta_p\) is the \(p\)-Laplacian and \(\lambda_i\) are given in \([-1, 1]\) with the order \(\lambda_i > \lambda_{i+1}\), and the joining Bernoulli conditions \[ F_i (x,| \nabla U_i| , | \nabla u_{i+1}| )=0,\qquad i=1,\dots,m \] are satisfied on the free boundaries \(\partial K_{i+1}\). This model arises in multi-layer fluid dynamics problems, but it also includes the steady state Stefan problem. Most of the paper deals with the case \(m=1\) (two-phase problem), but the extension to \(m>1\) is illustrated in the last section. In the two-phase problem one can take \(\lambda =0\) and give for instance Dirichlet boundary data \(u_1=1\) on \(\partial K_1, u_2=-1\) on \(\partial K_3\). The interface functions \(F_i\) are required to satisfy several conditions, which are illustrated in detail, that for instance include the case \(| \nabla u_i| ^\alpha -| \nabla u_{i+1}| ^\alpha =a_i (x)^\alpha\), with \(\alpha \geq 1, a_i >0, 1/a_i\) concave. Existence of classical solutions is shown, using a sub/super-solutions technique. The question of strict and uniform separation of interfaces is also dealt with.
scientific article; zbMATH DE number 2067195