A Variational Problem for Harmonic Functions in Ring-Shaped Domains with Partially Free Boundary
Andrea Colesanti · SIAM Journal on Mathematical Analysis · 1994
This paper considers two subsets $\Omega _0 $ and $\Omega $ of $\mathbb{R}^n $, $n = 2$ or $n = 3$, and two continuous real-valued functions $g_0 $ and g defined on $\partial \Omega $ and $\partial \Omega _0 $, respectively. The position of $\Omega $ is allowed to vary inside $\Omega _0 $ and the author looks for the minimum of the Dirichlet intergral of the function u, which is harmonic in $(\Omega _0 \backslash \Omega )$ and verifies the following boundary conditions: $u = g_0 $ on $\partial \Omega _0 $, $u = g$ on $\partial \Omega $ Under certain hypotheses on the regularity of $\partial \Omega _0 $ and $\partial \Omega $, and on $g_0 $ and g, an existence theorem is proved for the minimizing position of $\Omega $; it is shown through an example that the solution of the considered problem is not unique in general.