Higher dimensional problems with volume constraints -- existence and \(\Gamma \)-convergence (Q936144)

Summary: We study variational problems with volume constraints (also called level set constraints) of the form \[ \text{Minimize }E(u) := \int_{\Omega} f(u,\nabla u)dx, \] \[ |\{x\in \Omega : u(x) = a\}|= \alpha ,\qquad |\{x \in \Omega : u(x) = b\}| = \beta , \] on \(\Omega \subset \mathbb R^n\), where \(u \in H^1(\Omega )\) and \(\alpha + \beta < |\Omega |\). The volume constraints force a phase transition between the areas on which \(u = a\) and \(u = b\). We give some sharp existence results for the decoupled homogeneous and isotropic case \(f(u,\nabla u)=\psi(|\nabla u|) + \theta(u)\) under the assumption of \(p\)-polynomial growth and strict convexity of \(\psi\). We observe an interesting interaction between \(p\) and the regularity of the lower order term which is necessary to obtain existence and find a connection to the theory of dead cores. Moreover we obtain some existence results for the vector-valued analogue with constraints on \(|u|\). In the second part of this article we derive the \(\Gamma\)-limit of the functional \(E\) for a general class of functions \(f\) in the case of vanishing transition layers, i.e.\ when \(\alpha + \beta \to |\Omega |\). As limit functional we obtain a nonlocal free boundary problem.