Higher dimensional problems with volume constraints—Existence and <em>Γ</em>-convergence
Marc Oliver Rieger
University of Zürich, Switzerland

Abstract
We study variational problems with volume constraints (also called level set constraints) of the form \begin{eqnarray*} \mbox{Minimize }E(u):=\int_\G f(u,\nabla u)\,dx,\nonumber\\ |\{x\in\Omega,\;u(x)=a\}|=\alpha,\quad |\{x\in\Omega,\;u(x)=b\}|=\beta, \end{eqnarray*} on , where and . The volume constraints force a phase transition between the areas on which and .\\ We give some sharp existence results for the decoupled homogenous and isotropic case under the assumption of -polynomial growth and strict convexity of . We observe an interesting interaction between 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 .\\ In the second part of this article we derive the -limit of the functional for a general class of functions in the case of vanishing transition layers, i.e.\ when . As limit functional we obtain a nonlocal free boundary problem.
Cite this article
Marc Oliver Rieger, Higher dimensional problems with volume constraints—Existence and <em>Γ</em>-convergence. Interfaces Free Bound. 10 (2008), no. 2, pp. 155–172
DOI 10.4171/IFB/184