Higher dimensional problems with volume constraints—Existence and <em>Γ</em>-convergence

  • Marc Oliver Rieger

    University of Zürich, Switzerland


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 ΩRn\Omega\subset\R^n, where uH1(\G)u\in H^1(\G) and α+β<\G\alpha+\beta<|\G|. The volume constraints force a phase transition between the areas on which u=0u=0 and u=1u=1.\\ We give some sharp existence results for the decoupled homogenous and isotropic case f(u,u)=ψ(u)+θ(u)f(u,\nabla u)=\psi(|\nabla u|)+\theta(u) under the assumption of pp-polynomial growth and strict convexity of ψ\psi. We observe an interesting interaction between pp 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|u|.\\ In the second part of this article we derive the Γ\Gamma-limit of the functional EE for a general class of functions ff in the case of vanishing transition layers, i.e.\ when α+β\G\alpha+\beta\to|\G|. 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