We study variational problems with volume constraints (also called level set constraints) of the form
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 -convergence. Interfaces Free Bound. 10 (2008), no. 2, pp. 155–172DOI 10.4171/IFB/184