Convergence of solutions of bilateral problems in variable domains and related questions. (English) Zbl 1448.49020

Kovalevsky, Alexander A.

Ural Math. J. 3, No. 2, 51-66 (2017).

Summary: We discuss some results on the convergence of minimizers and minimum values of integral and more general functionals on sets of functions defined by bilateral constraints in variable domains. We consider the case of regular constraints, i.e., constraints lying in the corresponding Sobolev space, and the case where the lower constraint is zero and the upper constraint is an arbitrary nonnegative function. The first case concerns a larger class of integrands and requires the positivity almost everywhere of the difference between the upper and lower constraints. In the second case, this requirement is absent. Moreover, in the latter case, the exhaustion condition of an \(n\)-dimensional domain by a sequence of \(n\)-dimensional domains plays an important role. We give a series of results involving this condition. In particular, using the exhaustion condition, we prove a certain convergence of sets of functions defined by bilateral (generally irregular) constraints in variable domains.

MSC:

49J45	Methods involving semicontinuity and convergence; relaxation
46E35	Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems

Keywords:

integral functional; bilateral problem; minimizer; minimum value; \( \Gamma \)-convergence of functionals; strong connectedness of spaces; \( \mathcal{H}\)-convergence of sets; exhaustion condition

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