Variational problems with topological constraints
The authors are motivated by the goal of characterizing the topological characteristics of level sets of Sobolev functions that are preserved under weak limits. For example, they prove the following: If \(m>0\), \(\Omega\subset {\mathbb R}^2\) is open and bounded, \(g:{\mathbb R}\to{\mathbb R}\) is monotone increasing and left continuous with \(g=0\) on \((-\infty,0]\) and \(g>0\) on \((0,\infty)\), then \[ \begin{aligned} H^1_0(\Omega)&\cap C({\mathbb R}^2) \cap \Big\{ u : u\geq 0,\;\{u\leq t\} \text{\;and\;} \{u>t\} \text{\;are connected for\;}t\geq 0\Big\} \\ &\cap \Big\{u: \text{max} u =m,\;|\{u>t\}|\geq g(m-t) \text{\;for\;}t\geq 0\Big\} \end{aligned} \] is weakly closed in \(H_0^1(\Omega)\). This result can be applied to variational problems. In particular, the authors apply it to the problem of minimizing the Dirichlet integral in \[ H^1_0(\Omega)\cap C({\mathbb R}^2) \cap \Big\{ u : u\geq 0,\;\{u\leq t\} \text{\;and\;} \{u>t\} \text{\;are connected for\;}t\geq 0\Big\} \] among functions constrained to have a given decreasing rearrangement. Additional results and applications extending those described above are given. These are based on decompositions of Sobolev functions in the form \(\sum_{i=1}^{N}f_i(u_i)\) with the \(f_i\) Lipschitzian and the \(u_i\) chosen from spaces such as those above.
scientific article; zbMATH DE number 1455251