Variational problems with several volume constraints on the level sets

Abstract.

We consider functionals of the kind $I(u)= \int F(x,u,\nabla u)$ on $W^{1,p}$, and we study the problem $\min_{\mathcal{K}} I$, where $\mathcal{K}\subset W^{1,p}$ consists of those functions u whose level sets satisfy certain volume constraints $\Big \vert{\{u=l_i\}\Big \vert=\alpha_i>0$, where $\Big \vert \cdot\Big \vert$ denotes Lebesgue measure, and $\{l_i\}$, $\{\alpha_i\}$ are given numbers. Examples show that this problem may have no solution, even for simple smooth F. As a consequence, we relax the constraint $u\in\mathcal{K}$ to $u\in\mathcal{K}_+$, i.e. $\Big \vert \{u=l_i\}\Big \vert\geq \alpha_i$, and we show that the minimizers over $\mathcal{K}_+$ exist and are Hölder continuous. Then we prove several existence theorems for the original problem, showing that, under suitable assumptions on the integrand function F, every minimizer over $\mathcal{K}_+$ actually belongs to $\mathcal{K}$.