Calculus of Variations and Geometric Measure Theory

S. J. N. Mosconi - P. Tilli

Variational problems with several volume constraints on the level sets

created on 04 Dec 2000
modified on 17 Jun 2002


Published Paper

Inserted: 4 dec 2000
Last Updated: 17 jun 2002

Journal: Calculus of Variations
Year: 2001


We consider functionals of the kind $I(u)= \int F(x,u,\nabla u)$ on $W^{1,p}$, and we study the problem $\min_{K} I$, where $K\subset W^{1,p}$ consists of those functions $u$ whose level sets satisfy certain volume constraints $meas(\{u=l_i\})=\alpha_i>0$, where $meas(\cdot)$ 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 $u\in K$ to $u\in K_+$, i.e. $meas(\{u=l_i\})>= \alpha_i$, and we show that the minimizers over $K_+$ 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 $K_+$ actually belongs to $K$.

Keywords: constrained problem, variational problem, level set