*Published Paper*

**Inserted:** 4 dec 2000

**Last Updated:** 17 jun 2002

**Journal:** Calculus of Variations

**Year:** 2001

**Abstract:**

We consider functionals of the kind $I(u)= \int F(x,u,\nabla u)$ on $W^{1,p}$, and we study the problem $\min_{K} I$, 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 HÃ¶lder 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