We consider the problem of minimizing the energy $$E\left(u\right):={\int }_{\Omega }{\left|\nabla u\left(x\right)\right|}^2\mathrm{d}x+{\int }_{S_u\cap \Omega }\left(1+\left|\right[u\left]\right(x\left)\right|\right)\mathrm{d}{H}^{N-1}\left(x\right)$$ among all functions ∈ ²(Ω) for which two level sets $\{u=l_i\}$ have prescribed Lebesgue measure ${\alpha }_i$. Subject to this volume constraint the existence of minimizers for (.) is proved and the asymptotic behaviour of the solutions is investigated.

We consider higher order functionals of the form $F\left[u\right]=\underset{\Omega }{\int }f\left(D^{m}u\right)\mathrm{d}x\text{for}u:{ℝ}^n\supset \Omega \to {ℝ}^N,$ where the integrand $f:{⨀}^m({ℝ}^n,{ℝ}^N)\to ℝ$, m ≥ 1 is strictly quasiconvex and satisfies a non-standard growth condition. More precisely we assume that f fulfills the (p, q)-growth condition $${\gamma \left|A\right|}^p\le f\left(A\right)\le {L(1+|A|}^q)\text{for}\text{all}A\in {⨀}^m({ℝ}^n,{ℝ}^N),$$ with

We consider higher order functionals of the form $F\left[u\right]=\underset{\Omega }{\int }f\left(D^{m}u\right)\mathrm{d}x\text{for}u:{ℝ}^n\supset \Omega \to {ℝ}^N,$ where the integrand $f:{⨀}^m({ℝ}^n,{ℝ}^N)\to ℝ$, m ≥ 1 is strictly quasiconvex and satisfies a non-standard growth condition. More precisely we assume that f fulfills the (p, q)-growth condition $${\gamma \left|A\right|}^p\le f\left(A\right)\le {L(1+|A|}^q)\text{for}\text{all}A\in {⨀}^m({ℝ}^n,{ℝ}^N),$$ with

We consider minimization problems of the form ${\min }_{u\in \varphi +W_0^{1,1}\left(\Omega \right)}{\int }_{\Omega }[f\left(Du\left(x\right)\right)-u\left(x\right)]\mathrm{d}x$ where $\Omega \subseteq {ℝ}^N$ is a bounded convex open set, and the Borel function $f:{ℝ}^N\to [0,+\infty ]$ is assumed to be neither convex nor coercive. Under suitable assumptions involving the geometry of and the zero level set of , we prove that the viscosity solution of a related Hamilton–Jacobi equation provides a minimizer for the integral functional.

Let be a real Hilbert space, ${\Phi }_1:H\to$ a convex function of class ${𝒞}^1$ that we wish to minimize under the convex constraint . A classical approach consists in following the trajectories of the generalized steepest descent system (  Brézis [CITE]) applied to the non-smooth function ${\Phi }_1+{\delta }_S$. Following Antipin [1], it is also possible to use a continuous gradient-projection system. We propose here an alternative method as follows: given a smooth convex function ${\Phi }_0:H\to$ whose critical points coincide with  and...