On the radiality of constrained minimizers to the Schroedinger-Poisson-Slater energy

We study the radial symmetry of minimizers to the Schroedinger-Poisson-Slater (S-P-S) energy

Lower semicontinuity of supremal functional under differential constraint

We study the weak* lower semicontinuity of functionals of the form $$F(V)=supess_{x in Om} f(x,V (x)),,$$ where $Omsubset R^N$ is a bounded open set, $Vin L^{infty}(Omega;MM)cap Ker A$ and $A$ is a constant-rank partial differential operator. The notion of $A$-Young quasiconvexity, which is introduced here, provides a sufficient condition when $f(x,cdot)$ is only lower semicontinuous. We also establish necessary conditions for weak* lower semicontinuity. Finally, we discuss the divergence and curl-free cases and, as an application, we characterise the strength set in the context of electrical resistivity

Semicontinuity and supremal representation in the Calculus of Variations

We study the weak * lower semicontinuity properties of functionals of the form F (u) = ess sup f(x, Du(x)) x∈Ω where Ω is a bounded open set of R N and u ∈ W 1, ∞ (Ω). Without a continuity assumption on f(·, ξ) we show that the supremal functional F is weakly * lower semicontinuous if and only if it is a level convex functional (i.e. it has convex sub levels). In particular if F is weakly * lower semicontinuous, than it can be represented through a level convex function. Finally a counterexample shows that it is not possible to represent F through the level convex envelope of f

Semicontinuity and relaxation of LinftyL^{infty}-functionals

Fixed a bounded open set \Og of $\R^N$, we completely characterize the weak* lower semicontinuity of functionals of the form F(u,A)=\supess_{x \in A} f(x,u(x),Du (x)) defined for every $u \in W^{1,\infty}(\Omega)$ and for every open subset A\subset \Om. Without a continuity assumption on $f( \cdot,u,\xi)$ we show that the {\sl supremal} functional $F$ is weakly* lower semicontinuous if and only if it can be represented through a {\sl level convex} function. Then we study the properties of the lower semicontinuous envelope $\overline F$ of $F$. A complete relaxation theorem is shown in the case where $f$ is a continuous function. In the case $f=f(x,\xi)$ is only a Carath\'eodory function, we show that $\overline F$ coincides with the level convex envelope of $F$

On the lower semicontinuity and approximation of LinftyL^infty functionals

In this paper we show that if the supremal functional F(V,B) = ess sup x∈B f(x, DV (x)) is sequentially weak* lower semicontinuous on W1,∞(B, Rd) for every open set B ⊆ Ω (where Ω is a fixed open set of RN ), then f(x, ·) is rank-one level convex for a.e x ∈ Ω. Next, we provide an example of a weak Morrey quasiconvex function which is not strong Morrey quasiconvex. Finally we discuss the Lp-approximation of a supremal functional F via Γ-convergence when f is a non-negative and coercive Carath´eodory function