Abstract.In this paper, we prove that the L p approximants naturally associated to a supremal functional Γ-converge to it. This yields a lower semicontinuity result for supremal functionals whose supremand satisfy weak coercivity assumptions as well as a generalized Jensen inequality. The existence of minimizers for variational problems involving such functionals (together with a Dirichlet condition) then easily follows. In the scalar case we show the existence of at least one absolute minimizer (i.e. local solution) among these minimizers. We provide two different proofs of this fact relying on different assumptions and techniques.Mathematics Subject Classification. 49J45, 49J99.

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 W 1,∞ (B, R d) for every open set B ⊆ Ω (where Ω is a fixed open set of R N), 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 L p-approximation of a supremal functional F via Γ-convergence when f is a non-negative and coercive Carathéodory function.

We study the weak* lower semicontinuity of functionals of the formThe notion of A-Young quasiconvexity, which is introduced here, provides a sufficient condition when f (x, •) 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.

In this paper we consider positively 1-homogeneous supremal functionals of the type F(u) := sup f (x, ∇u(x)). We prove that the relaxationF is a difference quotient, that iswhere d F is a geodesic distance associated to F. Moreover we prove that the closure of the class of 1-homogeneous supremal functionals with respect toconvergence is given exactly by the class of difference quotients associated to geodesic distances. This class strictly contains supremal functionals, as the class of geodesic distances strictly contains intrinsic distances.

We consider shape functionals of the form $$F_q(\Omega )=P(\Omega )T^q(\Omega )$$ F q ( Ω ) = P ( Ω ) T q ( Ω ) on the class of open sets of prescribed Lebesgue measure. Here $$q>0$$ q > 0 is fixed, $$P(\Omega )$$ P ( Ω ) denotes the perimeter of $$\Omega $$ Ω and $$T(\Omega )$$ T ( Ω ) is the torsional rigidity of $$\Omega $$ Ω . The minimization and maximization of $$F_q(\Omega )$$ F q ( Ω ) is considered on various classes of admissible domains $$\Omega $$ Ω : in the class $$\mathcal {A}_{all}$$ A all of all domains, in the class $$\mathcal {A}_{convex}$$ A convex of convex domains, and in the class $$\mathcal {A}_{thin}$$ A thin of thin domains.

We study the radial symmetry\ud of minimizers to the Schr\"odinger-Poisson-Slater\ud (S-P-S) energy:\ud $$\inf_{\substack{u\in H^1(\R^3)\\\ud \|u\|_{L^2(\R^3)}=\rho}}\ud \frac 12 \int_{\R^3} |\nabla u|^2 + \frac 14\ud \int_{\R^3} \int_{\R^3} \frac{|u(x)|^2|u(y)|^2}{|x-y|}dxdy\ud - \frac 1p \int_{\R^3} |u|^p dx$$\ud provided that $2

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Fixed a bounded open set of R N , we completely characterize the weak* lower semicontinuity of functionals of the form