BV functions with respect to a measure and relaxation of metric integral functionals.
Lower semicontinuity of multiple -quasiconvex integrals
Lower semicontinuity results are obtained for multiple integrals of the kind , where is a given positive measure on , and the vector-valued function belongs to the Sobolev space associated with . The proofs are essentially based on blow-up techniques, and a significant role is played therein by the concepts of tangent space and of tangent measures to . More precisely, for fully general , a notion of quasiconvexity for along the tangent bundle to , turns out to be necessary for lower...
Fenomeni di concentrazione per energie di tipo Ginzburg-Landau
Si discute il comportamento asintotico di energie di tipo Ginzburg-Landau, per funzioni da in , e sotto l'ipotesi che l'esponente di crescita sia strettamente maggiore di . In particolare, si illustra un risultato di compattezza e di -convergenza, rispetto a una opportuna topologia sui Jacobiani, visti come correnti -dimensionali. L'energia limite è definita sulla classe degli -bordi interi , e la sua densità dipende localmente dalla molteplicità di tramite una famiglia di costanti di...
Lower semicontinuity of multiple µ-quasiconvex integrals
Lower semicontinuity results are obtained for multiple integrals of the kind , where is a given positive measure on , and the vector-valued function belongs to the Sobolev space associated with . The proofs are essentially based on blow-up techniques, and a significant role is played therein by the concepts of tangent space and of tangent measures to . More precisely, for fully general , a notion of quasiconvexity for along the tangent bundle to , turns out to be necessary for lower semicontinuity;...
Optimal convex shapes for concave functionals
Motivated by a long-standing conjecture of Pólya and Szegö about the Newtonian capacity of convex bodies, we discuss the role of concavity inequalities in shape optimization, and we provide several counterexamples to the Blaschke-concavity of variational functionals, including capacity. We then introduce a new algebraic structure on convex bodies, which allows to obtain global concavity and indecomposability results, and we discuss their application to isoperimetric-like inequalities. As a byproduct...
Optimal convex shapes for concave functionals
Motivated by a long-standing conjecture of Pólya and Szegö about the Newtonian capacity of convex bodies, we discuss the role of concavity inequalities in shape optimization, and we provide several counterexamples to the Blaschke-concavity of variational functionals, including capacity. We then introduce a new algebraic structure on convex bodies, which allows to obtain global concavity and indecomposability results, and we discuss their application...
Existence and nonexistence results for anisotropic quasilinear elliptic equations
Mean curvature of a measure and related variational problems
Optimal convex shapes for concave functionals
Motivated by a long-standing conjecture of Pólya and Szegö about the Newtonian capacity of convex bodies, we discuss the role of concavity inequalities in shape optimization, and we provide several counterexamples to the Blaschke-concavity of variational functionals, including capacity. We then introduce a new algebraic structure on convex bodies, which allows to obtain global concavity and indecomposability results, and we discuss their application...
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