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Lower semicontinuity of multiple μ -quasiconvex integrals

Ilaria Fragalà — 2003

ESAIM: Control, Optimisation and Calculus of Variations

Lower semicontinuity results are obtained for multiple integrals of the kind n f ( x , μ u ) d μ , where μ is a given positive measure on n , and the vector-valued function u belongs to the Sobolev space H μ 1 , p ( n , m ) associated with μ . 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 μ . More precisely, for fully general μ , a notion of quasiconvexity for f along the tangent bundle to μ , turns out to be necessary for lower...

Fenomeni di concentrazione per energie di tipo Ginzburg-Landau

Ilaria Fragalà — 2005

Bollettino dell'Unione Matematica Italiana

Si discute il comportamento asintotico di energie di tipo Ginzburg-Landau, per funzioni da R n + k in R k , e sotto l'ipotesi che l'esponente di crescita p sia strettamente maggiore di k . In particolare, si illustra un risultato di compattezza e di Γ -convergenza, rispetto a una opportuna topologia sui Jacobiani, visti come correnti n -dimensionali. L'energia limite è definita sulla classe degli n -bordi interi M , e la sua densità dipende localmente dalla molteplicità di M tramite una famiglia di costanti di...

Lower semicontinuity of multiple µ-quasiconvex integrals

Ilaria Fragalà — 2010

ESAIM: Control, Optimisation and Calculus of Variations

Lower semicontinuity results are obtained for multiple integrals of the kind n f ( x , μ u ) d μ , where is a given positive measure on n , and the vector-valued function belongs to the Sobolev space H μ 1 , p ( n , m ) 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

Dorin BucurIlaria FragalàJimmy Lamboley — 2012

ESAIM: Control, Optimisation and Calculus of Variations

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

Dorin BucurIlaria FragalàJimmy Lamboley — 2012

ESAIM: Control, Optimisation and Calculus of Variations

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...

Optimal convex shapes for concave functionals

Dorin BucurIlaria FragalàJimmy Lamboley — 2012

ESAIM: Control, Optimisation and Calculus of Variations

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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