Nicola Fusco

A regularity theorem for minimizers of quasiconvex integrals

SummaryWe prove C1,α partial regularity for minimizers of functionals with quasiconvex integrand f(x, u, Du) depending on vector-valued functions u. The integrand is required to be twice continuously differentiable in Du, and no assumption on the growth of the derivatives of f is made: a polynomial growth is required only on f itself.

Minimality via Second Variation for a Nonlocal Isoperimetric Problem

We discuss the local minimality of certain configurations for a nonlocal isoperimetric problem used to model microphase separation in diblock copolymer melts. We show that critical configurations with positive second variation are local minimizers of the nonlocal area functional and, in fact, satisfy a quantitative isoperimetric inequality with respect to sets that are L1-close. The link with local minimizers for the diffuse-interface Ohta-Kawasaki energy is also discussed. As a byproduct of the quantitative estimate, we get new results concerning periodic local minimizers of the area functional and a proof, via second variation, of the sharp quantitative isoperimetric inequality in the standard Euclidean case. As a further application, we address the global and local minimality of certain lamellar configurations.

The sharp Sobolev inequality in quantitative form

A quantitative version of the sharp Sobolev inequality in W (R), 1 < p < n, is established with a remainder term involving the distance from extremals.

Sobolev imbedding theorems in borderline cases

An imbedding theorem is given for functions whose gradient belongs to a class slightly larger than Ln(Ω), Ω ⊂ Rn.

C1, ? Partial regularity of functions minimising quasiconvex integrals

We prove C1, α almost everywhere regularity for minimisers of functionals of the form ∫F(x,u,Du), where F is uniformly strictly quasiconvex. This extends a recent result of Evans in which F is allowed only to depend on Du.

Partial regularity for minimisers of certain functionals having nonquadratic growth

SummaryWe prove partial regularity results for local minimisers of

A transmission problem in the calculus of variations

\begin{array}{*{20}c} {\int\limits_\Omega {(G^{\alpha \beta } (x,u)g_{ij} (x,u)D_\alpha u^i D_\beta u^j )^{{p \mathord{\left/ {\vphantom {p 2}} \right. \kern-\nulldelimiterspace} 2}} ,} } & {where p \geqslant 2.} \\ \end{array}

A strong form of the quantitative isoperimetric inequality

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