Emilio Acerbi

Gradient estimates for a class of parabolic systems

We establish local Calderon-Zygmund-type estimates for a class of parabolic problems whose model is the nonhomogeneous, degenerate/singular parabolic p-Laplacian system ut − div(|Du|p−2Du) = div(|F |p−2F ), proving that F ∈ Lqloc =⇒ Du ∈ Lqloc, ∀ q ≥ p. We also treat systems with discontinuous coefficients of vanishing mean oscillation (VMO) type.

Gradient estimates for the pðxÞ-Laplacean system

Abstract We prove Calderón and Zygmund type estimates for a class of elliptic problems whose model is the non-homogeneous p (x )-Laplacean system Under optimal continuity assumptions on the function p (x ) > 1 we prove that Our estimates are motivated by recent developments in non-Newtonian fluidmechanics and elliptic problems with non-standard growth conditions, and are the natural, ‘‘non-linear’’ counterpart of those obtained by Diening and Růžička [L. Diening and M. Růžička, Calderón-Zygmund operators on generalized Lebesgue spaces L p (‧) and problems related to fluid dynamics, J. reine angew. Math. 563 (2003), 197–220] in the linear case.

A variational definition of the strain energy for an elastic string

Using the variational point of view, the constitutive equations of an elastic one-dimensional string are deduced from the stress-strain relations of nonlinear three-dimensional elasticity, by passing to the limit when the other dimensions go to zero. The assumptions made on the three-dimensional model are not very restrictive.

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.

New lower semicontinuity results for polyconvex integrals

SummaryWe study integral functionals of the formF(u, Ω)=∫Ωf(▽u)dx, defined foru ∈ C1(Ω;Rk), Ω⊑Rn. The functionf is assumed to be polyconvex and to satisfy the inequalityf(A) ≥c0¦ℳ(A)¦ for a suitable constant c0 > 0, where ℳ(A) is then-vector whose components are the determinants of all minors of thek×n matrixA. We prove thatF is lower semicontinuous onC1(Ω;Rk) with respect to the strong topology ofL1(Ω;Rk). Then we consider the relaxed functional ℱ, defined as the greatest lower semicontinuous functional onL1(Ω;Rk) which is less than or equal toF on C1(Ω;Rk). For everyu ∈ BV(Ω;Rk) we prove that ℱ (u,Ω) ≥ ∫Ωf(▽u)dx+c0¦Dsu¦(Ω), whereDu=▽u dx+Dsu is the Lebesgue decomposition of the Radon measureDu. Moreover, under suitable growth conditions onf, we show that ℱ (u,Ω)=∫Ωf(▽u)dx for everyu ∈ W1,p(Ω;Rk), withp ≥ min{n,k}. We prove also that the functional ℱ (u, Ω) can not be represented by an inte- gral for an arbitrary functionu ∈ BVloc(Rn;Rk). In fact, two examples show that, in general, the set functionΩ → ℱ (u, Ω) is not subadditive whenu ∈ BVloc(Rn;Rk), even ifu ∈ Wloc1,p(Rn;Rk) for everyp < min{n,k}. Finally, we examine in detail the properties of the functionsu ∈ BV(Ω;Rk) such that ℱ (u, Ω)=∫Ωf(▽u)dx, particularly in the model casef(A)=¦ℳ(A)¦.

Reinforcement problems in the calculus of variations(

Abstract We study the torsion of an elastic bar surrounded by an increasingly thin layer made of increasingly hard material. In the model problem the ellipticity constant tends to zero in the outer layer; the equations considered may be fully nonlinear. Depending on the link between thickness and hardness we obtain three different expressions of the limit problem.

A transmission problem in the calculus of variations

AbstractWe study hölder regularity of minimizers of the functional

Relaxation of convex functionals : the gap problem

\int_\Omega {\left| {Du} \right|^{p\left( x \right)} dx}