Carlo Sbordone

Semicontinuity problems in the calculus of variations

Two meaningful cases in which (1.2b) is satisfied are the following: (i)f = f(x, 5) is measurable in x and upper semicontinuous in t; (ii)f is a Caratheodory function, i.e. measurable in x and continuous in (s, 5). Several authors have studied (and proved, under suitable hypothesis) the sequential lower semicontinuity (s.1.s.) of F(Q, . ) in the weak topology of HlsP(Q). A well known theorem of Serrin [l] assures the s.1.s. of F under the assumption that f is a non negative continuous function, convex in r. More recently some improvements of Serrin’s theorem have been given by De Giorgi [2], Berkowitz [3], Cesari [4], Ioffe [S], Olech [6] by considering the s.1.s. of the functional

RIESZ TRANSFORMS AND ELLIPTIC PDES WITH VMO COEFFICIENTS

AbstractThe present paper is concerned withLp-theory of the uniformly elliptic differential operator

Degree formulas for maps with nonintegrable Jacobian

Lu = \sum\limits_{i,j} {\frac{\partial }{{\partial x_i }}(a_{i,j} } (x)\frac{\partial }{{\partial x_i }}\,)

Variable exponents and grand Lebesgue spaces: Some optimal results

inRn with coefficients of vanishing mean oscillation. Recent estimates for the Riesz transform combined with Fredholm index theory enable us to establish invertibility of the map L:W-1,pRn→W1,pRn, for every 1<p<∞. As a side benefit, we obtain the existence and uniqueness theorem for the equationLu=µ with a signed measure in the right hand side. Within the framework of quasiconformal mappings we give a fairly general method of constructing solutions to the homogeneous equationLu=0.

EXPLICIT BOUNDS FOR COMPOSITION OPERATORS PRESERVING BMO(R)

We prove the local boundedness of minimizers of a functional with anisotropic polynomial growth. The result here obtained is optimal if compared with previously know counterexamples.

Div-curl fields of finite distortion

This paper arose from a discussion sparked between the authors after the lecture of Louis Nirenberg at the Conference in Naples on June 1, 1995. He presented a joint work with Haim Brezis [BN] on the degree theory for VMO (vanishing mean oscillation) mappings f : X → Y between n-dimensional smooth manifolds. Their results include a variety of discontinuous maps. We soon realized that we can contribute to their work by studying some Orlicz– Sobolev classes weaker than W (X,Y ). Our approach relies on new estimates for the Jacobians [IS], [GIM] and most recent improvements [I] concerning nonlinear commutators. Also L-Hodge theory [S], [ISS] plays a crucial role in this paper. Let us begin with the well known formula for the degree of a C-map f : X → Y :

ON THE MODULUS OF CONTINUITY OF SOLUTIONS TO THE n-LAPLACE EQUATION

Abstract A sharp integrability condition on the right-hand side of the p-Laplace system for all its solutions to be continuous is exhibited. Their uniform continuity is also analyzed and estimates for their modulus of continuity are provided. The relevant estimates are shown to be optimal as the right-hand side ranges in classes of rearrangement-invariant spaces, such as Lebesgue, Lorentz, Lorentz–Zygmund, and Marcinkiewicz spaces, as well as some customary Orlicz spaces.

On the Γ-convergence of matrix fields related to the adjugate Jacobian

Consider p : Ω → [1, +∞[, a measurable bounded function on a bounded set O with decreasing rearrangement p* : [0, |Ω|] → [1, +∞[. We construct a rearrangement invariant space with variable exponent p* denoted by . According to the growth of p*, we compare this space to the Lebesgue spaces or grand Lebesgue spaces. In particular, if p*(⋅) satisfies the log-Holder continuity at zero, then it is contained in the grand Lebesgue space Lp*(0))(Ω). This inclusion fails to be true if we impose a slower growth as at zero. Some other results are discussed.