Andrea Cianchi

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.

Global Lipschitz Regularity for a Class of Quasilinear Elliptic Equations

The Lipschitz continuity of solutions to Dirichlet and Neumann problems for nonlinear elliptic equations, including the p-Laplace equation, is established under minimal integrability assumptions on the data and on the curvature of the boundary of the domain. The case of arbitrary bounded convex domains is also included. The results have new consequences even for the Laplacian.

Optimal Orlicz-Sobolev embeddings

An embedding theorem for the Orlicz-Sobolev space W 1,A 0 (G), G ⊂ Rn, into a space of Orlicz-Lorentz type is established for any given Young function A. Such a space is shown to be the best possible among all rearrangement invariant spaces. A version of the theorem for anisotropic spaces is also exhibited. In particular, our results recover and provide a unified framework for various well-known Sobolev type embeddings, including the classical inequalities for the standard Sobolev space W 1,p 0 (G) by O’Neil and by Peetre (1 ≤ p < n), and by Brezis-Wainger and by Hansson (p = n).

Sobolev embeddings into BMO, VMO, andL∞

LetX be a rearrangement-invariant Banach function space onRn and letV1X be the Sobolev space of functions whose gradient belongs toX. We give necessary and sufficient conditions onX under whichV1X is continuously embedded into BMO or intoL∞. In particular, we show thatLn, ∞ is the largest rearrangement-invariant spaceX such thatV1X is continuously embedded into BMO and, similarly,Ln, 1 is the largest rearrangement-invariant spaceX such thatV1X is continuously embedded intoL∞. We further show thatV1X is a subset of VMO if and only if every function fromX has an absolutely continuous norm inLn, ∞. A compact inclusion ofV1X intoC0 is characterized as well.

Strong and Weak Type Inequalities for Some Classical Operators in Orlicz Spaces

Inequalities involving classical operators of harmonic analysis, such as maximal functions, fractional integrals and singular integrals of convolutive type have been extensively investigated in various function spaces. Results on weak and strong type inequalities for operators of this kind in Lebesgue spaces are classical and can be found for example in [ 4 , 20 , 24 ]. Generalizations of these results to Zygmund spaces are presented in [ 4 ]. An exhaustive treatment of the problem of boundedness of such operators in Lorentz and Lorentz–Zygmund spaces is given in [ 3 ]. See also [ 8 , 9 ] for further extensions in the framework of generalized Lorentz–Zygmund spaces. As far as Orlicz spaces are concerned, a characterization of Young functions A having the property that the Hardy–Littlewood maximal operator or the Hilbert and Riesz transforms are of weak or strong type from the Orlicz space L A into itself is known (see for example [ 13 ]). In [ 17 , 23 ] conditions on Young functions A and B are given for the fractional integral operator to be bounded from L A into L B under some restrictions involving the growths and certain monotonicity properties of A and B . The main purpose of this paper is to find necessary and sufficient conditions on general Young functions A and B ensuring that the above-mentioned operators are of weak or strong type from L A into L B . Our results for (fractional) maximal operators are presented in Section 2, while Section 3 deals with fractional and singular integrals. In particular, we re-cover a result concerning the standard Hardy–Littlewood maximal operator which has recently been proved in [ 2 , 11 , 12 ]. Finally, in Section 4, the resolvent operator of some differential problems is taken into account and a priori bounds for Orlicz norms of solutions to elliptic boundary value problems in terms of Orlicz norms of the data are established. Let us mention that part of the results of the present paper were announced in [ 6 ].

Hardy inequalities in Orlicz spaces

We establish a sharp extension, in the framework of Orlicz spaces, of the (n-dimensional) Hardy inequality, involving functions defined on a domain G, their gradients and the distance function from the boundary of G.

Gradient regularity via rearrangements for

A sharp estimate for the decreasing rearrangement of the length of the gradient of solutions to a class of nonlinear Dirichlet and Neumann elliptic boundary value problems is established under weak regularity assumptions on the domain. As a consequence, the problem of gradient bounds in norms depending on global integrability properties is reduced to one-dimensional Hardy-type inequalities. Applications to gradient estimates in Lebesgue, Lorentz, Zygmund, and Orlicz spaces are presented.

p

A comparison theorem for solutions to Dirichlet problems for nonlinear, fully anisotropic, elliptic equations is established via symmetrization. Applications to a priori estimates are also derived.

-Laplacian type elliptic boundary value problems

The local boundedness of minimizers of functionals is proved under growth conditions depending on the full gradient.

Symmetrization in Anisotropic Elliptic Problems

Non-linear elliptic Neumann problems, possibly in irregular domains and with data affected by low integrability properties, are taken into account. Existence, uniqueness and continuous dependence on the data of generalized solutions are established under a suitable balance between the integrability of the datum and the (ir)regularity of the domain. The latter is described in terms of isocapacitary inequalities. Applications to various classes of domains are also presented