It is known that, if u is a real valued function on R^n of bounded variation, then its total variation decreases under polarization. In this paper we identify the difference between the total variation of u and that one of its polar u_{\pi}.

A pointwise inequality between the radially decreasing symmetrals of minimizers of (possibly) anisotropic variational problems and the minimizers of suitably symmetrized problems is established. As a consequence, a priori sharp estimates for norms of the relevant minimizers are derived.

Sharp constants are exhibited in exponential inequalities corresponding to the limiting case of the Sobolev inequalities in Lorentz-Sobolev spaces of arbitrary order.

Sharp constants are exhibited in exponential inequalities corresponding to the limiting case of the Sobolev inequalities in Lorentz-Sobolev spaces of arbitrary order.

A priori estimates for solutions to homogeneous Neumann problems for uniformly elliptic equations in open subsets ­ of R^n are established, with data in the limiting space L^{n\2} or, more generally, in the Lorentz spaces L^{n\2, q}(\Omega). These estimates are optimal as far as either constants or norms are concerned.

The strongest possible norms in a priori estimates for solutions to a class of second-order nonlinear elliptic boundary value problems are exhibited.

In this paper we are concerned with the pointwise behaviour of functions in certain classes of weakly differentiable functions. We focus on first order Orlicz-Sobolev spaces and we establish optimal theorems of Rademacher type and of approximate differentiability type in this setting.

Sharp exponential estimates for solutions to homogeneous Neumann problems for nonlinear elliptic equations in open subsets ! of Rn are established, with data from limiting Lebesgue spaces, or, more generally, Lorentz spaces.

A priori estimates for solutions to homogeneous Neumann problems for uniformly elliptic equa- tions in open subsets \Omega­ of IR^n are established when the datum on right-hand side is in the limiting space L^{n/2}(\Omega), or, more generally, in the Lorentz spaces L^{n/2, q}(\Omega). These estimates are optimal as far as either constants or norms are concerned.