Continuity properties of solutions to the p-Laplace system

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.

Weighted inequalities for iterated Copson integral operators

“…in which h is a non-negative measurable function on (0, ∞) and C is independent of h. The task is to characterize those parameters and weights for which a constant C, independent of the positive function h, can be found so that (1) holds. Our approach to this problem involves a mixture of discretization and anti-discretization techniques combined with certain duality principles with respect to the integral pairing.…”

“…It turns out that the problem is equivalent to a balance condition which can be transformed into an inequality of the form (1). In a different world, see for instance the recent work [1], weighted inequalities involving Copson integrals are used to investigate the continuity properties of local solutions to the p-Laplace equation…”

“…For example, one may notice that the expression ∞ t h(s) ds represents a nonincreasing function on (0, ∞) when h is non-negative. This may enable us to view (1) as an inequality working on the cone of monotone functions. Inequalities involving such functions on an interval have been seeing a constant boom ever since 1990 and a vast amount of literature is nowadays available, see e.g.…”

“…[15,25,26]), and the approximation techniques stood their ground. It should be mentioned that there is a significant demand for obtaining two-sided estimates of the optimal (that is, smallest possible) constant in (1). This often arises from tasks that involve, in some way, two different integral mean operators, and it appears, for example, in results concerning the optimality of function spaces in Sobolev-type embeddings, carried out e.g.…”

Gradient estimates for problems with Orlicz growth

“…Corresponding results for systems are provided in [40]. Let us refer also to some other attempts [5,7,21,12,23,28,30,43,44,46].…”