On the fractional regularity for degenerate equations with $(p,q)$-growth

TL;DR

This paper investigates the global fractional regularity of solutions to degenerate quasilinear equations with $(p,q)$-growth, revealing how the interplay of parameters and data influences solution smoothness up to the boundary.

Contribution

It provides new global fractional regularity results for solutions of $(p,q)$-Laplacian equations, extending known local regularity and analyzing the effects of data and parameters.

Findings

Fractional regularity depends on $p$, $q$, and data.

Established boundary regularity results without Lavrentiev phenomenon.

Developed new a priori estimates and approximation techniques.

Abstract

This paper addresses the gain of global fractional regularity in Nikolskii spaces for solutions of a class of quasilinear degenerate equations with $(p,q)$-growth. Indeed, we investigate the effects of the datum on the derivatives of order greater than one of the solutions of the $(p,q)$-Laplacian operator, under Dirichlet's boundary conditions. As it turns out, even in the absence of the so-called Lavrentiev phenomenon and without variations on the order of ellipticity of the equations, the fractional regularity of these solutions ramifies depending on the interplay between the growth parameters $p$, $q$ and the data. Indeed, we are going to exploit the absence of this phenomenon in order to prove the validity up to the boundary of some regularity results, which are known to hold locally, and as well provide new fractional regularity for the associated solutions. In turn, there are obtained certain global regularity results by means of the combination between new a priori estimates and approximations of the differential operators, whereas the nonstandard boundary terms are handled by means of a careful choice for the local frame.