E. Mascolo | Searcharxiv

arXiv2026

On the interplay between $(p,q)$-growth and $x$-dependence of the energy integrand: a limit case

We establish the local Lipschitz regularity of the local minimizers of non autonomous integral funtionals of the form \[ \int_\Omega F(x, Dz)\,dx, \] where $\Omega$ is a bounded open set of $\mathbb{R}^n$, $n \ge 2$. The energy density $F(x,\xi)$ satisfies $(p,q)-$growth conditions with respect to the gradient variable and belongs to the Sobolev class $W^{1,\phi}$, with $\phi(t)=t^r\log^\alpha(e+t),$ $r\ge n$, $\alpha\ge 0$, as a function of the $x$ variable, under the condition $$ 1\le\frac{q}{p} \le 1 + \frac{1}{n} - \frac{1}{r}. $$ We present a unified approach that covers the limit case $$ \frac{q}{p} = 1 + \frac{1}{n} - \frac{1}{r} $$ and retrieves the results in \cite{EMM16} and in \cite{CGHPdN20}.

math.AP↗

arXiv2021

Local boundedness for solutions of a class of nonlinear elliptic systems

In this paper we are concerned with the regularity of solutions to a nonlinear elliptic system of $m$ equations in divergence form, satisfying $p$ growth from below and $q$ growth from above, with $p \leq q$; this case is known as $p, q$-growth conditions. Well known counterexamples, even in the simpler case $p=q$, show that solutions to systems may be singular; so, it is necessary to add suitable structure conditions on the system that force solutions to be regular. Here we obtain local boundedness of solutions under a componentwise coercivity condition. Our result is obtained by proving that each component $u^α$ of the solution $u=(u^1,...,u^m)$ satisfies an improved Caccioppoli's inequality and we get the boundedness of $u^α$ by applying De Giorgi's iteration method, provided the two exponents $p$ and $q$ are not too far apart. Let us remark that, in dimension $n=3$ and when $p=q$, our result works for $\frac{3}{2} < p < 3$, thus it complements the one of Bjorn whose technique allowed her to deal with $p \leq 2$ only. In the final section, we provide applications of our result.

math.AP↗