Continuity estimates for variable growth variational problems in the Heisenberg group

Abstract

We study regularity results for local minimizers of variable growth variational problem in Heisenberg groups under suitable integrability assumption on the horizontal gradient of the exponent function. More precisely, our main focus is on the continuity properties of the horizontal gradient $\mathfrak{X} u$, where $u \in HW_{\text{loc}}^{1,1}$ is a local minimizer of the functional \begin{align*} I [u]:= \int_{\Omega} \frac{1}{p(x)}\left\lvert \mathfrak{X} u \right\rvert^{p(x)}\ \mathrm{d}x \end{align*} in a domain of $\Omega \subset \mathbb{H}_{n},$ where $\mathbb{H}_{n}$ is the Heisenberg group with homogeneous dimension $Q=2n+2,$ where $p \in HW^{1,1}\left( \Omega\right)$ and we assume suitable integrability hypothesis on $\mathfrak{X} p.$ We prove (a) if $\mathfrak{X} p \in L^{q}\left( \Omega; \mathbb{R}^{2n}\right)$ with $q>Q,$ then $\mathfrak{X} u$ is H\"{o}lder continuous and (b) if $\mathfrak{X} p \in L^{(Q,1)}\log L \left( \Omega; \mathbb{R}^{2n}\right),$ then $\mathfrak{X} u$ is continuous. In fact, in the non-borderline case $(a)$, we prove H\"{o}lder continuity of the horizontal gradient for the minima of more general variational problems, assuming $p$ to be H\"{o}lder continuous, i.e. without any assumption on the weak derivative of $p.$ To the best of our knowledge, the present work is the first regularity result for minimizers of variable growth variational problems in the setting of Heisenberg groups.