Continuity estimates for variable growth variational problems in the Heisenberg group
Arka Mallick, Swarnendu Sil
TL;DR
The paper addresses regularity for local minimizers of a variable-growth functional on the Heisenberg group, focusing on the horizontal gradient $\mathfrak{X}u$ and its continuity properties. It develops a two-pronged regularity theory: (i) Hölder continuity of $\mathfrak{X}u$ under $\mathfrak{X}p \in L^{q}(\Omega)$ with $q>Q$, and (ii) borderline continuity under $\mathfrak{X}p \in L^{(Q,1)}\log L(\Omega)$, using an excess-to-excess decay and a linearization framework in the subelliptic setting. The work also proves higher integrability and Hölder continuity in non-borderline cases when $p$ is Hölder continuous, and extends the analysis to Lorentz-Zygmund-type integrability for $p$ and its derivatives. This constitutes the first regularity result of variable-growth variational problems in the subelliptic Heisenberg group, providing sharp a priori estimates and a robust method that links variable exponent theory with subelliptic geometric analysis.
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_Ω \frac{1}{p(x)}\left\lvert \mathfrak{X} u \right\rvert^{p(x)}\ \mathrm{d}x \end{align*} in a domain of $Ω\subset \mathbb{H}_{n},$ where $\mathbb{H}_{n}$ is the Heisenberg group with homogeneous dimension $Q=2n+2,$ where $p \in HW^{1,1}\left( Ω\right)$ and we assume suitable integrability hypothesis on $\mathfrak{X} p.$ We prove (a) if $\mathfrak{X} p \in L^{q}\left( Ω; \mathbb{R}^{2n}\right)$ with $q>Q,$ then $\mathfrak{X} u$ is Hölder continuous and (b) if $\mathfrak{X} p \in L^{(Q,1)}\log L \left( Ω; \mathbb{R}^{2n}\right),$ then $\mathfrak{X} u$ is continuous. In fact, in the non-borderline case $(a)$, we prove Hölder continuity of the horizontal gradient for the minima of more general variational problems, assuming $p$ to be Hölder 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.