Abstract
In this chapter we will study the regularity of weak solution to the non-degenerate p-Laplace equation (10.1007/978-3-319-23790-9_3) divHδ2+|∇Hu|2p-22∇Hu=0inΩ\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \text{ div }_{\mathbb {H}}{\left( \left( \delta ^2+|\nabla _{\mathbb {H}}{u}|^2\right) ^{\frac{p-2}{2}}\nabla _{\mathbb {H}}{u}\right) }=0 \quad \text {in} \,\; \varOmega \end{aligned}$$\end{document}for 1<p<∞\documentclass[12pt]{minimal}
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\begin{document}$$1< p <\infty $$\end{document}. We will use the results of the previous chapter, in particular that u∈Lip(D)\documentclass[12pt]{minimal}
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\begin{document}$$u\in Lip(D)$$\end{document} for a domain D satisfying (10.1007/978-3-319-23790-9_3) with 4.1∇HuL∞(D)≤M.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \left\| \nabla _{\mathbb {H}}{u} \right\| _{L^\infty (D)} \le M. \end{aligned}$$\end{document}For this reason the results will be stated in terms of D instead of Ω\documentclass[12pt]{minimal}
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\begin{document}$$\varOmega $$\end{document}, but since we are studying interior regularity and all the results are local, there is no loss of generality. We will use difference quotients to establish summability of derivatives in the vertical direction and then in the horizontal direction. Once we prove that Tu and ∇Hu\documentclass[12pt]{minimal}
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\begin{document}$$\nabla _{\mathbb {H}}{u}$$\end{document} are in HWloc1,2\documentclass[12pt]{minimal}
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\begin{document}$$HW^{1,2}_{loc}$$\end{document} and satisfy certain linear equations we can apply well known regularity results to prove higher integrability of the solution u. We stress the fact the our estimates depend on the Lipschitz constant M and on the non degeneracy parameter δ\documentclass[12pt]{minimal}
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\begin{document}$$\delta $$\end{document}, and blow up when δ\documentclass[12pt]{minimal}
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\begin{document}$$\delta $$\end{document} goes to zero.