New results for \(p\) harmonic functions
Let \(G=\{(y^{\prime },y_{n})\in \mathbb{R}^{n-1}\times \mathbb{R}:y_{n}>\phi (y^{\prime })\}\), where \(\phi :\mathbb{R}^{n-1}\rightarrow \mathbb{R}\) is Lipschitz. Further, let \(p>1\), let \(w\in \partial G\) and \(r>0\), and suppose that \(u,v\) are positive \(p\)-harmonic functions on \(B(w,r)\cap G\) that vanish continuously on \(B(w,r)\cap \partial G\). It is then shown that there exist \(c\geq 1\) and \(\alpha \in (0,1)\), depending only on \(p\), \(n\) and \(\phi \), such that \(\left| \log (u/v)(y_{1})-\log (u/v)(y_{2})\right| \leq c(\left| y_{1}-y_{2}\right| /r)^{\alpha }\) whenever \(y_{1},y_{2}\in B(w,r/c)\cap G\). This implies a result of the authors [Ann. Sci. Éc. Norm. Supér. (4) 40, No. 5, 765--813 (2007; Zbl 1134.31008)] concerning the boundary Harnack principle, and also implies the Hölder continuity up to the boundary of the ratio of two positive \(p\)-harmonic functions that vanish on a portion of a Lipschitz boundary. In addition the authors indicate applications to the Martin boundary for \(p\)-harmonic functions and to free boundary regularity questions.
scientific article; zbMATH DE number 5995324