Geodesic
Quasiconformal solutions to elliptic partial differential equations
David Kalaj1
1University of Montenegro, Faculty of Natural Sciences and Mathematics
Annales Fennici Mathematici, Tome 48 (2023) no. 1, pp. 361-374

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In this paper, we assume that $G$ and $\Omega$ are two Jordan domains in $\mathbb{R}^n$ with $\mathcal{C}^2$ boundaries, where $n\ge 2$, and prove that every quasiconformal mapping $f\in\mathcal{W}^{2,1+\epsilon}_{\mathrm{loc}}$ of $G$ onto $\Omega$, satisfying the elliptic partial differential inequality $|L_ A[f]|\lesssim (\|Df\|^2+|g|)$, with $g\in\mathcal{L}^p(G)$, where $p>n$, is Lipschitz continuous. The result is sharp since for $p=n$, the mapping $f$ is not necessarily Lipschitz continuous. This extends several results for harmonic quasiconformal mappings.  
DOI : 10.54330/afm.129643
Keywords: Quasiconformal mappings, elliptic PDE, Lipschitz continuity

David Kalaj  1

1 University of Montenegro, Faculty of Natural Sciences and Mathematics 
David Kalaj. Quasiconformal solutions to elliptic partial differential equations. Annales Fennici Mathematici, Tome 48 (2023) no. 1, pp. 361-374. doi: 10.54330/afm.129643
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