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Equivalence of solutions of eikonal equation in metric spaces
Qing Liu, Nageswari Shanmugalingam, Xiaodan Zhou
In this paper we prove the equivalence between some known notions of solutions to the eikonal equation and more general analogs of the Hamilton-Jacobi equations in complete and rec…
Horizontal convex envelope in the Heisenberg group and applications to sub-elliptic equations
Qing Liu, Xiaodan Zhou
This paper introduces in a natural way a notion of horizontal convex envelopes of continuous functions in the Heisenberg group. We provide a convexification process to find the env…
Strong Comparison Principle for harmonic functions in Carnot-Caratheodory spaces
Luca Capogna, Xiaodan Zhou
We extend Bony's propagation of support argument \cite{Bony} to solutions of the non-homogeneous sub-elliptic Laplacian associated to a system of smooth vector fields sat…
Lipschitz continuity and convexity preserving for solutions of semilinear evolution equations in the Heisenberg group
Qing Liu, Juan Manfredi, Xiaodan Zhou
In this paper we study viscosity solutions of semilinear parabolic equations in the Heisenberg group. We show uniqueness of viscosity solutions with exponential growth at space inf…