2 citations · 4 across the 17 of their papers we have counts for
17 papers
Superposition Property in Disjoint Variables for the Infinity Laplace Equation
Qing Liu, Juan J. Manfredi, Xiaodan Zhou
We establish a superposition principle in disjoint variables for the inhomogeneous infinity-Laplace equation. We show that the sum of viscosity solutions of the inhomogeneous infin…
A second-order operator for horizontal quasiconvexity in the Heisenberg group and application to convexity preserving for horizontal curvature flow
Antoni Kijowski, Qing Liu, Ye Zhang +1
This paper is concerned with a PDE approach to horizontally quasiconvex (h-quasiconvex) functions in the Heisenberg group based on a nonlinear second order elliptic operator. We di…
Discontinuous eikonal equations in metric measure spaces
Qing Liu, Nageswari Shanmugalingam, Xiaodan Zhou
In this paper, we study the eikonal equation in metric measure spaces, where the inhomogeneous term is allowed to be discontinuous, unbounded and merely -integrable in the domai…
Hamilton-Jacobi equations in metric spaces
Qing Liu, Xiaodan Zhou
These are lecture notes for our minicourse at OIST Summer Graduate School "Analysis and Partial Differential Equations" on June 12-17, 2023. We give an overview and collect a few i…
Green function in metric measure spaces
Mario Bonk, Luca Capogna, Xiaodan Zhou
We study existence and uniqueness of Green functions for the Cheeger -Laplacian in metric measure spaces that are Ahlfors -regular and support a -Poincaré inequality with…
A characterization of BV and Sobolev functions via nonlocal functionals in metric spaces
Panu Lahti, Andrea Pinamonti, Xiaodan Zhou
We study a characterization of BV and Sobolev functions via nonlocal functionals in metric spaces equipped with a doubling measure and supporting a Poincaré inequality. Compared wi…
Horizontally quasiconvex envelope in the Heisenberg group
Antoni Kijowski, Qing Liu, Xiaodan Zhou
This paper is concerned with a PDE-based approach to the horizontally quasiconvex (h-quasiconvex for short) envelope of a given continuous function in the Heisenberg group. We prov…
Absolutely continuous mappings on doubling metric measure spaces
Panu Lahti, Xiaodan Zhou
Following Malý's definition of absolutely continuous functions of several variables, we consider -absolutely continuous mappings between a doubling metric measu…
Metric quasiconformality and Sobolev regularity in non-Ahlfors regular spaces
Panu Lahti, Xiaodan Zhou
Given a homeomorphism between -dimensional spaces , we show that satisfying the metric definition of quasiconformality outside suitable exceptional set…
Quasiconformal and Sobolev mappings in non-Ahlfors regular metric spaces
Panu Lahti, Xiaodan Zhou
We show that a mapping satisfying the metric condition of quasiconformality outside suitable exceptional sets is in the Newton-Sobolev class $N_{\textrm{loc}}^{1,1…
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…
Functions of bounded variation on complete and connected one-dimensional metric spaces
Panu Lahti, Xiaodan Zhou
In this paper, we study functions of bounded variation on a complete and connected metric space with finite one-dimensional Hausdorff measure. The definition of BV functions on a c…
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…
Sobolev functions in the critical case are uniformly continuous in -Ahlfors regular metric spaces when
Xiaodan Zhou
We prove that functions in the Hajłasz-Sobolev space on an -Ahlfors regular metric space are uniformly continuous when .
Sobolev Embedding of a Sphere Containing An Arbitrary Cantor Set in the image
Piotr Hajłasz, Xiaodan Zhou
We construct a large class of pathological -dimensional topological spheres in by showing that for any Cantor set there is a top…