Optimality of a Gradient Bound for Polyhedral Wachspress Coordinates

2015 ◽  
pp. 210-215
Author(s):  
Michael S. Floater
Keyword(s):  
Gradient Bound ◽  
Wachspress Coordinates

scholarly journals A Liouville-Type Theorem for an Elliptic Equation with Superquadratic Growth in the Gradient

2020 ◽  
Vol 20 (2) ◽  
pp. 245-251
Author(s):  
Roberta Filippucci ◽  
Patrizia Pucci ◽  
Philippe Souplet
Keyword(s):  
Elliptic Equation ◽  
Harmonic Functions ◽  
Bounded Solution ◽  
Type Theorem ◽  
Elliptic Systems ◽  
Bounded Solutions ◽  
Liouville Type ◽  
Liouville Type Theorem ◽  
Superquadratic Growth ◽  
Gradient Bound

AbstractWe consider the elliptic equation {-\Delta u=u^{q}|\nabla u|^{p}} in {\mathbb{R}^{n}} for any {p>2} and {q>0}. We prove a Liouville-type theorem, which asserts that any positive bounded solution is constant. The proof technique is based on monotonicity properties for the spherical averages of sub- and super-harmonic functions, combined with a gradient bound obtained by a local Bernstein argument. This solves, in the case of bounded solutions, a problem left open in [2], where the case {0<p<2} is considered. Some extensions to elliptic systems are also given.

Gradient estimates for Neumann boundary value problem of Monge–Ampère type equations

2016 ◽  
Vol 19 (04) ◽  
pp. 1650041 ◽  
Author(s):  
Feida Jiang ◽  
Ni Xiang ◽  
Jinju Xu
Keyword(s):  
Matrix Function ◽  
Symmetric Matrix ◽  
Type Equation ◽  
Neumann Boundary ◽  
Gradient Estimates ◽  
Neumann Boundary Value Problem ◽  
The Matrix ◽  
Elliptic Solutions ◽  
Monge Ampere ◽  
Gradient Bound

This paper concerns the gradient estimates for Neumann problem of a certain Monge–Ampère type equation with a lower order symmetric matrix function in the determinant. Under a one-sided quadratic structure condition on the matrix function, we present two alternative full discussions of the global gradient bound for the elliptic solutions.

An improved gradient bound for 2D MBE

2020 ◽  
Vol 269 (12) ◽  
pp. 11165-11171
Author(s):  
Dong Li ◽  
Fan Wang ◽  
Kai Yang
Keyword(s):  
Gradient Bound
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