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Critical points of the solution to the $H_R=H_L$ surface equation

Alma L. Albujer, Magdalena Caballero

Abstract

Spacelike surfaces with the same mean curvature in $\mathbb{R}^3$ and $\mathbb{L}^3$ are locally described as the graph of the solutions to the $H_R=H_L$ surface equation, which is an elliptic partial differential equation except at the points at which the gradient vanishes, because the equation degenerates. In this paper we study precisely the critical points of the solutions to such equation. Specifically, we give a necessary geometrical condition for a point to be critical, we obtain a new uniqueness result for the Dirichlet problem related to the $H_R=H_L$ surface equation and we get a Heinz-type bound for the inradius of the domain of any solution to such equation, improving a previous result by the authors. Finally, we also get a bound for the inradius of the domain of any function of class $\mathcal{C}^2$ in terms of the curvature of its level curves.

Critical points of the solution to the $H_R=H_L$ surface equation

Abstract

Spacelike surfaces with the same mean curvature in and are locally described as the graph of the solutions to the surface equation, which is an elliptic partial differential equation except at the points at which the gradient vanishes, because the equation degenerates. In this paper we study precisely the critical points of the solutions to such equation. Specifically, we give a necessary geometrical condition for a point to be critical, we obtain a new uniqueness result for the Dirichlet problem related to the surface equation and we get a Heinz-type bound for the inradius of the domain of any solution to such equation, improving a previous result by the authors. Finally, we also get a bound for the inradius of the domain of any function of class in terms of the curvature of its level curves.
Paper Structure (4 sections, 5 theorems, 17 equations, 2 figures)

This paper contains 4 sections, 5 theorems, 17 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. On the critical points of the solutions to the $H_R=H_L$ surface equation
  4. A bound for the inradius of a graph depending on its level curves

Key Result

Theorem 1

Let $u$ be a solution to the $H_R=H_L$ surface equation defined on a domain $\Omega\subseteq\mathds{R}^2$ and let $(x_0,y_0)$ be a point in $\Omega$. Then, $H_R(x_0,y_0,u(x_0,y_0))=0$ for any critical point $(x_0,y_0)$ of $u$ in $\Omega$.

Figures (2)

  • Figure 1: Construction of $(x_1,y_1)$ and $(x_3,y_3)$
  • Figure 2: Level curve at the maximum $p$

Theorems & Definitions (6)

  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Corollary 2
  • Theorem 3
  • Remark 1