[Paper Review] Existence of a capillary surface with prescribed contact angle in $M imes\R$

Maria Calle, Leili Shahriyari|arXiv (Cornell University)|Dec 25, 2010

Geometric Analysis and Curvature Flows7 references3 citations

TL;DR

This paper establishes a priori gradient estimates for solutions to the prescribed mean curvature equation with a prescribed contact angle condition in the product manifold $M \times \mathbb{R}$, where $M^n$ is a Riemannian submanifold of $\mathbb{R}^{n+1}$. From these estimates, it derives the long-time existence of solutions, providing a foundational existence result for capillary surfaces with given boundary contact angles in this geometric setting.

ABSTRACT

We study the prescribed mean curvature equation with a prescribed boundary contact angle condition in $M imes\R$ where $M^n$ is a Riemannian submanifold in $\R^{n+1}$. The main purpose is to establish a priori gradient estimates for solutions, from which the long time existence of the solution are derived.

Motivation & Objective

• To investigate the existence of capillary surfaces in $M \times \mathbb{R}$ with a prescribed contact angle at the boundary.
• To address the challenge of controlling the gradient of solutions to the prescribed mean curvature equation under non-constant boundary contact angle conditions.
• To derive a priori gradient estimates that are essential for proving long-time existence of solutions.
• To extend existence theory for capillary surfaces beyond symmetric or flat ambient spaces to general Riemannian submanifolds $M^n \subset \mathbb{R}^{n+1}$.
• To establish a theoretical framework for solving geometric boundary value problems with natural contact angle constraints in product manifolds.

Proposed method

• Analyzes the prescribed mean curvature equation in the product space $M \times \mathbb{R}$, where $M^n$ is a Riemannian submanifold of $\mathbb{R}^{n+1}$.
• Imposes a boundary condition specifying the contact angle between the surface and the boundary of $M \times \mathbb{R}$, modeling physical capillary phenomena.
• Derives a priori $L^\infty$ gradient estimates for solutions using maximum principle techniques and geometric constraints on the ambient manifold.
• Applies the method of continuity and parabolic regularization to establish long-time existence of solutions from the gradient bounds.
• Utilizes the structure of the product manifold and the induced metric to control curvature and boundary behavior.
• Relies on comparison principles and barrier constructions adapted to the contact angle condition to control the solution's behavior near the boundary.

Experimental results

Research questions

• RQ1 Under what conditions does a capillary surface with a prescribed contact angle exist in $M \times \mathbb{R}$ for a general Riemannian submanifold $M^n \subset \mathbb{R}^{n+1}$?
• RQ2 How can one derive uniform gradient estimates for solutions to the prescribed mean curvature equation under a non-constant contact angle condition?
• RQ3 What geometric and analytic conditions on $M$ ensure the long-time existence of solutions to the boundary value problem?
• RQ4 Can the method of a priori estimates be extended to manifolds with non-trivial curvature to handle contact angle constraints?
• RQ5 How does the contact angle condition influence the regularity and existence of solutions in product Riemannian spaces?

Key findings

• A priori gradient estimates are established for solutions to the prescribed mean curvature equation in $M \times \mathbb{R}$ under a prescribed contact angle condition.
• The derived gradient bounds are independent of time, enabling the application of the method of continuity for long-time existence.
• The existence of a solution to the boundary value problem is guaranteed under the derived estimates, extending existence results to general Riemannian submanifolds.
• The method successfully handles non-constant contact angles by incorporating geometric constraints into the maximum principle framework.
• The results confirm that the structure of $M \times \mathbb{R}$ and the curvature of $M$ play a critical role in controlling solution behavior near the boundary.
• The approach provides a general framework for studying capillary surfaces with natural boundary conditions in geometrically non-symmetric settings.