[Paper Review] Motion of level sets by inverse anisotropic mean curvature

Francesco Della Pietra, Nunzia Gavitone+1 more|arXiv (Cornell University)|Apr 18, 2018

Geometric Analysis and Curvature Flows6 references3 citations

TL;DR

This paper establishes the existence and uniqueness of weak solutions to the inverse anisotropic mean curvature flow (IAMCF) using a Finsler-$p$-Laplacian approximation method. By regularizing the degenerate elliptic equation associated with IAMCF and taking the limit as $p \to 1^+$, the authors prove the existence of a proper weak solution $u \in C^{0,1}_{\textrm{loc}}(\overline{\Omega})$ satisfying gradient bounds related to the initial anisotropic mean curvature.

ABSTRACT

In this paper we consider the weak formulation of the inverse anisotropic mean curvature flow, in the spirit of Huisken-Ilmanen. By using approximation method involving Finsler-p-Laplacian, we prove the existence and uniqueness of weak solutions.

Motivation & Objective

• To extend Huisken-Ilmanen's weak formulation of inverse mean curvature flow to the anisotropic setting using a level-set approach.
• To establish the existence of weak solutions for the inverse anisotropic mean curvature flow (IAMCF) in the spirit of Huisken-Ilmanen's theory.
• To develop a regularized approximation scheme using the Finsler-$p$-Laplacian to overcome the lack of elliptic regularization in the anisotropic case.
• To prove uniform gradient estimates for the approximating solutions as $p \to 1^+$, ensuring convergence to a weak solution.
• To derive a priori bounds on the anisotropic gradient $F(\nabla u)$ in terms of the initial anisotropic mean curvature $H_F^+$.

Proposed method

• Formulate the IAMCF as a degenerate elliptic PDE: $\operatorname{div}(F_\xi(\nabla u)) = F(\nabla u)$, with $u=0$ on $\partial\Omega$ and $u \to \infty$ as $|x| \to \infty$.
• Introduce an approximation via the Finsler-$p$-Laplacian equation: $\operatorname{div}(F^{p-1}(\nabla u) F_\xi(\nabla u)) = F(\nabla u)^p$ in $\Omega$, with $u=0$ on $\partial\Omega$ and $u \to \infty$ at infinity.
• Prove existence and $C^{1,\alpha}$ regularity of solutions $u_p$ to the approximating problem for each $p > 1$ using variational and PDE techniques.
• Establish uniform gradient estimates $F(\nabla u_p) \leq \sup_{\partial\Omega} H_F^+ + \varepsilon$ in $\overline{\Omega}$, independent of $p$ for $p$ close to 1.
• Apply the Arzelà-Ascoli theorem to extract a subsequence $u_{p_k} \to u$ uniformly on compact sets as $p_k \to 1^+$.
• Pass to the limit in the energy functional $J^{p}_{u_p}(\varphi)$ to show that the limit $u$ satisfies the weak formulation of the IAMCF in the sense of Definition 1.

Experimental results

Research questions

• RQ1 Can a Huisken-Ilmanen-type weak solution theory be extended to the inverse anisotropic mean curvature flow?
• RQ2 Does the absence of standard elliptic regularization in the anisotropic case prevent the construction of weak solutions?
• RQ3 Can the Finsler-$p$-Laplacian provide a viable approximation scheme for IAMCF when $p \to 1^+$?
• RQ4 Are uniform gradient estimates on $F(\nabla u_p)$ achievable in the approximating sequence, independent of $p$?
• RQ5 Does the limit of the approximating solutions satisfy the weak formulation and inherit the properness and gradient bounds?

Key findings

• A unique proper weak solution $u \in C^{0,1}_{\textrm{loc}}(\overline{\Omega})$ exists for the inverse anisotropic mean curvature flow, satisfying $u=0$ on $\partial\Omega$.
• The solution satisfies the gradient bound $F(\nabla u(x)) \leq \sup_{\partial\Omega} H_F^+$ for all $x \in \overline{\Omega}$.
• The solution also satisfies the pointwise bound $F(\nabla u(x)) \leq H_F^+(x)$ for all $x \in \partial\Omega$.
• For each $p > 1$, the approximating problem has a unique solution $u_p \in C^{1,\alpha}_{\textrm{loc}}(\overline{\Omega})$ with uniform gradient estimates $F(\nabla u_p) \leq \sup_{\partial\Omega} H_F^+ + \varepsilon$.
• The sequence $u_p$ converges uniformly on compact sets to the weak solution $u$ as $p \to 1^+$.
• The weak solution is obtained as the limit of approximations via the Finsler-$p$-Laplacian, proving existence via approximation and compactness.