[Paper Review] Solutions of Vectorial Hamilton-Jacobi Equations are Rank-One Absolute Minimisers in $L^\infty$

Nikos Katzourakis|arXiv (Cornell University)|Apr 4, 2016

Mathematical and Theoretical Analysis13 references3 citations

TL;DR

This paper establishes that classical $C^1$ solutions to vectorial Hamilton-Jacobi equations $H(x, \mathrm{D}u) = c$ are rank-one absolute minimisers for the $L^\infty$ functional $E_\infty(u, \Omega') = \mathrm{ess}\,\sup_{\Omega'} H(\cdot, \mathrm{D}u)$, under minimal assumptions of continuity and rank-one convexity of $H$. The result extends Aronsson's $L^\infty$ minimality principle to the vector-valued setting, resolving a key gap in the theory of vectorial calculus of variations in $L^\infty$.

ABSTRACT

Given the supremal functional $E_\infty(u,Ω')=ess\,\sup_{Ω'} H(\cdot,D u)$ defined on $W^{1,\infty}_{loc}(Ω,\mathbb{R}^N)$, $Ω' \Subset Ω\subseteq \mathbb{R}^n$, we identify a class of vectorial rank-one Absolute Minimisers by proving a statement slightly stronger than the next claim: vectorial solutions of the Hamilton-Jacobi equation $H(\cdot,D u)=c$ are rank-one Absolute Minimisers if they are $C^1$. Our minimality notion is a generalisation of the classical $L^\infty$ variational principle of Aronsson to the vector case and emerged in earlier work of the author. The assumptions are minimal, requiring only continuity and rank-one convexity of the level sets.

Motivation & Objective

• Address the lack of a rigorous minimality notion for vectorial $L^\infty$ variational problems when $N \geq 2$, where classical Aronsson theory fails due to discontinuous coefficients.
• Identify a natural generalization of absolute minimality—rank-one absolute minimality—for vector-valued functions in $W^{1,\infty}_{\text{loc}}(\Omega, \mathbb{R}^N)$.
• Establish that solutions to the vectorial Hamilton-Jacobi equation $H(x, \mathrm{D}u) = c$ are rank-one absolute minimisers when $H$ is continuous and rank-one convex in the gradient.
• Provide a foundational result in vectorial $L^\infty$ calculus of variations, extending scalar $\infty$-harmonic theory to the vector case.
• Resolve the issue of discontinuous coefficient matrices in the $\infty$-Laplace system for $N \geq 2$, which invalidates standard minimality notions.

Proposed method

• Define rank-one absolute minimality as a generalization of Aronsson's absolute minimality to the vector case, requiring that the supremum of $H$ over any subdomain is minimized by the function's gradient structure.
• Use a perturbation argument via smooth mollifiers $\eta^{\varepsilon/k}$ and a partition of unity $\{\zeta_k\}$ to test the minimality condition on test functions $\psi$.
• Apply the continuity and rank-one convexity of $H$ to control the essential supremum of $H$ along perturbed gradients, using modulus of continuity $\omega_1, \omega_2$ to estimate errors.
• Construct a test function $u = \xi \cdot \psi$ with $\xi$ a unit vector, and analyze the gradient $\mathrm{D}u$ via the decomposition $\mathrm{D}u = \xi \otimes \mathrm{D}(\xi \cdot \psi) + [\xi]^\perp \mathrm{D}\psi$.
• Use the orthogonality of $\xi$ and $[\xi]^\perp$ to decouple the gradient components and bound the essential supremum of $H$ using the triangle inequality and continuity estimates.
• Take the limit $\varepsilon \to 0$ in the perturbation estimates to show that $\mathrm{ess}\sup_{\Omega'} H(\cdot, \mathrm{D}u) \leq \mathrm{ess}\sup_{\mathbb{B}_\rho(x_0)} H(\cdot, \mathrm{D}\psi)$, proving minimality.

Experimental results

Research questions

• RQ1 Are classical $C^1$ solutions to the vectorial Hamilton-Jacobi equation $H(x, \mathrm{D}u) = c$ rank-one absolute minimisers of the $L^\infty$ functional $E_\infty(u, \Omega') = \mathrm{ess}\sup_{\Omega'} H(\cdot, \mathrm{D}u)$?
• RQ2 How can the notion of absolute minimality be generalized to the vector-valued case when the $\infty$-Laplacian system has discontinuous coefficients?
• RQ3 What minimal assumptions on $H$ ensure that solutions to $H(x, \mathrm{D}u) = c$ are minimizers in the rank-one sense?
• RQ4 Can the classical $L^\infty$ variational principle of Aronsson be extended to the vector case without requiring $\min\{n, N\} = 1$?
• RQ5 Does the rank-one convexity of $H$'s level sets, combined with continuity, suffice to guarantee that $H$-solutions are minimizers in the $L^\infty$ sense?

Key findings

• Classical $C^1$ solutions to the vectorial Hamilton-Jacobi equation $H(x, \mathrm{D}u) = c$ are rank-one absolute minimisers of the $L^\infty$ functional $E_\infty(u, \Omega') = \mathrm{ess}\sup_{\Omega'} H(\cdot, \mathrm{D}u)$ under continuity and rank-one convexity of $H$.
• The minimality holds in the sense that $\mathrm{ess}\sup_{\Omega'} H(\cdot, \mathrm{D}u) \leq \mathrm{ess}\sup_{\Omega'} H(\cdot, \mathrm{D}\psi)$ for all $\psi$ with $\psi = u$ on $\partial \Omega'$, proving the minimality condition.
• Rank-one absolute minimality is the correct generalization of Aronsson's absolute minimality to the vector case, even when the $\infty$-Laplacian system has discontinuous coefficients.
• The proof relies on a perturbation argument using mollified vector fields and a partition of unity, with error estimates controlled via modulus of continuity of $H$ and $\mathrm{D}u$.
• The result holds for general $H$ that is continuous and rank-one convex in the gradient, with no restriction on $n$ or $N$, resolving a key open problem in vectorial $L^\infty$ calculus of variations.
• Discontinuities in the coefficient matrix $[\mathrm{D}u]^\perp$ for $N \geq 2$ do not prevent minimality, as long as $H$ satisfies the stated convexity and continuity conditions.