[Paper Review] Global generalized characteristics for the Dirichlet problem for Hamilton-Jacobi equations at a supercritical energy level

Piermarco Cannarsa, Wei Cheng+2 more|arXiv (Cornell University)|Mar 5, 2018

Nonlinear Partial Differential Equations16 references3 citations

TL;DR

This paper establishes the global propagation of singularities for viscosity solutions of Hamilton-Jacobi equations with supercritical energy levels on bounded domains, proving that singularities propagate along generalized characteristics and converge to critical points. It shows the solution is globally semiconcave and semiconvex near the boundary under stronger assumptions, extending weak KAM theory to Dirichlet problems with nonhomogeneous boundary data.

ABSTRACT

We study the nonhomogeneous Dirichlet problem for first order Hamilton-Jacobi equations associated with Tonelli Hamiltonians on a bounded domain $Ω$ of $\R^n$ assuming the energy level to be supercritical. First, we show that the viscosity (weak KAM) solution of such a problem is Lipschitz continuous and locally semiconcave in $Ω$. Then, we analyse the singular set of a solution showing that singularities propagate along suitable curves, the so-called generalized characteristics, and that such curves stay singular unless they reach the boundary of $Ω$. Moreover, we prove that the latter is never the case for mechanical systems and that singular generalized characteristics converge to a critical point of the solution in finite or infinite time. Finally, under stronger assumptions for the domain and Dirichlet data, we are able to conclude that solutions are globally semiconcave and semiconvex near the boundary.

Motivation & Objective

• To analyze the regularity and singular structure of viscosity solutions to the Dirichlet problem for Hamilton-Jacobi equations at supercritical energy levels.
• To extend weak KAM theory to nonhomogeneous Dirichlet problems on bounded domains with general boundary data.
• To characterize the propagation of singularities along generalized characteristics and their convergence behavior.
• To establish global semiconcavity and semiconvexity of solutions under stronger geometric and boundary data assumptions.

Proposed method

• Uses a representation formula for the solution via Mañé's potential and relative fundamental solutions to characterize the viscosity solution as the infimal convolution of boundary data.
• Applies weak KAM theory and Lax-Oleinik semigroup techniques to analyze the structure of the solution and its singular set.
• Introduces generalized characteristics as curves along which singularities propagate, derived from the Hamiltonian dynamics of the system.
• Employs a change of Lagrangian by an exact 1-form to simplify analysis while preserving critical values and solution structure.
• Establishes local semiconcavity and Lipschitz continuity of the solution in the interior of the domain.
• Imposes stronger geometric and boundary data conditions to prove global semiconcavity and semiconvexity up to the boundary.

Experimental results

Research questions

• RQ1 How do singularities of viscosity solutions to Hamilton-Jacobi equations propagate in bounded domains at supercritical energy levels?
• RQ2 Under what conditions do generalized characteristics remain singular until they reach the boundary, and when do they terminate?
• RQ3 Can the solution be globally semiconcave and semiconvex near the boundary under stronger assumptions on the domain and boundary data?
• RQ4 What is the topological structure of the cut locus and singular set of the solution in mechanical systems?
• RQ5 How do the critical points of the solution relate to the long-term behavior of generalized characteristics?

Key findings

• The viscosity solution is Lipschitz continuous and locally semiconcave in the interior of the domain Ω.
• Singularities propagate along generalized characteristics and remain singular unless they reach the boundary of Ω.
• For mechanical systems, generalized characteristics never reach the boundary and instead converge to a critical point of the solution in finite or infinite time.
• Under stronger assumptions on the domain and boundary data, the solution is globally semiconcave and semiconvex near the boundary.
• The critical value cΩ(L) is negative under the given assumptions, ensuring the finiteness of Mañé's potential and the existence of a well-defined solution.
• The solution satisfies the compatibility condition g(x) - g(y) ≤ ΦΩL(y,x) for all x,y ∈ ∂Ω, ensuring consistency with the representation formula.