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Uniqueness of semi-concave weak solutions for Hamilton-Jacobi equations

Victor Issa

Abstract

It is well known that when the nonlinearity is convex, the Hamilton-Jacobi PDE admits a unique semi-convex weak solution, which is the viscosity solution. In this paper, motivated by problems arising from spin glasses, we show that if the Hamilton-Jacobi PDE with strictly convex nonlinearity and regular enough initial condition admits a semi-concave weak solution, then this solution is the viscosity solution.

Uniqueness of semi-concave weak solutions for Hamilton-Jacobi equations

Abstract

It is well known that when the nonlinearity is convex, the Hamilton-Jacobi PDE admits a unique semi-convex weak solution, which is the viscosity solution. In this paper, motivated by problems arising from spin glasses, we show that if the Hamilton-Jacobi PDE with strictly convex nonlinearity and regular enough initial condition admits a semi-concave weak solution, then this solution is the viscosity solution.
Paper Structure (6 sections, 11 theorems, 78 equations, 4 figures)

This paper contains 6 sections, 11 theorems, 78 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Proof Sketch
  3. Lower bound
  4. Construction of the characteristics
  5. Proof of the main result
  6. A counter-example for non-convex nonlinearities

Key Result

Theorem 1.1

Let $T^* \in [0,+\infty)$, $H : \mathbb{R}^d \to \mathbb{R}$ be a $\mathcal{C}^2$ strictly convex function and let $f : [0,T^*] \times \mathbb{R}^d \to \mathbb{R}$ be a Lipschitz function. Assume that $u_0 = f(0,\cdot)$ is $\mathcal{C}^{1,1}$, the function $f$ satisfies $\partial_ t f - H(\nabla_x

Figures (4)

  • Figure 1: Graph of the non-convex function $H$
  • Figure 2: Graph of the Lipschitz initial condition $v_0$ for $L = 2$
  • Figure 3: Graph of $(x,t) \mapsto (x+tH(v_0(x)),t)$ when $L=at_1$
  • Figure :

Theorems & Definitions (20)

  • Theorem 1.1
  • Theorem 2.1
  • proof : Proof of Theorem \ref{['t.uniqueness']} using Theorem \ref{['t.wuniqueness']}
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • Lemma 4.1
  • proof
  • ...and 10 more