[Paper Review] Herglotz' variational principle and Lax-Oleinik evolution

Piermarco Cannarsa, Wei Cheng+3 more|arXiv (Cornell University)|Jul 12, 2019

Geometric Analysis and Curvature Flows41 references4 citations

TL;DR

This paper establishes a Lipschitz regularity estimate for minimizers in Herglotz' variational principle under time-dependent, non-autonomous settings, using a generalized du Bois-Reymond lemma to derive Erdmann’s condition and the Euler-Lagrange equation. It further provides a representation formula for viscosity solutions of the Hamilton-Jacobi equation via Lax-Oleinik evolution, extending the theory to Riemannian manifolds with global existence and regularity results.

ABSTRACT

We develop an elementary method to give a Lipschitz estimate for the minimizers in the problem of Herglotz' variational principle proposed in \cite{CCWY2018} in the time-dependent case. We deduce Erdmann's condition and the Euler-Lagrange equation separately under different sets of assumptions, by using a generalized du Bois-Reymond lemma. As an application, we obtain a representation formula for the viscosity solution of the Cauchy problem for the Hamilton-Jacobi equation \begin{align*} D_tu(t,x)+H(t,x,D_xu(t,x),u(t,x))=0 \end{align*} and study the related Lax-Oleinik evolution.

Motivation & Objective

• To establish a Lipschitz regularity estimate for minimizers in Herglotz’ variational principle in the time-dependent case.
• To derive Erdmann’s condition and the Euler-Lagrange equation using a generalized du Bois-Reymond lemma under different assumptions.
• To provide a representation formula for viscosity solutions of the Cauchy problem for the Hamilton-Jacobi equation.
• To extend the existence and regularity theory of minimizers to compact, connected, and closed C² Riemannian manifolds.
• To analyze the Lax-Oleinik evolution in the context of Herglotz’ variational principle with global and local geometric constraints.

Proposed method

• A generalized du Bois-Reymond lemma is employed to derive necessary conditions for optimality, including Erdmann’s condition and the Euler-Lagrange equation.
• The paper uses a priori Lipschitz bounds on minimizers derived from structural assumptions on the Lagrangian L, including superlinear growth and bounded derivatives.
• A local-to-global argument via partitioning the time interval and using a finite atlas of normal coordinate charts reduces the global problem to local ones.
• The Lax-Oleinik evolution is constructed through a dynamic programming principle, using infimal convolution of local fundamental solutions.
• The existence of minimizers is proven via compactness and semiconcavity arguments on compact manifolds, leveraging uniform bounds on the state and velocity trajectories.
• The viscosity solution is represented via an infimum over piecewise C¹ curves, with the value functional defined by solving a Carathéodory ODE along each curve.

Experimental results

Research questions

• RQ1 Under what conditions does a minimizer of Herglotz’ variational principle remain Lipschitz continuous in the time-dependent setting?
• RQ2 How can Erdmann’s condition and the Euler-Lagrange equation be derived independently under different assumptions using a generalized du Bois-Reymond lemma?
• RQ3 What is the precise representation of the viscosity solution to the Hamilton-Jacobi equation using the Lax-Oleinik formula in the context of Herglotz’ principle?
• RQ4 How can the existence and regularity of minimizers be extended from Euclidean domains to compact Riemannian manifolds?
• RQ5 What is the role of the local chart structure and the 'broken geodesic' argument in constructing global minimizers on manifolds?

Key findings

• A uniform Lipschitz bound on the velocity of minimizers is established, with the bound depending only on the time interval, the initial value u, and the geometry of the manifold.
• The minimizer satisfies the Herglotz equation in local coordinates, and the dual arc p(s) = L_v is C² when L is C².
• The viscosity solution of the Hamilton-Jacobi equation is represented via the Lax-Oleinik formula as an infimum over piecewise C¹ curves with state and control constraints.
• The fundamental solution h_L is semiconcave locally, and the infimum in the Lax-Oleinik formula is attained due to compactness of the state and control trajectories.
• On a compact Riemannian manifold, the global minimizer exists and is C¹-Lipschitz, with uniform bounds on |u_ξ(s)| and |ẋ(s)| depending only on u, x, y, and the time interval.
• The Lax-Oleinik evolution is well-defined and semiconcave globally, with the value function satisfying the Hamilton-Jacobi equation in the viscosity sense.