Abstract
In this paper we develop the Aubry-Mather theory for Lagrangians in which the potential energy can be discontinuous. Namely we assume that the Lagrangian is lower semicontinuous in the state variable, piecewise smooth with a (smooth) discontinuity surface, as well as coercive and convex in the velocity. We establish existence of Mather measures, various approximation results, partial regularity of viscosity solutions away from the singularity, invariance by the Euler–Lagrange flow away from the singular set, and further jump conditions that correspond to conservation of energy and tangential momentum across the discontinuity.
Abbreviations
- \({\{e_i\}_{i=1}^N}\) :
-
The canonical basis of \({\mathbb{R}^N}\)
- x i :
-
The i-th component of a vector \({x \in \mathbb{R}^N}\)
- |x|:
-
The norm of a vector \({x \in \mathbb{R}^N}\)
- [x, y]:
-
The line segment \({\{tx + (1 - t)y, t \in [0,1]\}, x, y \in \mathbb{R}^N}\)
- \({\frac{\partial}{\partial x_i} \psi = \partial_{x_i} \psi}\) :
-
The partial derivative of the function \({\psi}\) with respect to the variable x i
- \({D_x \psi}\) :
-
Gradient of the function \({\psi}\) with respect to x, that is \({(\partial_{x_1} \psi, \ldots, \partial_{x_N} \psi)}\)
- \({\dot x(t), \ddot x(t)}\) :
-
The first and second derivative of a function \({x: I \subset \mathbb{R} \rightarrow \mathbb{R}^N}\)
- \({|\psi|_\infty}\) :
-
The \({L^{\infty}}\) -norm of a function \({\psi}\)
- \({{\bar{\Omega}}}\) :
-
Closure of an open set \({\Omega \subset \mathbb{R}^N}\)
- \({\partial \Omega}\) :
-
Boundary of an open set \({\Omega \subset \mathbb{R}^N}\)
- B(x, r):
-
The Euclidean ball in \({\mathbb{R}^N}\) of radius r > 0 around x
- \({\Omega_\delta}\) :
-
For any open set \({\Omega \subset \mathbb{R}^N}\) , any \({\delta > 0}\) , the set \({\{x \in \mathbb{R}^N| {\rm dist}(x, \Omega) < \delta \}}\)
- T x M :
-
The tangent space of a smooth manifold M at the point \({x \in M}\)
- \({\delta_{x_0}}\) :
-
The Dirac mass concentrated at x 0
- \({\mu_{X}(x)}\) :
-
The projection on X of a measure \({\mu(x,y)}\) on \({X \times Y}\)
- 〚\({{\ r(t)}_{t_0}}\)〛:
-
The jump of the function \({r: I \subset \mathbb{R}\rightarrow \mathbb{R}}\) at t 0, that is \({\lim_{t \rightarrow t_0^+}r(t) - \lim_{t \rightarrow t_0^-}r(t)}\)
References
Bardi, M., Capuzzo-Dolcetta, I.: Optimal Control and Viscosity Solutions of Hamilton–Jacobi–Bellman Equations. Systems and Control: Foundations and Applications. Birkhäuser Boston Inc., Boston (1997) (With appendices by Maurizio Falcone and Pierpaolo Soravia)
Barles, G.: Solutions de viscosité des équations de Hamilton–Jacobi. Mathématiques and Applications (Berlin) [Mathematics and Applications], vol. 17. Springer, Paris (1994)
Barles, G., Briani, A., Chasseigne, E.: A Bellman approach for two-domains optimal control problems in \({{\mathbb{R}}^n}\) . COCV.http://hal.archives-ouvertes.fr/hal-00652406
Biryuk A., Gomes D.A.: An introduction to the Aubry-Mather theory. São Paulo J. Math. Sci. 4(1), 17–63 (2010)
Bolza, O.: Lectures on the Calculus of Variations, 2nd edn. Chelsea Publishing Co., New York (1961)
Briani, A., Davini, A.: Monge solutions for discontinuous Hamiltonians. ESAIM Control Optim. Calc. Var. 11(2), 229–251 (2005, electronic)
Caffarelli, L., Crandall, M.G., Kocan, M., Swie ̦ch, A.: On viscosity solutions of fully nonlinear equations with measurable ingredients. Commun. Pure Appl. Math. 49(4), 365–397 (1996)
Camilli F.: An Hopf-Lax formula for a class of measurable Hamilton–Jacobi equations. Nonlinear Anal. 57(2), 265–286 (2004)
Camilli F., Siconolfi A.: Time-dependent measurable Hamilton–Jacobi equations. Commun. Partial Differ. Equ. 30(4–6), 813–847 (2005)
Dal Maso G., Frankowska H.: Autonomous integral functionals with discontinuous nonconvex integrands: Lipschitz regularity of minimizers, DuBois-Reymond necessary conditions, and Hamilton–Jacobi equations. Appl. Math. Optim. 48(1), 39–66 (2003)
Davini A.: Bolza problems with discontinuous Lagrangians and Lipschitz-continuity of the value function. SIAM J. Control Optim. 46(5), 1897–1921 (2007)
Evans, L.C.: Weak Convergence Methods for Nonlinear Partial Differential Equations. CBMS Regional Conference Series in Mathematics, vol. 74. Published for the Conference Board of the Mathematical Sciences, Washington, DC (1990)
Evans L.C., Gomes D.: Effective Hamiltonians and averaging for Hamiltonian dynamics. I. Arch. Ration. Mech. Anal. 157(1), 1–33 (2001)
Fathi A., Siconolfi A.: Existence of C 1 critical subsolutions of the Hamilton–Jacobi equation. Invent. Math. 155(2), 363–388 (2004)
Fathi A., Siconolfi A.: PDE aspects of Aubry-Mather theory for quasiconvex Hamiltonians. Calc. Var. Partial Differ. Equ. 22(2), 185–228 (2005)
Figalli A., Mandorino V.: Fine properties of minimizers of mechanical lagrangians with sobolev potentials. Discrete Contin. Dyn. Syst. Ser. A 31(4), 1325–1346 (2011)
Gomes, D.: Viscosity solutions of Hamilton–Jacobi equations. Publicações Matemáticas do IMPA. [IMPA Mathematical Publications]. Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2009. 27o Colóquio Brasileiro de Matemática. [27th Brazilian Mathematics Colloquium]
Mañé R.: On the minimizing measures of Lagrangian dynamical systems. Nonlinearity 5(3), 623–638 (1992)
Mather J.N.: Action minimizing invariant measures for positive definite Lagrangian systems. Math. Z 207(2), 169–207 (1991)
Newcomb R.T., Su J.: Eikonal equations with discontinuities. Differ. Integral Equ. 8(8), 1947–1960 (1995)
Soner, H.M.: Optimal control with state-space constraint I & II. SIAM J. Control Optim., 24(3 & 6): 552–561, 1110–1122 (1986)
Soravia P.: Boundary value problems for Hamilton–Jacobi equations with discontinuous Lagrangian. Indiana Univ. Math. J 51(2), 451–477 (2002)
Soravia, P.: Degenerate eikonal equations with discontinuous refraction index. ESAIM Control Optim. Calc. Var. 12(2), 216–230 (2006, electronic)
Villani, C.: Topics in Optimal Transportation. Graduate Studies in Mathematics, vol. 58. American Mathematical Society, Providence (2003)
Author information
Authors and Affiliations
Corresponding author
Additional information
D. Gomes was partially supported by CAMGSD-LARSys through FCT-Portugal and by grants PTDC/MAT/114397/2009, UTAustin/MAT/0057/2008, and UTA-CMU/MAT/ 0007/2009.
G. Terrone was supported by the UTAustin-Portugal partnership through the FCT post-doctoral fellowship SFRH/BPD/40338/2007, CAMGSD-LARSys through FCT Program POCTI - FEDER and by grants PTDC/MAT/114397/2009, UTAustin/MAT/0057/2008, and UTA-CMU/MAT/0007/2009.
Rights and permissions
About this article
Cite this article
Gomes, D.A., Terrone, G. The Mather problem for lower semicontinuous Lagrangians. Nonlinear Differ. Equ. Appl. 21, 167–217 (2014). https://doi.org/10.1007/s00030-013-0243-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00030-013-0243-0
Mathematics Subject Classification (2010)
- Primary 35F21
- 37J50
- Secondary 35D40
- 35R05
- 49K20
Keywords
- Discontinuous Lagrangians
- Action minimizing measures
- Hamilton–Jacobi equations
- Viscosity solutions