Abstract
In this paper we develop the AubryMather 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 ith 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 \({\lim_{t \rightarrow t_0^+}r(t)  \lim_{t \rightarrow t_0^}r(t)}\)
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D. Gomes was partially supported by CAMGSDLARSys through FCTPortugal and by grants PTDC/MAT/114397/2009, UTAustin/MAT/0057/2008, and UTACMU/MAT/ 0007/2009.
G. Terrone was supported by the UTAustinPortugal partnership through the FCT postdoctoral fellowship SFRH/BPD/40338/2007, CAMGSDLARSys through FCT Program POCTI  FEDER and by grants PTDC/MAT/114397/2009, UTAustin/MAT/0057/2008, and UTACMU/MAT/0007/2009.
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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/s0003001302430
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DOI: https://doi.org/10.1007/s0003001302430
Mathematics Subject Classification (2010)
 Primary 35F21
 37J50
 Secondary 35D40
 35R05
 49K20
Keywords
 Discontinuous Lagrangians
 Action minimizing measures
 Hamilton–Jacobi equations
 Viscosity solutions