The Mather problem for lower semicontinuous Lagrangians

Abstract

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. Mathematics Subject Classification (2010). List of symbolsThe canonical basis of R . , ∂ xN ψ) x(t),ẍ(t)The first and second derivative of a functionThe tangent space of a smooth manifold M at the pointThe Dirac mass concentrated at x 0 μ X (x)The projection on X of a measure μ(x, y) onThe jump of the function r : I ⊂ R → R at t 0 , that is lim t→t

A novel approach for reconstructing water surface using shape from shading technique

“…Utilizing the linear wave theory a wave function can be specified as follows u(x, y) = α sin (k.r + ωt + ϕ) (2) where α, k, r, ω and ϕ are amplitude, wave number, position vector, angular frequency and phase of the traveling wave respectively. Here we have k = (k cos θ, k sin θ) and |k| = 2π λ where λ is wavelength.…”

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