These notes record and slightly modify my 5 lectures from the CIME conference on “Calculus of variations and nonlinear partial differential equations”, held in Cetraro during the week of June 27 - July 2, 2005, organized by Bernard Dacorogna and Paolo Marcellini. I am proud to brag that this was the third CIME course I have given during the past ten years, the others at the meetings on “Viscosity solutions and applications” (Montecatini Terme, 1995) and on “Optimal transportation and applications” (Martina Franca, 2001).
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References
L. Ambrosio, Lecture notes on optimal transport problems, in Mathemat- ical Aspects of Evolving Interfaces 1-52, Lecture Notes in Mathematics 1812, Springer, 2003.
N. Anantharaman, Gibbs measures and semiclassical approximation to action-minimizing measures, Transactions AMS, to appear.
E. N. Barron, Viscosity solutions and analysis in L∞ , in Nonlinear Analy-sis, Differential Equations and Control, Dordrecht, 1999.
P. Bernard and B. Buffoni, The Monge problem for supercritical Mañé potentials on compact manifolds, to appear.
P. Bernard and B. Buffoni, Optimal mass transportation and Mather theory, to appear.
P. Bernard and B. Buffoni, The Mather-Fathi duality as limit of optimal transportation problems, to appear.
M. Concordel, Periodic homogenization of Hamilton-Jacobi equations I: additive eigenvalues and variational formula, Indiana Univ. Math. J. 45 (1996),1095-1117.
M. Concordel, Periodic homogenization of Hamilton-Jacobi equations II: eikonal equations, Proc. Roy. Soc. Edinburgh 127 (1997), 665-689.
L. C. Evans, Partial Differential Equations, American Mathematical Society, 1998. Third printing, 2002.
L. C. Evans, Periodic homogenization of certain fully nonlinear PDE, Proc Royal Society Edinburgh 120 (1992), 245-265.
L. C. Evans, Some new PDE methods for weak KAM theory, Calculus of Variations and Partial Differential Equations, 17 (2003), 159-177.
L. C. Evans, Towards a quantum analogue of weak KAM theory, Communications in Mathematical Physics, 244 (2004), 311-334.
L. C. Evans, Three singular variational problems, in Viscosity Solutions of Differential Equations and Related Topics, Research Institute for the Mathematical Sciences, RIMS Kokyuroku 1323, 2003.
L. C. Evans, A survey of partial differential equations methods in weak KAM theory, Communications in Pure and Applied Mathematics 57 (2004),445-480.
L. C. Evans and D. Gomes, Effective Hamiltonians and averaging for Hamiltonian dynamics I, Archive Rational Mech and Analysis 157 (2001), 1-33.
L. C. Evans and D. Gomes, Effective Hamiltonians and averaging for Hamiltonian dynamics II, Archive Rational Mech and Analysis 161 (2002),271-305.
A. Fathi, Théorème KAM faible et theorie de Mather sur les systemes lagrangiens, C. R. Acad. Sci. Paris Sr. I Math. 324 (1997), 1043-1046.
A. Fathi, Solutions KAM faibles conjuguees et barrieres de Peierls, C. R. Acad. Sci. Paris Sr. I Math. 325 (1997), 649-652.
A. Fathi, Orbites heteroclines et ensemble de Peierls, C. R. Acad. Sci. Paris Sr. I Math. 326 (1998), 1213-1216.
A. Fathi, Sur la convergence du semi-groupe de Lax-Oleinik, C. R. Acad. Sci. Paris Sr. I Math. 327 (1998), 267-270.
A. Fathi, The Weak KAM Theorem in Lagrangian Dynamics (Cambridge Studies in Advanced Mathematics), to appear.
A. Fathi and A. Siconolfi, Existence of C 1 critical subsolutions of the Hamilton-Jacobi equation, Invent. Math. 155 (2004), 363-388.
A. Fathi and A. Siconolfi, PDE aspects of Aubry-Mather theory for quasi- convex Hamiltonians, Calculus of Variations and PDE 22 (2005), 185-228.
A. Fathi and A. Siconolfi, Existence of solutions for the Aronsson-Euler equation, to appear.
G. Forni and J. Mather, Action minimizing orbits in Hamiltonian systems, in Transition to Chaos in Classical and Quantum Mechanics, Lecture Notes in Math 1589, edited by S. Graffi, Springer, 1994.
H. Goldstein, Classical mechanics (2nd ed), Addison-Wesley, 1980.
D. Gomes, Viscosity solutions of Hamilton-Jacobi equations and asymptotics for Hamiltonian systems, Calculus of Variations and PDE 14 (2002), 345-357.
D. Gomes, Perturbation theory for Hamilton-Jacobi equations and stability of Aubry-Mather sets, SIAM J. Math. Analysis 35 (2003), 135-147.
D. Gomes, A stochastic analog of Aubry-Mather theory, Nonlinearity 10 (2002),271-305.
D. Gomes and A. Oberman, Computing the effective Hamiltonian using a variational approach, SIAM J. Control Optim. 43 (2004), 792-812.
R. Iturriaga and H. Sanchez-Morgado, On the stochastic Aubry-Mather theory, to appear.
V. Kaloshin, Mather theory, weak KAM and viscosity solutions of Hamilton-Jacobi PDE, preprint.
P.-L. Lions, G. Papanicolaou, and S. R. S. Varadhan, Homogenization of Hamilton-Jacobi equations, unpublished, circa 1988.
R. Mañé, Global Variational Methods in Conservative Dynamics, Instituto de Matemática Pura e Aplicada, Rio de Janeiro.
P. Marcellini, Regularity for some scalar variational problems under general growth conditions, J Optimization Theory and Applications 90 (1996),161-181.
P. Marcellini, Everywhere regularity for a class of elliptic systems without growth conditions, Annali Scuola Normale di Pisa 23 (1996), 1-25.
E. Mascolo and A. P. Migliorini, Everywhere regularity for vectorial functionals with general growth, ESAIM: Control, Optimization and Calculus of Variations 9 (2003), 399-418.
J. Mather, Minimal measures, Comment. Math Helvetici 64 (1989), 375-394.
J. Mather, Action minimizing invariant measures for positive definite La-grangian systems, Math. Zeitschrift 207 (1991), 169-207.
I. Percival and D. Richards, Introduction to Dynamics, Cambridge University Press, 1982.
C. Villani, Topics in Optimal Transportation, American Math Society, 2003.
C. E. Wayne, An introduction to KAM theory, in Dynamical Systems and Probabilistic Methods in Partial Differential Equations, 3-29, Lectures in Applied Math 31, American Math Society, 1996.
Y. Yu, L∞ Variational Problems, Aronsson Equations and Weak KAM Theory, thesis, University of California, Berkeley, 2005.
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Evans, L.C. (2008). Weak KAM Theory and Partial Differential Equations. In: Dacorogna, B., Marcellini, P. (eds) Calculus of Variations and Nonlinear Partial Differential Equations. Lecture Notes in Mathematics, vol 1927. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-75914-0_4
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