Lecturer: Nikolaos KATZOURAKIS, University of Reading, Reading, Berkshire, UK Webpages: [email protected] | http://www.reading.ac.uk/maths-and-stats/ Title: AN INTRODUCTION TO VISCOSITY SOLUTIONS FOR FULLY NONLINEAR PDES AND APPLICATIONS TO CALCULUS OF VARIATIONS IN L∞ Date and time: Mon, May 6 to Fri, May 10, 2013, 9:00 to 11:00 Abstract: In this course we will present the main concepts and results of the so-called theory of viscosity solutions which applies to fully nonlinear 1st and 2nd order PDEs. For such equations, solutions generally are nonsmooth and standard approaches do not apply: weak, strong, measure-valued and distributional solutions either do not exist or can not even be defined. The name of this theory is after the "vanishing viscosity method" where it first originated, but now constitutes an independent theory of "weak" solutions which applies to nondivergence form PDEs. Interesting PDEs arise for example in Geometry and Geometric Evolution (Monge-Ampere PDEs, Motion by Mean Curvature, Optimal Lipschitz Extensions), Optimal Control and Game Theory (Hamilton-Jacobi-Bellman PDEs, Isaacs PDEs) and Calculus of Variations in Lp and L∞ (Euler- Lagrange PDEs, $p$-Laplacian, Aronsson PDEs, ∞-Laplacian). We will focus in particular on applications in Calculus of Variations in L∞. No previous knowledge will be assumed for the audience and basic graduate-level mathematical maturity will suffice for the main core of this course. Bibliography: [1] M. G. Crandall A Visit with the $\infty$-Laplacian, in Calculus of Variations and Non-Linear Partial Differential Equations, Springer Lecture notes in Mathematics 1927, CIME, Cetraro Italy 2005. [2] M. G. Crandall, L. C. Evans, R. Gariepy, Optimal Lipschitz extensions and the infinity Laplacian, Calc. Var. 13, 123 - 139 (2001). [3] M. G. Crandall, L. C. Evans, P. L. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 282 (1984), 487 - 502. [4] M. G. Crandall, H. Ishii, P.-L. Lions, User's Guide to Viscosity Solutions of 2nd Order Partial Differential Equations, Bulletin of the AMS, Vol. 27, Nr 1, Pages 1 - 67, 1992. [5] M. G. Crandall, P. L. Lions,Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 277 (1983), 1 - 42. [6] N. I. Katzourakis, Explicit Singular Viscosity Solutions of the Aronsson Equation, C. R. Acad. Sci. Paris, Ser. I 349, No. 21 - 22, 1173-1176 (2011). [7] N. I. Katzourakis, Contact Solutions for Nonlinear Systems of Partial Differential Equations, preprint, 2012. 8] P. L. Lions, Generalized solutions of Hamilton-Jacobi equations, Research Notes in Math. 69, Pitman, London, 1982.

Title: AN INTRODUCTION TO VISCOSITY SOLUTIONS …€¦ · Title: AN INTRODUCTION TO VISCOSITY SOLUTIONS FOR FULLY NONLINEAR ... M. G. Crandall, P. L. Lions,Viscosity solutions of

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Page 1: Title: AN INTRODUCTION TO VISCOSITY SOLUTIONS …€¦ · Title: AN INTRODUCTION TO VISCOSITY SOLUTIONS FOR FULLY NONLINEAR ... M. G. Crandall, P. L. Lions,Viscosity solutions of

Lecturer: Nikolaos KATZOURAKIS, University of Reading, Reading, Berkshire, UK Webpages: [email protected] | http://www.reading.ac.uk/maths-and-stats/ Title: AN INTRODUCTION TO VISCOSITY SOLUTIONS FOR FULLY NONLINEAR PDES AND APPLICATIONS TO CALCULUS OF VARIATIONS IN L∞ Date and time: Mon, May 6 to Fri, May 10, 2013, 9:00 to 11:00 Abstract: In this course we will present the main concepts and results of the so-called theory of viscosity solutions which applies to fully nonlinear 1st and 2nd order PDEs. For such equations, solutions generally are nonsmooth and standard approaches do not apply: weak, strong, measure-valued and distributional solutions either do not exist or can not even be defined. The name of this theory is after the "vanishing viscosity method" where it first originated, but now constitutes an independent theory of "weak" solutions which applies to nondivergence form PDEs. Interesting PDEs arise for example in Geometry and Geometric Evolution (Monge-Ampere PDEs, Motion by Mean Curvature, Optimal Lipschitz Extensions), Optimal Control and Game Theory (Hamilton-Jacobi-Bellman PDEs, Isaacs PDEs) and Calculus of Variations in Lp and L∞ (Euler-Lagrange PDEs, $p$-Laplacian, Aronsson PDEs, ∞-Laplacian). We will focus in particular on applications in Calculus of Variations in L∞. No previous knowledge will be assumed for the audience and basic graduate-level mathematical maturity will suffice for the main core of this course. Bibliography: [1] M. G. Crandall A Visit with the $\infty$-Laplacian, in Calculus of Variations and Non-Linear Partial Differential Equations, Springer Lecture notes in Mathematics 1927, CIME, Cetraro Italy 2005. [2] M. G. Crandall, L. C. Evans, R. Gariepy, Optimal Lipschitz extensions and the infinity Laplacian, Calc. Var. 13, 123 - 139 (2001). [3] M. G. Crandall, L. C. Evans, P. L. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 282 (1984), 487 - 502. [4] M. G. Crandall, H. Ishii, P.-L. Lions, User's Guide to Viscosity Solutions of 2nd Order Partial Differential Equations, Bulletin of the AMS, Vol. 27, Nr 1, Pages 1 - 67, 1992. [5] M. G. Crandall, P. L. Lions,Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 277 (1983), 1 - 42. [6] N. I. Katzourakis, Explicit Singular Viscosity Solutions of the Aronsson Equation, C. R. Acad. Sci. Paris, Ser. I 349, No. 21 - 22, 1173-1176 (2011). [7] N. I. Katzourakis, Contact Solutions for Nonlinear Systems of Partial Differential Equations, preprint, 2012. 8] P. L. Lions, Generalized solutions of Hamilton-Jacobi equations, Research Notes in Math. 69, Pitman, London, 1982.