Syllabus: Sobolev spaces. Weak solutions in Sobolev spaces of elliptic PDE in divergence form. Solvability of linear elliptic equations in divergence form and regularity of the weak solutions. “Bootstrap” methods for regularity of the weak solutions of nonlinear equations. Variational characterization of the eigenvalues of a self-adjoint elliptic operator of second order. Variational formulation of solutions of divergence-form PDE. Methods for searching critical points - direct minimization, “mountain-pass” and “linking” type theorems. Notion of weak viscosity solution of an elliptic PDE. Existence and regularity of the viscosity solutions of general nonlinear elliptic PDE.
1) Gilbarg, D.; Trudinger, N.S. Elliptic partial differential equations of second order. Classics in Mathematics. Springer-Verlag, Berlin, 2001.
2) Evans, L.C. Partial differential equations. Second edition. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010.
3) Han, Qing; Lin, Fanghua. Elliptic partial differential equations. Second edition. Courant Lecture Notes in Mathematics, 1. Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2011.
1) Willem, M. Minimax theorems. Progress in Nonlinear Differential Equations and their Applications, 24. Birkhäuser Boston, Inc., Boston, MA, 1996.
2) Struwe, M. Variational methods. Applications to nonlinear partial differential equations and Hamiltonian systems. Fourth edition. A Series of Modern Surveys in Mathematics, 34. Springer-Verlag, Berlin, 2010.
3) Caffarelli, L.A.; Cabré, X. Fully nonlinear elliptic equations. American Mathematical Society Colloquium Publications, 43. American Mathematical Society, Providence, RI, 1995.
4) Crandall, M.G.; Ishii, H.i; Lions, P.-L. User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 1, 1–67.