Strongly Continuous Semigroups of Bounded Linear Operators.
The Hille-Yosida theorem. The Lumer-Phillips theorem. Stone's theorem. The Abstract Cauchy Problem.
Applications, including a wave equation and a Shrödinger equation.
Nonvariational Techniques for Nonlinear PDE's.
Monotonicity Methods. Schauder's and Shaefer's fixed point theorems. Subsolutions and Supersolutions.
Pohozaev's identity. Geometric properties of solutions. The Gidas, Ni, Nirenberg theorem (bounded and unbounded domains). Brouwer degree. Leray-Schauder degree. Applications.
Bibliography
A. Pazy. Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences 44, Springer-Verlag, 1983.
L. C. Evans. PDE's, AMS Graduate Studies in Mathematics, vol. 19, 1998.
Gidas, Ni, Nirenberg. Symmetry and Related Principles via the Maximum Principle, Comm. Math. Phys., 68, pp. 209-243, 1979.
Gidas, Ni, Nirenberg. Symmetry of Positive Solutions of Nonlinear Elliptic Equations in Rn. Mathematical Analysis and Applications, Part A, pp. 369-402, 1981.
I. Fonseca, W. Gangbo. Degree Theory in Analysis and Applications, Oxford Lecture Series in Mathematics and Applications, 1995.