March 15/16: I proved existence ¡­

March 15/16: I proved existence of weak solutions to a class of nonlinear elliptic equations for which there is no variational formulation. The key point was a monotonicity condition on the nonlinearity, which allowed to use the Browder-Minty argument. I tried to convey an important strategy for proving existence of solutions to nonlinear PDEs without a variational structure, which can be summarized as follows: 1) Construct approximate solutions. 2) Compactness (convergence analysis). 3) Identify the limit from Step 2 as a solution to the (nonlinear) PDE.

The topics next week are uniqueness and fixed point theorems.