Viscosity solutions for PDEs

This thesis deals with the theory of \textit{viscosity solutions}, a powerful and elegant framework for analyzing partial differential equations (PDEs) that may not admit solutions in the classical sense. The need for such a theory arises naturally when dealing with nonlinear PDEs, where traditional notions of solutions—requiring a certain degree of smoothness—often fall short. Viscosity solutions provide a robust weak solution concept that prioritizes the local geometric behavior of the function, allowing for a much broader and more flexible analysis.