# Properties of discontinuous viscosity solutions to PDEs with discontinuous coefficients

I am currently learning viscosity solutions. In the lecture notes by N. Katzourakis, he introduced the theories of continuous viscosity solution to PDE with continuous nonlinearitiy. And in Chapter 9, he mentioned that neither the solutions nor the coefficients is actually needed to be continuous. He pointed out that all the results hold true if the solution and the coefficient are replaced by their semi-continuous envelopes.

For example, for degenerate elliptic PDE $F(\cdot, u,Du,D^2u)=0$ with $$F(x,r,p,X)\le F(x,r,p,Y),\quad \forall X\le Y\in \mathbb{S}(n),$$ a function $u:\Omega\to \mathbb{R}$ is a viscosity subsolution on $\Omega$ if for all $x\in \Omega$, and $(p,X)\in \mathcal{J}^{2,+}u^*(x)$, we have $$F^*(x,u^*(x),p,X)\ge 0.$$ Here $u^*$ and $F^*$ denote the upper-semicontinuous envelops of $u$ and $F$, respectively.

I was wondering what does he mean by "all the results"? Does he mean all the properties of Jets, the stability results and well-posedness holds for this definition? Can you recommend a reference which discuss the properties of viscosity solutions in this sense?

• Yes, I think he means properties of jets, stability results and so on. The standard reference is probably the "Users' guide" by Crandall, Ishii and Lions. For first order equations you can also see the book by Bardi and Capuzzo-Dolcetta. – Rigel May 31 '17 at 7:36