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?
This is a peculiar claim, since all of the results certainly do not hold in the discontinuous case. The definition of viscosity solution easily extends to the discontinuous case, and one can often prove existence of solutions (say, by a representation formula), but uniqueness is only known in some very special cases. There is definitely not a standard general uniqueness result, and hence no stability, for discontinuous problems in the same way there is for the continuous version.
For example, it is possible to prove uniqueness for certain types of discontinuous first order equations provided the discontinuities are very mild (and usually the solution itself must be continuous). One paper you could look at is Deckelnick, Elliot, 2004 (referenced below), but there are several others. The original definitions for discontinuous Hamiltonians were introduced by Ishii in 1985 in the paper referenced below. This is not covered in the User's guide or many of the standard reference books, as far as I know.
Deckelnick, Klaus, and Charles M. Elliott. "Uniqueness and error analysis for Hamilton-Jacobi equations with discontinuities." Interfaces and free boundaries 6.3 (2004): 329-349.
H. Ishii, Hamilton-Jacobi equations with discontinuous Hamiltonians on arbitrary open sets, Bull. Fac. Sci. Eng. Chuo Univ 28 (28) (1985)