I have a conceptual misunderstanding about Chapter 10 of Evans' book on PDEs. It seems the author wants to show that for convex Hamiltonians, the Hamilton-Jacobi equation $u_t+H(Du,x)=0$ has a viscous solution and ends up showing that the equation $u_t+\min\limits_{a\in A}\{f(x,a)\cdot Du+r(x,a)\}=0$ has a viscous solution. Does this prove that there is a viscous solution of the Hamilton-Jacobi equation for every convex Hamiltonians or just for the particular case $H(p,x)=\min\limits_{a\in A}\{f(x,a)\cdot p+r(x,a)\}$? I don't know much about the subject so I might have missed something here.
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