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A basic PDE I would like to understand much better is the viscous Hamilton-Jacobi equation, such as: \begin{equation*} u - \epsilon \Delta u + H(Du) = f(x) \end{equation*} or \begin{equation*} u_{t} - \epsilon \Delta u + H(Du) = f(x) \end{equation*} with or without boundary conditions in the stationary case, or the Cauchy problem in the time-dependent case. I'm interested in the case when $\epsilon > 0$.

Very general viscosity solutions theory implies these equations have continuous solutions under mild assumptions on $H$ and $f$. However, my understanding is the Laplacian term should give us much better regularity than just continuity.

This is a relatively basic example and quite well-motivated from the point of view of stochastic control theory, but nonetheless I'm having trouble finding a down-to-earth reference that shows how to establish regularity for these equations without throwing in the kitchen sink. (In other words, I'm looking for a reference at the level of lecture notes so that I can avoid a little longer wading through Gilbarg-Trudinger or the parabolic equivalent.). It would be particularly nice if the reference in question used a fixed point theorem argument to get existence and regularity simultaneously, but I'm open to an alternative approach.

Is anyone aware of lectures notes that explain how to establish regularity for these equations? Alternatively, are there papers where this is explained in a compact way? My complaint as a student here is this is touched on only very briefly in Evans (in the discussion of fixed point theorems) and the more advanced textbooks on this strike me as extremely dense and somewhat old-fashioned. I may as well start working my way into those books, but if I can get a head start with something more concrete it would be nice.

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  • $\begingroup$ I'm not aware of any notes along these lines, and I don't think you will find a proof that avoids diving a bit into Gilbarg-Trundinger. The basic thing to try is a fixed point argument in, say, $C^{k,\alpha}$, solving iteratively $u_{k+1}-\epsilon \Delta u_{k+1} + H(Du_{k}) = f$. You will need to use the Schauder estimates from Gilbarg-Trudinger, and some a priori gradient estimate (via the maximum principle). $\endgroup$ – Jeff Jun 24 '18 at 11:53
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You may consult the following lecture notes:

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