The now famous infnity laplacian is the equations $$ \langle D^2u Du,Du\rangle=0 $$ and the normalized infnity laplacian is $$ \langle D^2u Du/|Du|,Du/|Du|\rangle=0. $$ Is a viscosity solution of one PDE a solution of another PDE? And what about the respective inhomogenuous equations? I would appreciate a proof or references.
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$\begingroup$ It's a bit hard to understand what exactly you are asking. For the RHS $0$ case, they are automatically equivalent under the normal definitions of viscosity solution (touch by any smooth function for top, one having nonzero gradient for bottom): if you touch from one side by a smooth function, the two inequalities are equivalent for smooth functions; this is clear if the gradient is nonzero, while if it is zero the top one is vacuous while the bottom one isn't checked. With a nonzero RHS, they are just two different PDE, the difference has nothing to do with viscosity solutions per se. $\endgroup$– user378654Commented Aug 25, 2023 at 17:07
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