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Solvability of uniformly elliptic fully nonlinear PDE. (English) Zbl 1193.35054

The author studies uniqueness and non-uniqueness of viscosity solutions of uniformly elliptic fully nonlinear equations of the Hamilton-Jacobi-Bellman-Isaacs type

$F\left({D}^{2}u,Du,u,x\right)=f\left(x\right)\phantom{\rule{4pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}{\Omega },\phantom{\rule{1.em}{0ex}}u=\psi \left(x\right)\phantom{\rule{4pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\text{on}\phantom{\rule{4.pt}{0ex}}\partial {\Omega }·$

with unbounded ingredients and quadratic growth in the gradient without hypotheses of convexity or properness. Some of the presented results are new even for equations in divergence form.

##### MSC:
 35J60 Nonlinear elliptic equations 35J25 Second order elliptic equations, boundary value problems 35D40 Viscosity solutions of PDE
##### References:
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