Pointwise properties of <inline-formula><tex-math id="M1">\begin{document}$ L^p $\end{document}</tex-math></inline-formula>-viscosity solutions of uniformly elliptic equations with quadratically growing gradient terms

Abstract

<p style='text-indent:20px;'>We prove various pointwise properties of <inline-formula><tex-math id="M2">\begin{document}$ L^p $\end{document}</tex-math></inline-formula>-viscosity solutions of fully nonlinear uniformly elliptic equations with unbounded measurable terms and possibly quadratically growing gradient terms. These include differentiability properties and "pointwise maximum principle". In particular we discuss an equivalent pointwise definition of <inline-formula><tex-math id="M3">\begin{document}$ L^p $\end{document}</tex-math></inline-formula>-viscosity solution which, together with pointwise properties, allows to operate on <inline-formula><tex-math id="M4">\begin{document}$ L^p $\end{document}</tex-math></inline-formula>-viscosity solutions almost as if they were strong solutions and move them freely from one equation to another. Such results were proved before in [<xref ref-type="bibr" rid="b6">6</xref>,<xref ref-type="bibr" rid="b8">8</xref>,<xref ref-type="bibr" rid="b30">30</xref>] for equations with linearly growing gradient terms, however it appears that they have not been widely used. Here we generalize pointwise properties of <inline-formula><tex-math id="M5">\begin{document}$ L^p $\end{document}</tex-math></inline-formula>-viscosity solutions to a larger class of equations and show that they create a very powerful set of tools which can be used for instance to prove regularity results for equations and differential inequalities.