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Zbl 0734.35033
Crandall, Michael G.
Semidifferentials, quadratic forms and fully nonlinear elliptic equations of second order. (English)
[J] Ann. Inst. Henri Poincaré, Anal. Non Linéaire 6, No.6, 419-435 (1989). ISSN 0294-1449

Let $\Omega$ be a bounded open subset of $\bbfR\sp n$, $S\sp{n\times n}$ be the set of real symmetric $n\times n$ matrices and $u,v: {\bar\Omega}\to \bbfR$. The main result of this paper is the following theorem. Let u be bounded and upper-semicontinuous and v be bounded and lower-semicontinuous. Let $\lambda >0$ and $(\hat x,\hat y)\in {\bar\Omega}\times {\bar\Omega}$ satisfy $$ u(x)-v(y)- \frac{\lambda}{2}\Vert x-y\Vert\sp 2\le u(\hat x)-v(\hat y)- \frac{\lambda}{2}\Vert \hat x-\hat y\Vert\sp 2 \text{ for } (x,y)\in {\bar \Omega}\times {\bar \Omega}. $$ Then there are $X,Y\in S\sp{n\times n}$ such that $$ (1)\quad -4\lambda \left( \matrix I & 0\\ 0 & I\endmatrix \right)\le \pmatrix X & 0\\ 0 & -Y \endpmatrix\le 2\lambda \pmatrix I & -I\\ -I & I \endpmatrix $$ and $(u(\hat x)$, $\lambda(\hat x-\hat y),X)\in \bar D\sp{2,+}u(\hat x)$, $(v(\hat y)$, $\lambda(\hat x- \hat y),Y)\in \bar D\sp{2,-}v(\hat y)$, where $\bar D\sp{2,+}u(\hat x)$ $(\bar D\sp{2,-}u(\hat y))$ denotes the closure of the set of second order superdifferentials (respectively, sub-differentials) of u (respectively, v) at $\hat x$ (respectively, $\hat y$). Moreover, there is a $Z\in S\sp{n\times n}$ such that (1) holds with $X=Y=Z$ and $- \lambda I\le Z\le \lambda I$. (Here orderings are in the sense of quadratic forms). \par From this theorem the author obtains comparison result for viscosity solutions of fully nonlinear second order elliptic equations. He also formulates a version of the above theorem appropriate to the discussion of fully nonlinear parabolic equations.
[G.V.Jaiani (Tbilisi)]

MSC 2000:: *35J65 (Nonlinear) BVP for (non)linear elliptic equations
26B05 Continuity and differentiation questions (several real variables)
35B05 General behavior of solutions of PDE
35J70 Elliptic equations of degenerate type
35K60 (Nonlinear) BVP for (non)linear parabolic equations
35K65 Parabolic equations of degenerate type

Keywords: superdifferentials; sub-differentials; viscosity solutions; fully nonlinear

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