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Semidifferentials, quadratic forms and fully nonlinear elliptic equations of second order. (English) Zbl 0734.35033

Let ${\Omega }$ be a bounded open subset of ${ℝ}^{n}$, ${S}^{n×n}$ be the set of real symmetric $n×n$ matrices and $u,v:\overline{{\Omega }}\to ℝ$. 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 $\left(\stackrel{^}{x},\stackrel{^}{y}\right)\in \overline{{\Omega }}×\overline{{\Omega }}$ satisfy

$u\left(x\right)-v\left(y\right)-\frac{\lambda }{2}{\parallel x-y\parallel }^{2}\le u\left(\stackrel{^}{x}\right)-v\left(\stackrel{^}{y}\right)-\frac{\lambda }{2}{\parallel \stackrel{^}{x}-\stackrel{^}{y}\parallel }^{2}\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\left(x,y\right)\in \overline{{\Omega }}×\overline{{\Omega }}·$

Then there are $X,Y\in {S}^{n×n}$ such that

$\left(1\right)\phantom{\rule{1.em}{0ex}}-4\lambda \left(\begin{array}{cc}I& 0\\ 0& I\end{array}\right)\le \left(\begin{array}{cc}X& 0\\ 0& -Y\end{array}\right)\le 2\lambda \left(\begin{array}{cc}I& -I\\ -I& I\end{array}\right)$

and $\left(u\left(\stackrel{^}{x}\right)$, $\lambda \left(\stackrel{^}{x}-\stackrel{^}{y}\right),X\right)\in {\overline{D}}^{2,+}u\left(\stackrel{^}{x}\right)$, $\left(v\left(\stackrel{^}{y}\right)$, $\lambda \left(\stackrel{^}{x}-\stackrel{^}{y}\right),Y\right)\in {\overline{D}}^{2,-}v\left(\stackrel{^}{y}\right)$, where ${\overline{D}}^{2,+}u\left(\stackrel{^}{x}\right)$ $\left({\overline{D}}^{2,-}u\left(\stackrel{^}{y}\right)\right)$ denotes the closure of the set of second order superdifferentials (respectively, sub-differentials) of u (respectively, v) at $\stackrel{^}{x}$ (respectively, $\stackrel{^}{y}$). Moreover, there is a $Z\in {S}^{n×n}$ 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).

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.

##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 26B05 Continuity and differentiation questions (several real variables) 35B05 Oscillation, zeros of solutions, mean value theorems, etc. (PDE) 35J70 Degenerate elliptic equations 35K60 Nonlinear initial value problems for linear parabolic equations 35K65 Parabolic equations of degenerate type