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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 \(\mathbb{R}^ n\), \(S^{n\times n}\) be the set of real symmetric \(n\times n\) matrices and \(u,v: {\bar\Omega}\to \mathbb{R}\). 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 \((\hat x,\hat y)\in {\bar\Omega}\times {\bar\Omega}\) satisfy \[ u(x)-v(y)- \frac{\lambda}{2}\| x-y\|^ 2\leq u(\hat x)-v(\hat y)- \frac{\lambda}{2}\| \hat x-\hat y\|^ 2 \text{ for } (x,y)\in {\bar \Omega}\times {\bar \Omega}. \] Then there are \(X,Y\in S^{n\times n}\) such that \[ (1)\quad -4\lambda \left( \begin{matrix} I & 0\\ 0 & I\end{matrix} \right)\leq \begin{pmatrix} X & 0\\ 0 & -Y \end{pmatrix}\leq 2\lambda \begin{pmatrix} I & -I\\ -I & I \end{pmatrix} \] and \((u(\hat x)\), \(\lambda(\hat x-\hat y),X)\in \bar D^{2,+}u(\hat x)\), \((v(\hat y)\), \(\lambda(\hat x- \hat y),Y)\in \bar D^{2,-}v(\hat y)\), where \(\bar D^{2,+}u(\hat x)\) \((\bar D^{2,-}u(\hat y))\) 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 \(Z\in S^{n\times n}\) such that (1) holds with \(X=Y=Z\) and \(- \lambda I\leq Z\leq \lambda I\). (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
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35J70 Degenerate elliptic equations
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35K65 Degenerate parabolic equations
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References:
[1] Aleksandrov, A. D., Almost everywhere existence of the second differential of a convex function and some properties of convex functions, Lenigrad University Annals (Mathematical Series), Vol 37, 3-35, (1939), In Russian
[2] H. Ishii, On Existence and Uniqueness of Viscosity Solutions of Fully Nonlinear Second-Order Elliptic PDE’s, to appear in Comm. Pure Appl. Math. · Zbl 0645.35025
[3] H. Ishii, and P.-L. Lions, Viscosity Solutions of Fully Nonlinear Second-Order Elliptic Partial Differential Equations, to appear in J. Diff. Equa. · Zbl 0708.35031
[4] Jensen, R., The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations, Arch. Rat. Mech. Anal., 101, 1-27, (1988) · Zbl 0708.35019
[5] R. Jensen, Uniqueness Criteria for Viscosity Solutions of Fully Nonlinear Elliptic Partial Differential Equations, preprint. · Zbl 0838.35037
[6] Jensen, R.; Lions, P.-L.; Souganidis, P. E., A uniqueness result for viscosity solutions of second order fully nonlinear partial differential equations, Proc. A.M.S., 102, 975-978, (1988) · Zbl 0662.35048
[7] Lions, P.-L.; Souganidis, P. E., Viscosity solutions of second-order equations, stochastic control and stochastic differential games, Proceedings of Workshop on Stochastic Control and PDE’s, (June 1986), I.M.A., University of Minnesota
[8] N. S. Trudinger, Comparison Principles and Pointwise Estimates for Viscosity Solutions of Nonlinear Elliptic Equations, preprint. · Zbl 0695.35007
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