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

Let Ω be a bounded open subset of n , S n×n be the set of real symmetric n×n matrices and u,v:Ω ¯. 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 λ>0 and (x ^,y ^)Ω ¯×Ω ¯ satisfy

u(x)-v(y)-λ 2x-y 2 u(x ^)-v(y ^)-λ 2x ^-y ^ 2 for(x,y)Ω ¯×Ω ¯·

Then there are X,YS n×n such that

(1)-4λI00IX00-Y2λI-I-II

and (u(x ^), λ(x ^-y ^),X)D ¯ 2,+ u(x ^), (v(y ^), λ(x ^-y ^),Y)D ¯ 2,- v(y ^), where D ¯ 2,+ u(x ^) (D ¯ 2,- u(y ^)) denotes the closure of the set of second order superdifferentials (respectively, sub-differentials) of u (respectively, v) at x ^ (respectively, y ^). Moreover, there is a ZS n×n such that (1) holds with X=Y=Z and -λIZλI. (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:
35J65Nonlinear boundary value problems for linear elliptic equations
26B05Continuity and differentiation questions (several real variables)
35B05Oscillation, zeros of solutions, mean value theorems, etc. (PDE)
35J70Degenerate elliptic equations
35K60Nonlinear initial value problems for linear parabolic equations
35K65Parabolic equations of degenerate type