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A description of the stability cones generated by differential operators of Monge-Ampère type. (English. Russian original) Zbl 0556.35026
Math. USSR, Sb. 50, 259-268 (1985); translation from Mat. Sb., Nov. Ser. 122(164), No. 2, 265-275 (1983).
In a domain $$\Omega \subset R^ n$$ consider nonlinear second order differential operators. Let $$F_ m(u)=spur_ m(\partial^ 2u/\partial x_ i\partial x_ j)$$ be the sum of the main m-th order minors of the Hessian $$\det (\partial^ 2u/\partial x_ i\partial x_ j),$$ $$m=1,...,n$$. A set $$K\subset C^ 2(\Omega)$$ is called stable for an operator F if any function u(x,t)$$\in K$$ for all $$t>0$$ under the conditions u(x,0)$$\in K$$ and $$F[u(x,t)]>0.$$
Theorem 1. The cone $$K_ m=\{u(x):F_ i(u)>0$$, $$i=1,...,m\}$$ is a stable set for the operator $$F_ m(u)$$, $$1\leq m\leq n$$. - Let $$E_ m$$ be an ellipticity set for the operator $$F_ m$$. Theorem 2. $$K_ n=E_ n$$ and $$K_ m\subset E_ m$$ if $$m<n$$.
Reviewer: G.I.Laptev

##### MSC:
 35G30 Boundary value problems for nonlinear higher-order PDEs 35B35 Stability in context of PDEs
##### Keywords:
differential operators of Monge-Ampère type
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