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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
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