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Solution of the Dirichlet problem for some equations of Monge-Ampère type. (English. Russian original) Zbl 0609.35042
Math. USSR, Sb. 56, 403-415 (1987); translation from Mat. Sb., Nov. Ser. 128(170), No. 3, 403-415 (1985).
The main theorem is an existence theorem concerning the nonlinear partial differential equation \[ (1.1)\quad F_ m (u)=f(x,u,u_ x)\quad in\quad \Omega;\quad u\equiv 0\quad on\quad \partial \Omega, \] where \(\partial \Omega\) is a strictly convex surface in \({\mathbb{R}}^ n\), \(n\geq 2\) of the class \(C^{\ell +2+\alpha}\), \(\ell \geq 2\), \(0<\alpha <1\). The function f obeys the inequality \(0<\nu \leq f(x,u,p)\leq \psi (| p|)\) with \({\mathcal A}=\{x\in {\bar \Omega},u\in R^ 1,p\in R^ n\}\), \(f(x,u,p)\in C^{\ell +\alpha}({\mathcal A})\), and \(1<mC^{m-1}_{n-1}\kappa^ m_ 0\int^{\infty}_{0}(t^{m-1}/\psi (t))dt\); \(m=1,2\) or 3, while \(\kappa_ 0\) denotes the minimal value of one of the principal curvatures of \(\partial \Omega\), \[ -(n-m)(1-\delta^ 1_ m)(\partial^ 2f^{1/m}(x,u,p)/\partial p^ i\partial p^ j)\xi^ i\xi^ j\leq \epsilon f^{1+1/m}(x,u,p)| \xi |^ 2, \] plus certain growth condition on f for very large values of p. Then the equation (1.1) has a solution. Special cases of (1.1) include the Dirichlet problem for the equation \(\Delta u=f(x,u,u_ x)\), \(m=1\), and the Monge-Ampère equation \(\det (u_{xx})=f(x,u,u_ x)>0\) with \(m=n.\)
The author indicated in previous work that the natural setting is the cone \[ K_ m(\Omega)=\{u(x)\in C^ 2(\Omega),\quad F_ i(u)>0,\quad i=1,2,...,m\}, \] \(K_ m(\Omega)-convex\). The operator \(F_ m(u)\) is elliptic in \(K_ m(\Omega)\) and \[ (\partial F_ m(u)/\partial u_{ij})\xi^ i\xi^ j\geq (F_ m(u)/| u_{xx}|)| \xi |^ 2,\quad (\partial^ 2F_ m^{1/m}(u)/\partial u_{ij}\partial u_{k\ell})\xi^{ij}\xi^{k\ell}\leq 0, \]
\[ \xi =(\xi^ 1...\xi^ n),\quad | u_{xx}|^ 2=\sum^{n}_{i,j=1}u^ 2_{ij}. \] Using a priori inequalities discussed by the author [Zap. Nauchn. Semin. Leningr. Otd. Mat. Steklova 96, 69-79 (1980; Zbl 0472.35040)], several results are derived relating derivatives of solution to principal curvatures of \(\partial \Omega\). Also the following uniqueness theorem is proved. Let \(f(x,u,p)\in C^ 1({\mathcal A})\), \(\partial f/\partial u\geq 0\), \(f(x,u,p)>0\), \(m=1,2,3,n\), then there exists at most one solution in the cone \(K_ m(\Omega)\).
Reviewer: V.Komkov

MSC:
35J65 Nonlinear boundary value problems for linear elliptic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35G30 Boundary value problems for nonlinear higher-order PDEs
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