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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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