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Nonlinear second-order elliptic equations. V: The Dirichlet problem for Weingarten hypersurfaces. (English) Zbl 0672.35028

[For part IV see the preceding review.]
Here is studied the Dirichlet problem for a function u in a bounded domain \(\Omega\) in \({\mathbb{R}}^ n\) with smooth strictly convex boundary \(\partial \Omega\). At any point x in \(\Omega\) the principal curvatures \(\kappa =(\kappa_ 1,...,\kappa_ n)\) of the graph (x,u(x)) satisfy a relation \((1)\quad f(\kappa_ 1,...,\kappa_ n)=\psi (x)>0,\) where \(\psi\) is a given smooth positive function on \({\bar \Omega}\). The function u satisfies the Dirichlet boundary condition \((2)\quad u=0\quad on\quad \partial \Omega.\)
The existence and the uniqueness of the solution of (1), (2) with some special properties is proved under appropriate assumptions on f.
Reviewer: P.Drábek

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35B65 Smoothness and regularity of solutions to PDEs
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
53A05 Surfaces in Euclidean and related spaces

Citations:

Zbl 0654.35031
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References:

[1] and , Inequalities, Erg. der Math. und ihrer Grenzgebiete, Vol. 30, 1961, Springer Verlag, Berlin, Göttingen, New York.
[2] Caffarelli, Comm. Pure Appl. Math. 38 pp 209– (1985)
[3] Caffarelli, Acta Math. 155 pp 261– (1985)
[4] , and , Nonlinear second order equations IV. Starshaped compact Weingarten hypersurfaces. Current topics in partial differential equations, ed. by , , 1986, pp. 1–26, Kinokunize Co., Tokyo.
[5] , and , On a form of Bernstein’s theorem, to appear.
[6] A priori interior gradient bounds for solutions to elliptic Weingarten surfaces, to appear.
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