Caffarelli, Luis; Nirenberg, Louis; Spruck, Joel Nonlinear second-order elliptic equations. V: The Dirichlet problem for Weingarten hypersurfaces. (English) Zbl 0672.35028 Commun. Pure Appl. Math. 41, No. 1, 47-70 (1988). [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 Cited in 4 ReviewsCited in 87 Documents 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 Keywords:Weingarten hypersurfaces; Dirichlet problem; smooth strictly convex boundary; principal curvatures; Dirichlet boundary condition; existence; uniqueness Citations:Zbl 0654.35031 × Cite Format Result Cite Review PDF Full Text: DOI 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. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.