Nonlinear second order elliptic equations. IV. Starshaped compact Weingarten hypersurfaces. (English) Zbl 0672.35027

Caffarelli, L.; Nirenberg, L.; Spruck, J.

Current topics in partial differential equations, Pap. dedic. S. Mizohata Occas. 60th Birthday, 1-26 (1986).

[For the entire collection see Zbl 0604.00006.]
[For Part III see Acta Math. 155, 261-301 (1985; Zbl 0654.35031).]
The existence of the embedded Weingarten surface Y: \(S^ n\to R^{n+1}\) is studied, the principal curvatures \([k_ 1,...,k_ n]\) of which satisfy a relation (1) \(f(-k_ 1,...,-k_ n)=\psi (Y).\)
Under the suitable assumptions on f and \(\psi\), the localization of Y as a graph of function v (i.e., \(Y=[x,v(x)]\), \(x=[x_ 1,...,x_ n]\), \(x_{n+1}=v(x))\), transform (1) to the elliptic equation \(G(Dv,D^ 2v)=\psi (x,v)\) (Section 1).
It is proved that there exists a \(C^{\infty}\)-surface which solves (1), as well as the fact that any two solutions are endpoints of a one- parameter family of homothetic dilations, all of which are solutions (Theorem 1). The proof of this result is given by the continuity method (Section 2), which is based on a priori estimates, established in Sections 3,4.

Reviewer: O.John

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:

existence; embedded Weingarten surface; principal curvatures; homothetic dilations; continuity method; a priori estimates