The Dirichlet problem for the prescribed curvature equations. (English) Zbl 0721.35018

This paper considers the Dirichlet problem for a class of fully nonlinear elliptic equations. If \(\kappa =(\kappa_ 1,...,\kappa_ n)\) is the vector of principal curvatures of the graph of the unknown function u, these equations have the form \(f(\kappa)=\psi (x)\) for some smooth, symmetric function f and smooth positive function \(\psi\). Special cases include the equations of prescribed mean, scalar, Gauss, and harmonic curvatures which correspond to f(\(\kappa\)) having the forms \[ \sum^{n}_{i=1}\kappa_ i,\quad \sum^{n}_{i=1}\sum^{n}_{j=i+1}\kappa_ i\kappa_ j,\quad \prod^{n}_{i=1}\kappa_ i,\quad (\sum^{n}_{i=1}1/\kappa_ i)^{-1}, \] respectively. The precise conditions on f are rather technical, but they are analogous to conditions introduced by L. Caffarelli, L. Nirenberg, and J. Spruck [Acta Math. 155, 261-301 (1985; Zbl 0654.35031)], who considered the eigenvalues of the Hessian \(D^ 2u\) rather than \(\kappa\). Under a natural condition connecting the principal curvatures of the boundary of the domain and the equation, the author proves that the Dirichlet problem has a unique Lipschitz viscosity solution.


35J65 Nonlinear boundary value problems for linear elliptic equations
35B45 A priori estimates in context of PDEs


Zbl 0654.35031
Full Text: DOI


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