Solution to the mean curvature equation for nonparametric surfaces by fixed point methods

P. AMSTER, J.P. BORGNA, M. C. MARIANI Y D. RIAL

REVISTA DE LA UNIóN MATEMáTICA ARGENTINA

Año: 1999 vol. 41 p. 15 - 21

0041-6932

The authors consider the existence problem for graphs $z= u(x,y)$ having prescribed mean curvature $H= H(x,y,z)$ in some bounded domain $\Omega\subset R^2$ of class $C^2$ and satisfying the Dirichlet boundary condition $u=\varphi$ on $\partial\Omega$. By some fairly obvious fixed point argument they show the existence of a solution provided that the $C^0(\Omega \times[- \varepsilon ,+\varepsilon])$-norm of $ H $  and the $W^{2,p}(\Omega)$-norm of $\varphi$ are sufficiently small for some $\varepsilon> 0$ and $p>2$.