Existence and uniqueness of H-systems solutions with Dirichlet conditions

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

JOURNAL OF NONLINEAR ANALYSIS

Año: 2000 vol. 42 p. 673 - 677

0362-546X

Let $B= {(u, v)\in R^2: u^2+ v^2< 1}$ be the unit disc. The main purpose of this paper is to find a vector function $u:B \rightarrow R^3$ satisfying the prescribed mean curvature equation $\Delta X= 2H(u,v) X_\times X_v$ in $B$, and $X= g$ on $\partial B$, where $\times $ denotes the exterior product and $H:B \rightarrow R$ is a given continuous function, while $g$ is in a Sobolev space $W^{2,p}(B,R^3)$. Using Sobolev immersion the authors transform the above problem into a contraction mapping on the Sobolev space $W^{1,p}$, and then derive the existence and local uniqueness for a solution of the mean curvature equation