Existence and regularity of weak solutions to the prescribed mean curvature equation for a nonparametric surface

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

ABSTRACT AND APPLIED ANALYSIS

Año: 1999 vol. 4 p. 61 - 69

1085-3375

Let $Omega subset {Bbb R}^2$ be a bounded domain and $f in C^2(Omega)$ be given. The mean curvature, $h$, of the nonparametric surface graphed by $f$, $X(u, v) = (u, v, f(u,v))$ is given by $$ (1 + f_v^2) f_{uu} + (1+f_u^2) f_{vv} - 2 f_u f_v f_{uv} = 2h(u, v, f) (1 + | abla f|^2)^{3/2}. ag 1 $$ The authors considered the quasilinear elliptic PDE (1) with Dirichlet boundary condition and proved the existence of weak solutions using variational method. More precisely, for given $h$, the equation (1) is exactly the Euler-Lagrange equation of the following functional to $h$ $$ J_h(f) = int_Omega left((1+| abla f|^2)^{1/2} + H(u,v,f) ight) du dv. $$ Here $f in H^1(Omega), H(u,v,z) = int_0^z 2 h(u,v,t) dt$, and $H^1(Omega)$ is the usual Sobolev space. The authors proved that under some condition on $h$ and boundary data, the functional $J_h$ has a global minimum in a convex subset of $H^1(Omega)$, which provides a weak solution for (1). They also showed a standard regularity theroem for the equation (1)