On Surfaces of Prescribed Weighted Mean Curvature

Utilizing a weight matrix we study surfaces of prescribed weighted mean curvature which yield a natural generalisation to critical points of anisotropic surface energies. We first derive a differential equation for the normal of immersions with prescribed weighted mean curvature, generalising a result of Clarenz and von der Mosel. Next we study graphs of prescribed weighted mean curvature, for which a quasilinear elliptic equation is proved. Using this equation, we can show height and boundary gradient estimates. Finally, we solve the Dirichlet problem for graphs of prescribed weighted mean curvature. Introduction Given some open set U ⊂ Rn let X : U → Rn+1 be a smooth immersion. We denote by N : U → Sn its normal vector. In [4] Clarenz and von der Mosel studied critical points of the specific parametric functional A(X) = ∫ U ( F (N) + 〈Q(X), N〉 ) dA . (1) Requiring the homogeneity condition F (tp) = tF (p) for all p ∈ Rn+1 and t > 0, this functional becomes invariant under reparametrisation of the surface. The Euler equation of this functional leads to surfaces X whose weighted mean curvature HF is prescribed by HF = divQ. A simple example is the area functional with F (p) = |p| and Q ≡ 0, leading to surfaces whose mean curvature H vanishes, i.e. minimal surfaces. In case of F (p) = |p| together with some arbitrary Q one obtains surfaces of prescribed mean curvature H = divQ. We will now generalise the class of prescribed weighted mean curvature surfaces: We allow surfaces which do not necessarily arise as critical points of parametric functionals. To this end, let us consider a symmetric (n+ 1)× (n+ 1) weight matrix G = G(p) : R\{0} → R(n+1)×(n+1) . We require two conditions on the weight matrix: First, the ellipticity condition 〈G(p)y, y〉 > 0 for all y ∈ p⊥\{0} (2) i.e. G(p) restricted to the n-dimensional space p⊥ = {y ∈ Rn+1 | 〈y, p〉 = 0} is positive definite. Secondly we assume tG(tp) = G(p) and G(p)p = 0 for all p ∈ R\{0} , t > 0 (3) i.e. G(p) homogeneous of degree −1 and p belongs to the kernel of G(p). Critical points of the functional (1) will be included in our considerations. For that case we just have to define the