Stability of Critica of Parametric Elliptic Functionals

We study the stability of noncompact critica of parametric elliptic functionals with a volume constraint. Let F : U ⊂ S → R be a smooth function. For a smooth, oriented hypersurface X : Σ → E whose Gauss map ν : Σ → S is assumed to lie in U , we define the functional FΛ(X) = ∫ Σ F (ν)dΣ + ΛV (X), Λ ∈ R, (1) where V (X) denotes the algebraic n + 1 dimensional volume enclosed by X, V (X) := 1 n + 1 ∫ Σ 〈X, ν〉 dΣ. Such functionals are used to model anisotropic surface energies. Applications can be found in metallurgy and crystallography. We will impose a convexity condition on the functional: denote by DF and DF the gradient and Hessian of F on S. Then we require that at each point in U the matrix DF + F1 is positive definite. We will refer to the case when U = S as the uniform case. The functional appearing in (1) will be referred to as a parametric elliptic functional, (PEF). By parallel translation in E, DF may be considered as a smooth tangent vector field along X. The Euler-Lagrange equation for the functional FΛ is divΣDF − nHF + Λ = 0 (2) where H is the mean curvature of the immersion. The convexity condition implies that the EulerLagrange equation is absolutely elliptic in the sense of [7]. This implies that a maximum principle analogous to that constant mean curvature surfaces holds. In the uniform case the Euler-Lagrange equation is also uniformly elliptic. Another interpretation of the convexity condition is the following. Consider the embedding G : U → E defined by G(ν) = DF (ν) + Fν. Then the convexity condition implies that G defines a smooth, convex surface in E. The surface defined by G is called the Wulff shape of F . An important result known as Wulff’s theorem, though actually proved by Jean Taylor, is that the Wulff shape is the absolute minimizer of the functional F0 among all closed ‘hypersurfaces’ in E enclosing the same (n+1)-volume. Here the term hypersurface can be taken in the sense of the boundary of a set of positive Lebesgue measure. Thus the Wulff shape solves the isoprimetric problem for the functional F0. In [10] we showed that any closed critical hypersurface of a PEF FΛ with nonzero Λ , is, up to translations and homotheties, the Wulff shape. In this paper we will investigate the stability of non-compact critica. After deriving a second variation formula, we show that any complete, orientable, globally stable two dimensional critica of a uniformly elliptic FΛ with nonzero Λ is necessarily compact and is thus, by our previously mentioned result, essentially the Wulff shape.