We consider second order elliptic divergence form systems with complex measurable coefficients$A$that are independent of the transversal coordinate, and prove that the set of$A$for which the boundary value problem with$L$_{2}Dirichlet or Neumann data is well posed, is an open set. Furthermore we prove that these boundary value problems are well posed when$A$is either Hermitean, block or constant. Our methods apply to more general systems of partial differential equations and as an example we prove perturbation results for boundary value problems for differential forms.

We consider the inverse curvature flow of strictly convex hypersurfaces in the space form N of constant sectional curvature K_N with speed given by F^{−α}, where α∈(0,1] for K_N=0,−1 and α=1 for K_N=1, F is a smooth, symmetric homogeneous of degree one function which is inverse concave and has dual F_∗ approaching zero on the boundary of the positive cone Γ_+. We show that the ratio of the largest principal curvature to the smallest principal curvature of the flow hypersurface is controlled by its initial value. This can be used to prove the smooth convergence of the flows.

Let (M, g) be a complete 3-dimensional asymptotically flat manifold with everywhere positive scalar curvature. We prove that, given a compact subset K  M, all volume preserving stable constant mean curvature surfaces of sufficiently large area will avoid K. This complements the results of G. Huisken and S.-T. Yau [17] and of J. Qing and G. Tian [26] on the uniqueness of large volume preserving stable constant mean curvature spheres in initial data sets that are asymptotically close to Schwarzschild with mass m > 0. The analysis in [17] and [26] takes place in the asymptotic regime of M. Here we adapt ideas from the minimal surface proof of the positive mass theorem [32] by R. Schoen and S.-T. Yau and develop geometric properties of volume preserving stable constant mean curvature surfaces to handle surfaces that run through the part of M that is far from Euclidean.

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Let$H$^{∞}be the algebra of bounded analytic functions in the unit disk$D$. Let$I=I(f$_{1},...,$f$_{N}) be the ideal generated by$f$_{1},...,$f$_{N}∈$H$^{∞}and$J=J(f$_{1},...,$f$_{N}) the ideal of the functions$f∈H$^{∞}for which there exists a constant$C=C(f)$such that |$f(z)|≤C(|f$_{1}$(z)|+$...;$+|f$_{N}$(z)$|),$z$∈$D$. It is clear that $$I \subseteq J$$ , but an example due to J. Bourgain shows that$J$is not, in general, in the norm closure of$I$. Our first result asserts that$J$is included in the norm closure of$I$if$I$contains a Carleson-Newman Blaschke product, or equivalently, if there exists$s$>0 such that $$\mathop {\inf }\limits_{z \in D} \sum\limits_{k = 0}^s {(1 - |z|)^k } \sum\limits_{j = 1}^N {|f_j^{(k)} (z)| > 0.} $$