L^p-Harmonic Forms and Estimates on Minimal Hypersurfaces and Square Root of Elliptic Systems with Dvoretzky Theorem

Abstract

We study the minimal hypersurfaces and 𝐿2-, 𝐿𝑝- harmonic 1-forms on submanifolds with finite total curvature on minimal hypersurfaces with finite total curvature, finite index and first eigenvalue of a stable minimal hypersurface. We also study the 𝐿𝑝 p-harmonic 1-forms on submanifolds in a Hadamard manifold. We start by the Hardy inequality for functions vanishing on a part of the boundary to show the square roots of elliptic second order divergence operators on strongly Lipschitz domains on 𝐿2 and 𝐿𝑝 theory hence extended to 𝐿𝑝-estimates for the square root problem for second-order divergence form operators and of elliptic systems with mixed boundary conditions. The small ball probability and pointwise estimates for marginals of convex bodies with the Hastings additivity counterexample by Dvoretzky theorem are characterized. We deal with Dvoretzky theorem on almost spherical sections of convex bodies and for subspaces of the 𝐿𝑝-space