Abstract
We prove partial regularity for local minimisers of certain strictly quasiconvex integral functionals, over a class of Sobolev mappings into a compact Riemannian manifold, to which such mappings are said to be holonomically constrained. Our approach uses the lifting of Sobolev mappings to the universal covering space, the connectedness of the covering space, an application of Ekeland’s variational principle and a certain tangential $${\mathbb{A}}$$ -harmonic approximation lemma obtained directly via a Lipschitz approximation argument. This allows regularity to be established directly on the level of the gradient. Several applications to variational problems in condensed matter physics with broken symmetries are also discussed, in particular those concerning the superfluidity of liquid helium-3 and nematic liquid crystals.
Highlights
Introduction and Statement of Main ResultLet M be a compact submanifold of RN without boundary and ⊂ Rn be a Lipschitz domain
Let M be a compact submanifold of RN without boundary, there exists a tubular neighbourhood O of M in RN
By approximating M locally at a point x as the graph of a quadratic polynomial given by the second fundamental form, one can show for any z ∈ RN with 0 < |z| 1 sufficiently small, using the elementary geometry of the normal line at (x + z) through x + z and the normal line at x along the projection of z onto (Tx M)⊥, that η(x + z) = 1 |Qz|2 + O(|z|3), 2 where the matrix Q represents the orthogonal projection onto the normal space of M at the point x
Summary
Let M be a compact submanifold of RN without boundary and ⊂ Rn be a Lipschitz domain. If u minimises a functional of the form f (Du) dx, where f is a strictly quasiconvex C2-function with p-growth, p 2, that satisfies an additional bound on the second derivatives D2 f , the gradient Du is locally Hölder continuous outside a set of Lebesgue measure zero This result was subsequently generalised to integrals with x- and u-dependences in [39,46] and the bound for D2 f was removed in [3]. Our proof of Theorem 1.2 is direct via an application of Ekeland’s variational principle; a certain tangential A-harmonic approximate lemma; Caccioppoli inequalities using only minimality, the p-growth of (h1) and the strong quasiconvexity of (h2); the Sobolev lifting result of Theorem 4.1 below; and an extension-type result using the -connectedness of the compact covering space M.
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