A new proof of regularity of weak solutions of the H -surface equation

Abstract

We give a new proof of a theorem of Bethuel, asserting that arbitrary weak solutions u ∈ W 1,2 (IB, IR 3 ) of the H-surface system ∆u = 2H(u)ux 1 ∧ ux 2 are locally Hölder continuous provided that H is a bounded Lipschitz function. Contrary to Bethuel's, our proof completely omits Lorentz spaces. Estimates below natural exponents of integrability are used instead. (The same method yields a new proof of Hélein's theorem on regularity of harmonic maps from surfaces into arbitrary compact Riemannian manifolds.) We also prove that weak solutions with continuous trace are continuous up to the boundary, and give an extension of these results to the equation of hypersurfaces of prescribed mean curvature in IR n+1 , this time assuming in addition that |∇H(y)| decays at infinity like |y| −1 .

On weak solutions for $p$-Laplacian equations with weights

“…As concerns the results available in the literature, we mention the pioneering papers by Tomi [22], [23] and, among the most recent contributions, the paper [1], where Bethuel takes advantage of some properties of Lorentz spaces in order to get the regularity under the assumption that H is C 1 , bounded and globally Lipschitz on R 3 . The same result is proved with a method based on the Hodge decomposition theorem and on some Morreytype estimates in the recent paper [21] by Strzelecki. We also mention the papers [2] and [3] by Bethuel and Ghidaglia, in which the smooth and bounded curvature H depends only on two variables, or, more generally, it satisfies a suitable decay condition at infinity along a direction in R 3 .…”

“…The main tools in [2], [3] are the co-area formula and the theory of Hardy spaces. The duality between the Hardy space H 1 and the space BMO of functions having bounded mean oscillation is used in an essential way also in [21]. We also refer to this last paper for a complete list of references.…”

Weak limit and blowup of approximate solutions to H-systems

“…We remark that if H ∈ C 1 (R 3 , R) with H and ∇H bounded, the statement (i), which in our case needs a lot of work, would be a consequence of a result by Bethuel [2] (see also [24]). …”

Boundary regularity via Uhlenbeck–Rivière decomposition

“…The interior regularity was proved by T. Rivière in [Riv07] (for e ≡ 0) and our proof is based on Rivière's decomposition result combined with the Dirichlet growth approach by Rivière and Struwe in [RS08] as well as some additional arguments due to P. Strzelecki [Str03]. Remark 1.3 Let us emphasize that one can prove Theorem 1.1 also by reflection across ∂D 2 , whenever there is some ψ ∈ W 2,p (D 2 , R m ), p > 1, such that u = ψ on ∂D 2 .…”

“…In order to exploit this last relation, we have to estimate [v ] BM O appropriately. This can be done by exactly the same calculations as in Step 5 of Strzlecki's article [Str03]: Proposition 2.3 There is a constant C p such that…”