Boundary regularity and the Dirichlet problem for harmonic maps

“…This is indeed an equivalent question, because Lipschitz mappings can be approximated by smooth mappings in the Sobolev norm. If p ≥ dim M, then smooth mappings are dense in W 1,p (M, N ), by the theorem of Schoen and Uhlenbeck [33,34], but if p < dim M, the answer depends on the topology of manifolds M and N . The following necessary condition for the density is due to Bethuel and Zheng [3].…”

Section: Introductionmentioning

confidence: 99%

“…. = π n−1 (N ) = 0, and dim M ≤ n, then Proposition 1.3 gives density for 1 ≤ p < n, but if p ≥ n we always have density by the result of Schoen-Uhlenbeck [33,34]. Remark 1.19.…”

mentioning

confidence: 99%

Lipschitz homotopy and density of Lipschitz mappings in Sobolev spaces

Hajłasz¹,

Schikorra²

2014

Ann. Acad. Sci. Fenn. Math.

View full text Add to dashboard Cite

Abstract. We construct a smooth compact n-dimensional manifold Y with one point singularity such that all its Lipschitz homotopy groups are trivial, but Lipschitz mappings Lip (S n , Y ) are not dense in the Sobolev space W 1,n (S n , Y ). On the other hand we show that if a metric space Y is Lipschitz (n − 1)-connected, then Lipschitz mappings Lip (X, Y ) are dense in N 1,p (X, Y ) whenever the Nagata dimension of X is bounded by n and the space X supports the p-Poincaré inequality. Lipschitz (n − 1)-connectedness is a stronger condition than vanishing of the first n − 1 Lipschitz homotopy groups as it assumes quantitative estimates of Lipschitz constants.

show abstract

“…Energy-minimizing maps enjoy complete boundary regularity, [98,57,72] with e.g. Ω and g being C 1,α smooth, partly because the boundary tangent maps are necessarily constant.…”

Section: Some Minimizing Tangent Mapsmentioning

confidence: 99%