Bi-Sobolev homeomorphisms f with Df and Df(-1) of low rank using laminates

作者：Oliva Marcos

来源：Calculus of Variations and Partial Differential Equations, 2016, 55(6): 135.

DOI：10.1007/s00526-016-1080-x

Let Omega subset of R-n be a bounded open set. Given 1 <= m1, m2 <= n - 2, we construct a homeomorphism f : Omega -> Omega that is Holder continuous, f is the identity on partial derivative Omega, the derivative Df has rank m(1) a.e. in Omega, the derivative Df(-1) of the inverse has rank m(2) a.e. in Omega, Df is an element of W-1,W-p and Df(-1) is an element of W-1,W-q for p < min{m(1) + 1, n - m2}, q < min{m(2) + 1, n - m(1)}. The proof is based on convex integration and laminates. We also show that the integrability of the function and the inverse is sharp.

• 出版日期2016-12