The L^p-integrability of the partial derivatives of a quasiconformal mapping

Abstract

Suppose that f:D-• R n is an «-dimensional J^-quasi-conformal mapping. Then the first partial derivatives off are locally ZZ-integrable in D for/? e [ 1, n + c), where c is a positive constant which depends only on K and n. Suppose that D is a domain in Euclidean rc-space R n where n ^ 2, and that ƒ : D-* R n is a homeomorphism. For each xeDwe let L/x) = lim sup | ƒ (y)-f(x)\/\y-x\, J f (x) = lim sup m(f(B(x, r)))/m(B(x, r)), r->0 where B(x, r) denotes the open ^-dimensional ball of radius r about x and m denotes Lebesgue measure in R n. We call L f (x) and J f (x\ respectively , the maximum stretching and generalized Jacobian for the homeo-morphism ƒ at the point x. These functions are nonnegative and measurable in D, and Lebesgue's theorem implies that J f is locally LMntegrable there. Suppose in addition that ƒ is X-quasiconformal in D. Then L n f ^ KJ f a.e. in D, and thus L f is locally L n-integrable in D. Bojarski has shown in [1] that a little more is true in the case where n = 2, namely that L f is locally L p-integrable in D for p e [2, 2 + c), where c is a positive constant which depends only on K. His proof consists of applying the Calderón-Zygmund inequality [2] to the Hubert transform which relates the complex derivatives of a normalized plane quasiconformal mapping. Unfortunately this elegant two-dimensional argument does not suggest what the situation is when n > 2. The purpose of this note is to announce the following n-dimensional version of Bojarski's theorem. THEOREM. Suppose that D is a domain in R n and that f\D^R n isa K-quasiconformal mapping. Then L f is locally L p-integrable in D for p e [1, n + c\ where c is a positive constant which depends only on K and n. This result is derived from the following two lemmas. The first is an inequality relating the L 1-and L"-means of L f over small n-cubes, while AM S (MOS) subject classifications (1970). Primary 30A60; Secondary 30A86.

The Wiener test for degenerate elliptic equations

“…For example, all functions w(x) of the form \x f , a > -n satisfy either (A^) or (QC). For more details on the nature of these conditions see [2,5], and [6].…”

On the continuity of the polyconvex, quasiconvex and rank-oneconvex envelopes with respect to growth condition

“…In Section 3 we show that: (See [4].) The proof is based on Gehring's lemma on reverse Holder inequality in [6] and a result of Giaquinta and Modica in [7]. We deduce, as a byproduct, that there exist quasiconvex functions / : U NXM >-> R for every TV, M > 1 integers, that are not polyconvex.…”

A priori estimates, geometric effects and asymptotic behavior

“…The curious family of functions of bounded mean oscillations, introduced in [39], has been applied and studied extensively (see [40]- [46], and the references given there). More recently, it has been linked to another class of mappings u characterized by an a priori inequality for iï: H. M. Reimann [47] proves that the logarithm of the Jacobian determinant of a quasiconformai mapping is of bounded mean oscillation.…”