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

Abstract

ABSTRACT. Suppose that f:D -• Rn is an «-dimensional J^-quasiconformal 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 letwhere 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 homeomorphism 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. 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.

Lp-Mean coercivity, regularity and relaxation in the calculus of variations

“…The proof is complete. where Ä is a constant depending on f and p: By Gehring's reverse H older inequality estimates [13], we conclude that {h j } is bounded in L s loc (B) for some s ¿ r and hence {b j } is bounded in W 1;p+ 0 loc (B) for some 0 ¿ 0 depending only on f and p: We have thus proved Theorem 4.2. Let s = p + 0 =2: We claim that A ∈ Z[g] for any quasiconvex function g satisfying…”

“…This can be considered as a global version of the aforementioned results on the higher integrability of energy minimizers (see, e.g., [13][14][15]17,20,24]). …”

“…This type of higher integrability results, pioneered by Gehring's celebrated work [13], has been well-known for the energy minimizers of variational integrals under certain pointwise growth conditions (see [14,15,20,24,23]). Indeed, by adapting the Caccioppoli-type estimates as in Meyers and Elcrat [24] and Giaquinta and Giusti [15], Theorem 2.1 is proved by the well-known technique of reverse H older inequalities of Gehring [13]. The proof here, however, requires a careful treatment since f p does not satisfy the usual pointwise growth condition.…”

Doubling measures, monotonicity, and quasiconformality

“…For example, if f : R n → R n is quasiconformal, then Df n is a doubling weight, and moreover an A ∞ weight [10]. Consequently, Df is doubling as well.…”

Mappings of finite distortion: Reverse inequalities for the Jacobian

“…Indeed, relying on the fundamental work of Gehring [9], Elcrat and Meyers [5] and Martio [29] proved the reverse Hölder inequality for reverse inequalities different from (1.1). We establish the reverse inequality…”