散度-旋度场的正则性及其应用

摘要

The aim of the present paper is to prove higher integrability results for div-curl vector fields (B, E) ∈ Lq(1-ε)loc (Ω, Rn) × Lq(1-ε)loc (Ω, Rn), 1 ＜ p, q ＜ ∞, 1/p + 1/q = 1, ε sufficiently small, such that div B = 0, curl E = 0 satisfying a reverse inequality of the type ｜ B ｜q + ｜ E ｜p≤ C(B, E)+ ｜ F ｜q,with F ∈Lr(Ω, Rn),r ＞ q(l - ε). Applications to the theory of weak quasiregular mappings and very weak solutions of nonhomogeneous A-harmonic equations divA(x, ▽u) = div Fare given.%证明了散度-旋度向量场(B,E)∈Lq(1-ε)loc(Ω,Rn)×Lp(1-ε)loc(Ω,Rn)的高阶可积性,这里1＜p,q＜∞,1/p+1/q=1,ε充分小,divB=0,curlE=0满足逆不等式| B |q+| E |p≤ C〈B,E〉+| F |q,其中F∈Lr(Ω,Rn),r＞q(1-ε).给出了上述结果在弱拟正则映射和非齐次A-调和方程divA(x,▽u)=divF很弱解中的应用.