We call that λ(x)belongs to Muckenhoupt class A_p if λ(x)≥0 and satisfiesfor all balls in R,where|B_R| denotes the Lebesgue measure of a ball with centerx_0 and radius R,andLet be a bounded domain.Define normHere,d...We call that λ(x)belongs to Muckenhoupt class A_p if λ(x)≥0 and satisfiesfor all balls in R,where|B_R| denotes the Lebesgue measure of a ball with centerx_0 and radius R,andLet be a bounded domain.Define normHere,denotes the Banach space of all measurable functions defined展开更多
Let us consider the following elliptic systems of second order-D_α(A_i~α(x, u, Du))=B_4(x, u, Du), i=1, …, N, x∈Q(?)R^n, n≥3 (1) and supposeⅰ) |A_i~α(x, u, Du)|≤L(1+|Du|);ⅱ) (1+|p|)^(-1)A_i~α(x, u, p)are H(?...Let us consider the following elliptic systems of second order-D_α(A_i~α(x, u, Du))=B_4(x, u, Du), i=1, …, N, x∈Q(?)R^n, n≥3 (1) and supposeⅰ) |A_i~α(x, u, Du)|≤L(1+|Du|);ⅱ) (1+|p|)^(-1)A_i~α(x, u, p)are H(?)lder-continuous functions with some exponent δ on (?)×R^N uniformly with respect to p, i.e.ⅲ) A_i~α(x, u, p) are differentiable function in p with bounded and continuous derivativesⅳ)ⅴ) for all u∈H_(loc)~1(Ω, R^N)∩L^(n(γ-1)/(2-γ))(Ω, R^N), B(x, u, Du)is ineasurable and |B(x, u, p)|≤a(|p|~γ+|u|~τ)+b(x), where 1+2/n<γ<2, τ≤max((n+2)/(n-2), (γ-1)/(2-γ)-ε), (?)ε>0, b(x)∈L2n/(n+2), n^2/(n+2)+e(Ω), (?)ε>0.Remarks. Only bounded open set Q will be considered in this paper; for all p≥1, λ≥0, which is clled a Morrey Space.Let assumptions ⅰ)-ⅳ) hold, Giaquinta and Modica have proved the regularity of both the H^1 weak solutions of (1) under controllable growth condition |B|≤α(|p|~γ+|u|^((n+2)/(n-2))+b, 0<γ≤1+2/n and the H^1∩L~∞ weak solutions of (1) under natural growth condition |B|≤α|p|~2+b with a smallness condition 2aM<λ(|u|≤M), which implys that the H^1∩L~∞ weak solutions have the same regularty in the case of 1+2/n<γ<2. In the case of γ=2, many counterexamples (see [2] showed that u must be in H^1L~∞, while in the case of 1+2/n<γ<2, we consider the H^1∩L^n(γ-1)/(2-γ) weak solutions of (1), weaken the instability conditions upon them (from L~∞ to L^n(γ-1)/(2-γ) and obtain the same regularity results. Finally we show that the exponent n(γ-1)/(2-γ) can not be docreased anymore for the sake of the regularity results.Delinition 1. We call u∈H^1∩L^n(γ-1)/(2-γ)(Q, R^N) be a weak solution of (1), providod that where We use the convention that repeated indices are summed. i, j go from 1 to N ann α, β from 1 to n.展开更多
文摘We call that λ(x)belongs to Muckenhoupt class A_p if λ(x)≥0 and satisfiesfor all balls in R,where|B_R| denotes the Lebesgue measure of a ball with centerx_0 and radius R,andLet be a bounded domain.Define normHere,denotes the Banach space of all measurable functions defined
基金This work is supported in part by the Foundation of Zhongshan University, Advanced Research Center.
文摘Let us consider the following elliptic systems of second order-D_α(A_i~α(x, u, Du))=B_4(x, u, Du), i=1, …, N, x∈Q(?)R^n, n≥3 (1) and supposeⅰ) |A_i~α(x, u, Du)|≤L(1+|Du|);ⅱ) (1+|p|)^(-1)A_i~α(x, u, p)are H(?)lder-continuous functions with some exponent δ on (?)×R^N uniformly with respect to p, i.e.ⅲ) A_i~α(x, u, p) are differentiable function in p with bounded and continuous derivativesⅳ)ⅴ) for all u∈H_(loc)~1(Ω, R^N)∩L^(n(γ-1)/(2-γ))(Ω, R^N), B(x, u, Du)is ineasurable and |B(x, u, p)|≤a(|p|~γ+|u|~τ)+b(x), where 1+2/n<γ<2, τ≤max((n+2)/(n-2), (γ-1)/(2-γ)-ε), (?)ε>0, b(x)∈L2n/(n+2), n^2/(n+2)+e(Ω), (?)ε>0.Remarks. Only bounded open set Q will be considered in this paper; for all p≥1, λ≥0, which is clled a Morrey Space.Let assumptions ⅰ)-ⅳ) hold, Giaquinta and Modica have proved the regularity of both the H^1 weak solutions of (1) under controllable growth condition |B|≤α(|p|~γ+|u|^((n+2)/(n-2))+b, 0<γ≤1+2/n and the H^1∩L~∞ weak solutions of (1) under natural growth condition |B|≤α|p|~2+b with a smallness condition 2aM<λ(|u|≤M), which implys that the H^1∩L~∞ weak solutions have the same regularty in the case of 1+2/n<γ<2. In the case of γ=2, many counterexamples (see [2] showed that u must be in H^1L~∞, while in the case of 1+2/n<γ<2, we consider the H^1∩L^n(γ-1)/(2-γ) weak solutions of (1), weaken the instability conditions upon them (from L~∞ to L^n(γ-1)/(2-γ) and obtain the same regularity results. Finally we show that the exponent n(γ-1)/(2-γ) can not be docreased anymore for the sake of the regularity results.Delinition 1. We call u∈H^1∩L^n(γ-1)/(2-γ)(Q, R^N) be a weak solution of (1), providod that where We use the convention that repeated indices are summed. i, j go from 1 to N ann α, β from 1 to n.
