收稿日期: 2020-09-24
网络出版日期: 2021-01-06
基金资助
Supported by the National Natural Science Foundation of China (Grant No.11726622);
Scientific Research Fund of Young Teachers in Longqiao College (Grant No. LQKJ2020-01).
Korn’s Inequality and Divergence Equations on Generalize Orlicz Spaces
Received date: 2020-09-24
Online published: 2021-01-06
Supported by
Supported by the National Natural Science Foundation of China (Grant No.11726622);
Scientific Research Fund of Young Teachers in Longqiao College (Grant No. LQKJ2020-01).
Let ϕ be a generalized Orlicz function satisfying (A0), (A1), (A2), (aInc) and (aDec). We prove that the mapping
f →f #:=supB 1/\int|B||f(x)-fB|dx is continuous on Lϕ(·)(Rn) by extrapolation. Based on this result we generalize Korn’s inequality to the setting of generalized Orlicz spaces, i.e., ||\triangledown f||L^{ϕ(·)}(Ω) \lesssim||DF|||L^{ϕ}(Ω) . Using the Calder´on–Zygmund theory on generalized Orlicz spaces, we obtain that the divergence equation divu=f has a solution u∈(W01,ϕ(·)(Ω))n such that ||\triangledown u||L^{ϕ(·)}(Ω) \lesssim ||f||L^{ϕ}(Ω).
吴芮民, 王松柏 . 广义的Orlicz空间上的Korn不等式与散度方程[J]. 数学季刊, 2020 , 35(4) : 344 -353 . DOI: 10.13371/j.cnki.chin.q.j.m.2020.04.002
Let ϕ be a generalized Orlicz function satisfying (A0), (A1), (A2), (aInc) and (aDec). We prove that the mapping
f →f #:=supB 1/\int|B||f(x)-fB|dx is continuous on Lϕ(·)(Rn) by extrapolation. Based on this result we generalize Korn’s inequality to the setting of generalized Orlicz spaces, i.e., ||\triangledown f||L^{ϕ(·)}(Ω) \lesssim||DF|||L^{ϕ}(Ω) . Using the Calder´on–Zygmund theory on generalized Orlicz spaces, we obtain that the divergence equation divu=f has a solution u∈(W01,ϕ(·)(Ω))n such that ||\triangledown u||L^{ϕ(·)}(Ω) \lesssim ||f||L^{ϕ}(Ω).
Key words： Generalized Orlicz spaces; Korn’s inequality; Extrapolation.
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