Rigidity of pairs of quasiregular mappings whose symmetric part of gradient are close

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• 作者：Andrew Lorent ^{lorentaw@uc.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：30C65 ; 26B99
• 刊名：Annales de l'Institut Henri Poincare (C) Non Linear Analysis
• 出版年：2016
• 出版时间：January-February 2016
• 年：2016
• 卷：33
• 期：1
• 页码：23-65
• 全文大小：602 K

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For hmlsrc">hImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0294144914000808&_mathId=si1.gif&_user=111111111&_pii=S0294144914000808&_rdoc=1&_issn=02941449&md5=af43f2dca8bc7dbe3db6083caa351f16" title="Click to view the MathML source">A∈M^2×2hContainer hidden">hCode">h altimg="si1.gif" overflow="scroll">A∈M2×2h> let hmlsrc">he MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0294144914000808&_mathId=si2.gif&_user=111111111&_pii=S0294144914000808&_rdoc=1&_issn=02941449&md5=650e598ee5fd004b230a6171c858d4c3">height="18" width="101" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0294144914000808-si2.gif">hContainer hidden">hCode">h altimg="si2.gif" overflow="scroll">Shy="false">(Ahy="false">)=ATAh>, i.e. the symmetric part of the polar decomposition of A  . We consider the relation between two quasiregular mappings whose symmetric part of gradient are close. Our main result is the following. Suppose hmlsrc">hImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0294144914000808&_mathId=si3.gif&_user=111111111&_pii=S0294144914000808&_rdoc=1&_issn=02941449&md5=62d3d69e68c1acaaf46f6b2905a8c178" title="Click to view the MathML source">v,u∈W^1,2(B₁(0):R²)hContainer hidden">hCode">h altimg="si3.gif" overflow="scroll">v,u∈W1,2hy="false">(B1hy="false">(0hy="false">):hvariant="double-struck">R2hy="false">)h> are Q  -quasiregular mappings with hmlsrc">hImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0294144914000808&_mathId=si4.gif&_user=111111111&_pii=S0294144914000808&_rdoc=1&_issn=02941449&md5=9790ca53a00a73b0c36d292760913d3d" title="Click to view the MathML source">∫_B1(0)det⁡(Du)^−pdz≤C_phContainer hidden">hCode">h altimg="si4.gif" overflow="scroll">∫B1hy="false">(0hy="false">)hvariant="normal">det⁡hy="false">(Duhy="false">)−pdz≤Cph> for some hmlsrc">hImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0294144914000808&_mathId=si37.gif&_user=111111111&_pii=S0294144914000808&_rdoc=1&_issn=02941449&md5=85b43037246b579bafafc4c4b1820986" title="Click to view the MathML source">p∈(0,1)hContainer hidden">hCode">h altimg="si37.gif" overflow="scroll">p∈hy="false">(0,1hy="false">)h> and hmlsrc">hImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0294144914000808&_mathId=si6.gif&_user=111111111&_pii=S0294144914000808&_rdoc=1&_issn=02941449&md5=79b5574bb479f5360a6d8e59c92bc119" title="Click to view the MathML source">∫_B1(0)|Du|²dz≤πhContainer hidden">hCode">h altimg="si6.gif" overflow="scroll">∫B1hy="false">(0hy="false">)hy="false">|Duhy="false">|2dz≤πh>. There exists constant hmlsrc">hImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0294144914000808&_mathId=si7.gif&_user=111111111&_pii=S0294144914000808&_rdoc=1&_issn=02941449&md5=fdd2c212f24adcea123aa9de2bf6a33e" title="Click to view the MathML source">M>1hContainer hidden">hCode">h altimg="si7.gif" overflow="scroll">M>1h> such that if hmlsrc">hImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0294144914000808&_mathId=si8.gif&_user=111111111&_pii=S0294144914000808&_rdoc=1&_issn=02941449&md5=e0e2c58d84c0f8523225a7bf544e2e53" title="Click to view the MathML source">∫_B1(0)|S(Dv)−S(Du)|²dz=ϵhContainer hidden">hCode">h altimg="si8.gif" overflow="scroll">∫B1hy="false">(0hy="false">)hy="false">|Shy="false">(Dvhy="false">)−Shy="false">(Duhy="false">)hy="false">|2dz=ϵh> then Taking hmlsrc">hImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0294144914000808&_mathId=si10.gif&_user=111111111&_pii=S0294144914000808&_rdoc=1&_issn=02941449&md5=e8c4d62a440ab30353c4596b37602ee8" title="Click to view the MathML source">u=IdhContainer hidden">hCode">h altimg="si10.gif" overflow="scroll">u=hvariant="italic">Idh> we obtain a special case of the quantitative rigidity result of Friesecke, James and Müller [13]. Our main result can be considered as a first step in a new line of generalization of Theorem 1 of [13] in which Id is replaced by a mapping of non-trivial degree.