On the flow of non-axisymmetric perturbations of cylinders via surface diffusion

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• 作者：Jeremy LeCrone^a ; ^{lecronjs@ksu.edu" class="auth_mail" title="E-mail the corresponding author} ; ; Gieri Simonett^b ; ^{gieri.simonett@vanderbilt.edu" class="auth_mail" title="E-mail the corresponding author} ;
• 关键词：primary ; 35K93 ; 53C44 ; secondary ; 35B35 ; 35B32 ; 46T20
• 刊名：Journal of Differential Equations
• 出版年：2016
• 出版时间：15 March 2016
• 年：2016
• 卷：260
• 期：6
• 页码：5510-5531
• 全文大小：381 K

文摘

We study the surface diffusion flow acting on a class of general (non-axisymmetric) perturbations of cylinders 46&_mathId=si1.gif&_user=111111111&_pii=S0022039615006646&_rdoc=1&_issn=00220396&md5=267209d1f16610e830f6eb322837df32" title="Click to view the MathML source">C_r${\mathcal{C}}_r$ in 46&_mathId=si2.gif&_user=111111111&_pii=S0022039615006646&_rdoc=1&_issn=00220396&md5=f576c1763a459de389c786c58f4c8161" title="Click to view the MathML source">IR³$\mathrm{I}{\mathrm{R}}^3$. Using tools from parabolic theory on uniformly regular manifolds, and maximal regularity, we establish existence and uniqueness of solutions to surface diffusion flow starting from (spatially-unbounded) surfaces defined over 46&_mathId=si1.gif&_user=111111111&_pii=S0022039615006646&_rdoc=1&_issn=00220396&md5=267209d1f16610e830f6eb322837df32" title="Click to view the MathML source">C_r${\mathcal{C}}_r$ via scalar height functions which are uniformly bounded away from the central cylindrical axis. Additionally, we show that 46&_mathId=si1.gif&_user=111111111&_pii=S0022039615006646&_rdoc=1&_issn=00220396&md5=267209d1f16610e830f6eb322837df32" title="Click to view the MathML source">C_r${\mathcal{C}}_r$ is normally stable with respect to 2π  -axially-periodic perturbations if the radius 46&_mathId=si288.gif&_user=111111111&_pii=S0022039615006646&_rdoc=1&_issn=00220396&md5=eadf33b208a8a502925a9449bc120677" title="Click to view the MathML source">r>1$r>1$, and unstable if 46&_mathId=si325.gif&_user=111111111&_pii=S0022039615006646&_rdoc=1&_issn=00220396&md5=f226fe49d0a521d30633af1e9e72fe36" title="Click to view the MathML source">0<r<1$0<r<1$. Stability is also shown to hold in settings with axial Neumann boundary conditions.