Effective transmission conditions for Hamilton-Jacobi equations defined on two domains separated by an oscillatory interface

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• 作者：Yves Achdou^a ; ^{achdou@ljll.univ-paris-diderot.fr" class="auth_mail" title="E-mail the corresponding author} ; Salomé ; Oudet^b ; Nicoletta Tchou^b
• 关键词：35Q93 ; 35F21 ; 35B27 ; 49L25
• 刊名：Journal de mathematiques pures et appliquees
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：106
• 期：6
• 页码：1091-1121
• 全文大小：693 K

文摘

We consider a family of optimal control problems in the plane with dynamics and running costs possibly discontinuous across an oscillatory interface class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021782416300289&_mathId=si1.gif&_user=111111111&_pii=S0021782416300289&_rdoc=1&_issn=00217824&md5=06dc8b397fecd72a43dca779678bb38f" title="Click to view the MathML source">Γ_εclass="mathContainer hidden">class="mathCode">${\mathrm{\Gamma }}_{\epsilon }$. The oscillations of the interface have small period and amplitude, both of the order of ε  , and the interfaces class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021782416300289&_mathId=si1.gif&_user=111111111&_pii=S0021782416300289&_rdoc=1&_issn=00217824&md5=06dc8b397fecd72a43dca779678bb38f" title="Click to view the MathML source">Γ_εclass="mathContainer hidden">class="mathCode">${\mathrm{\Gamma }}_{\epsilon }$ tend to a straight line Γ. We study the asymptotic behavior as class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021782416300289&_mathId=si2.gif&_user=111111111&_pii=S0021782416300289&_rdoc=1&_issn=00217824&md5=f252ef21208d9a10c16bebaff72a9368" title="Click to view the MathML source">ε→0class="mathContainer hidden">class="mathCode">$\epsilon \to 0$. We prove that the value function tends to the solution of Hamilton–Jacobi equations in the two half-planes limited by Γ, with an effective transmission condition on Γ keeping track of the oscillations of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021782416300289&_mathId=si1.gif&_user=111111111&_pii=S0021782416300289&_rdoc=1&_issn=00217824&md5=06dc8b397fecd72a43dca779678bb38f" title="Click to view the MathML source">Γ_εclass="mathContainer hidden">class="mathCode">${\mathrm{\Gamma }}_{\epsilon }$.