Hölder bounds and regularity of emerging free boundaries for strongly competing Schrödinger equations with nontrivial grouping

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• 作者：Nicola Soave^a ; ^{nicola.soave@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; ^{nicola.soave@math.uni-giessen.de" class="auth_mail" title="E-mail the corresponding author} ; Hugo Tavares^b ; ^{htavares@math.ist.utl.pt" class="auth_mail" title="E-mail the corresponding author} ; Susanna Terracini^c ; ^{susanna.terracini@unito.it" class="auth_mail" title="E-mail the corresponding author} ; Alessandro Zilio^d ; ^{azilio@ehess.fr" class="auth_mail" title="E-mail the corresponding author} ; ^{alessandro.zilio@polimi.it" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 35B45 ; 35B65 ; 35R35 ; secondary ; 35B53 ; 35B25 ; 35J47 ; 35R06
• 刊名：Nonlinear Analysis
• 出版年：2016
• 出版时间：June 2016
• 年：2016
• 卷：138
• 期：Complete
• 页码：388-427
• 全文大小：840 K

文摘

We study interior regularity issues for systems of elliptic equations of the type set in domains formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X15003685&_mathId=si2.gif&_user=111111111&_pii=S0362546X15003685&_rdoc=1&_issn=0362546X&md5=6c6d96eab05626be4f63249988714656" title="Click to view the MathML source">Ω⊂R^N$\Omega \subset {\mathbb{R}}^N$, for formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X15003685&_mathId=si3.gif&_user=111111111&_pii=S0362546X15003685&_rdoc=1&_issn=0362546X&md5=d696d2d0ca5b63d35281e0a7c0b046a9" title="Click to view the MathML source">N⩾1$N⩾1$. The paper is devoted to the derivation of formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X15003685&_mathId=si4.gif&_user=111111111&_pii=S0362546X15003685&_rdoc=1&_issn=0362546X&md5=9ca61f74c4b026d277f3b9bdb9008589" title="Click to view the MathML source">C^0,α${\mathcal{C}}^{0,\alpha }$ estimates that are uniform in the competition parameter formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X15003685&_mathId=si5.gif&_user=111111111&_pii=S0362546X15003685&_rdoc=1&_issn=0362546X&md5=efdac9bc3e1acd27e832dd63c0c66243" title="Click to view the MathML source">β>0$\beta >0$, as well as to the regularity of the limiting free-boundary problem obtained for formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X15003685&_mathId=si6.gif&_user=111111111&_pii=S0362546X15003685&_rdoc=1&_issn=0362546X&md5=2c4cf03e416a5913218477d60a4ae226" title="Click to view the MathML source">β→+∞$\beta \to +\infty$.

The main novelty of the problem under consideration resides in the non-trivial grouping of the densities: in particular, we assume that the interaction parameters formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X15003685&_mathId=si7.gif&_user=111111111&_pii=S0362546X15003685&_rdoc=1&_issn=0362546X&md5=548fc00480025d3cc9ebeaf43d2c99f5" title="Click to view the MathML source">a_ij$a_{ij}$ are only non-negative, and thus may vanish for specific couples formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X15003685&_mathId=si8.gif&_user=111111111&_pii=S0362546X15003685&_rdoc=1&_issn=0362546X&md5=e283661e9e11d70906137e04af7a19f0" title="Click to view the MathML source">(i,j)$(i,j)$. As a main consequence, in the limit formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X15003685&_mathId=si6.gif&_user=111111111&_pii=S0362546X15003685&_rdoc=1&_issn=0362546X&md5=2c4cf03e416a5913218477d60a4ae226" title="Click to view the MathML source">β→+∞$\beta \to +\infty$, densities do not segregate pairwise in general, but are grouped in classes which, in turn, form a mutually disjoint partition. Moreover, with respect to the literature, we consider more general forcing terms, sign-changing solutions, and an arbitrary formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X15003685&_mathId=si10.gif&_user=111111111&_pii=S0362546X15003685&_rdoc=1&_issn=0362546X&md5=514214ef983644b9c47f461e84b68800" title="Click to view the MathML source">p>0$p>0$. In addition, we present a regularity theory of the emerging free-boundary, defined by the interface among different segregated groups.

These equations are very common in the study of Bose–Einstein condensates and are of key importance for the analysis of optimal partition problems related to high order eigenvalues.