One-dimensional solutions of non-local Allen-Cahn-type equations with rough kernels

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• 作者：Matteo Cozzi^a ; ^b ; ^{matteo.cozzi@unimi.it" class="auth_mail" title="E-mail the corresponding author} ; Tommaso Passalacqua^a ; ^{tommaso.passalacqua@unimi.it" class="auth_mail" title="E-mail the corresponding author}
• 关键词：47G10 ; 47B34 ; 35R11 ; 35B08
• 刊名：Journal of Differential Equations
• 出版年：2016
• 出版时间：15 April 2016
• 年：2016
• 卷：260
• 期：8
• 页码：6638-6696
• 全文大小：716 K

文摘

We are interested in the study of local and global minimizers for an energy functional of the type where W is a smooth, even double-well potential and K   is a non-negative symmetric kernel in a general class, which contains as a particular case the choice g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616000073&_mathId=si2.gif&_user=111111111&_pii=S0022039616000073&_rdoc=1&_issn=00220396&md5=be76e82ae3b4224d12900a0c435d16e4" title="Click to view the MathML source">K(z)=|z|^−N−2sgif" overflow="scroll">K(z)=|z|−N−2s, with g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616000073&_mathId=si3.gif&_user=111111111&_pii=S0022039616000073&_rdoc=1&_issn=00220396&md5=74e44ff2c4f8566548ac81332372a798" title="Click to view the MathML source">s∈(0,1)gif" overflow="scroll">s∈(0,1), related to the fractional Laplacian. We show the existence and uniqueness (up to translations) of one-dimensional minimizers in the full space g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616000073&_mathId=si323.gif&_user=111111111&_pii=S0022039616000073&_rdoc=1&_issn=00220396&md5=c29044b43cfd662eddb874188489c5f3" title="Click to view the MathML source">R^Ngif" overflow="scroll">RN and obtain sharp estimates for some quantities associated to it. In particular, we deduce the existence of solutions of the non-local Allen–Cahn equation which possess one-dimensional symmetry.

The results presented here were proved in ,  and  for the model case g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616000073&_mathId=si2.gif&_user=111111111&_pii=S0022039616000073&_rdoc=1&_issn=00220396&md5=be76e82ae3b4224d12900a0c435d16e4" title="Click to view the MathML source">K(z)=|z|^−N−2sgif" overflow="scroll">K(z)=|z|−N−2s. In our work, we consider instead general kernels which may be possibly non-homogeneous and truncated at infinity.