کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
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6417070 | 1338520 | 2016 | 59 صفحه PDF | دانلود رایگان |
We are interested in the study of local and global minimizers for an energy functional of the type14â¬R2Nâ(RNâΩ)2|u(x)âu(y)|2K(xây)dxdy+â«Î©W(u(x))dx, 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 K(z)=|z|âNâ2s, with sâ(0,1), related to the fractional Laplacian. We show the existence and uniqueness (up to translations) of one-dimensional minimizers in the full space 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 equationp.v.â«RN(u(x)âu(y))K(xây)dy+Wâ²(u(x))=0for any xâRN, which possess one-dimensional symmetry.The results presented here were proved in [9,10,36] for the model case K(z)=|z|âNâ2s. In our work, we consider instead general kernels which may be possibly non-homogeneous and truncated at infinity.
Journal: Journal of Differential Equations - Volume 260, Issue 8, 15 April 2016, Pages 6638-6696