Explicit minimizers of some non-local anisotropic energies: a short proof
نویسندگان
چکیده
Abstract In this paper we consider non-local energies defined on probability measures in the plane, given by a convolution interaction term plus quadratic confinement. The kernel is , $z=x+iy$?> with $-1<\alpha<1$?> . This anisotropic except for Coulomb case $\alpha=0$?> We present short compact proof of known surprising fact that unique minimizer energy normalized characteristic function domain enclosed an ellipse horizontal semi-axis $\sqrt{1-\alpha}$?> and vertical $\sqrt{1+\alpha}$?> Letting $\alpha \to 1^-$?> find semicircle law axis corresponding energy, result related to interacting dislocations, previously obtained some authors. devote first sections presenting well-known background material simplest way possible, so readers unfamiliar subject proofs accessible.
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Article history: Received 18 September 2014 Available online 4 November 2014 MSC: 49A50 49J45 58E12
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ژورنال
عنوان ژورنال: Izvestiya: Mathematics
سال: 2021
ISSN: ['1468-4810', '1064-5632']
DOI: https://doi.org/10.1070/im9048