We consider both the minimisation of a class of nonlocal interaction energies over non-negative measures with unit mass and a class of singular integral equations of the first kind of Fredholm type. Our setting covers applications to dislocation pile-ups, contact problems, fracture mechanics and random matrix theory. Our main result shows that both the minimisation problems and the related singular integral equations have the same unique solution, which provides new regularity results on the minimiser of the energy and new positivity results on the solutions to singular integral equations.
Mots-clés : Interaction energy, energy minimisation, regularity of the minimiser, singular integral equation
@article{COCV_2020__26_1_A27_0,
author = {Kimura, M. and van Meurs, P.},
title = {Regularity of the minimiser of one-dimensional interaction energies},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {26},
year = {2020},
doi = {10.1051/cocv/2019043},
mrnumber = {4072630},
zbl = {1434.74057},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2019043/}
}
TY - JOUR AU - Kimura, M. AU - van Meurs, P. TI - Regularity of the minimiser of one-dimensional interaction energies JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2020 VL - 26 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2019043/ DO - 10.1051/cocv/2019043 LA - en ID - COCV_2020__26_1_A27_0 ER -
%0 Journal Article %A Kimura, M. %A van Meurs, P. %T Regularity of the minimiser of one-dimensional interaction energies %J ESAIM: Control, Optimisation and Calculus of Variations %D 2020 %V 26 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2019043/ %R 10.1051/cocv/2019043 %G en %F COCV_2020__26_1_A27_0
Kimura, M.; van Meurs, P. Regularity of the minimiser of one-dimensional interaction energies. ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 27. doi : 10.1051/cocv/2019043. http://www.numdam.org/articles/10.1051/cocv/2019043/
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