We present a construction of harmonic functions on bounded domains for the spectral fractional Laplacian operator and we classify them in terms of their divergent profile at the boundary. This is used to establish and solve boundary value problems associated with nonhomogeneous boundary conditions. We provide a weak- theory to show how problems with measure data at the boundary and inside the domain are well-posed. We study linear and semilinear problems, performing a sub- and supersolution method. We finally show the existence of large solutions for some power-like nonlinearities.
Mots-clés : Spectral fractional Laplacian, Dirichlet problem, Boundary blow-up solutions, Large solutions
@article{AIHPC_2017__34_2_439_0,
author = {Abatangelo, Nicola and Dupaigne, Louis},
title = {Nonhomogeneous boundary conditions for the spectral fractional {Laplacian}},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
pages = {439--467},
publisher = {Elsevier},
volume = {34},
number = {2},
year = {2017},
doi = {10.1016/j.anihpc.2016.02.001},
mrnumber = {3610940},
zbl = {1372.35327},
language = {en},
url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2016.02.001/}
}
TY - JOUR AU - Abatangelo, Nicola AU - Dupaigne, Louis TI - Nonhomogeneous boundary conditions for the spectral fractional Laplacian JO - Annales de l'I.H.P. Analyse non linéaire PY - 2017 SP - 439 EP - 467 VL - 34 IS - 2 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.anihpc.2016.02.001/ DO - 10.1016/j.anihpc.2016.02.001 LA - en ID - AIHPC_2017__34_2_439_0 ER -
%0 Journal Article %A Abatangelo, Nicola %A Dupaigne, Louis %T Nonhomogeneous boundary conditions for the spectral fractional Laplacian %J Annales de l'I.H.P. Analyse non linéaire %D 2017 %P 439-467 %V 34 %N 2 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.anihpc.2016.02.001/ %R 10.1016/j.anihpc.2016.02.001 %G en %F AIHPC_2017__34_2_439_0
Abatangelo, Nicola; Dupaigne, Louis. Nonhomogeneous boundary conditions for the spectral fractional Laplacian. Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 2, pp. 439-467. doi : 10.1016/j.anihpc.2016.02.001. http://archive.numdam.org/articles/10.1016/j.anihpc.2016.02.001/
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