TY - JOUR
T1 - Real analytic families of harmonic functions in a planar domain with a small hole
AU - Dalla Riva, M.
AU - Musolino, P.
PY - 2015/2/1
Y1 - 2015/2/1
N2 - We consider a Dirichlet problem in a planar domain with a hole of diameter proportional to a real parameter ε and we denote by uε the corresponding solution. The behavior of uε for ε small and positive can be described in terms of real analytic functions of two variables evaluated at (ε, 1/log ε). We show that under suitable assumptions on the geometry and on the boundary data one can get rid of the logarithmic behavior displayed by uε for ε small and describe uε by real analytic functions of ε. Then it is natural to ask what happens when ε is negative. The case of boundary data depending on ε is also considered. The aim is to study real analytic families of harmonic functions which are not necessarily solutions of a particular boundary value problem.
AB - We consider a Dirichlet problem in a planar domain with a hole of diameter proportional to a real parameter ε and we denote by uε the corresponding solution. The behavior of uε for ε small and positive can be described in terms of real analytic functions of two variables evaluated at (ε, 1/log ε). We show that under suitable assumptions on the geometry and on the boundary data one can get rid of the logarithmic behavior displayed by uε for ε small and describe uε by real analytic functions of ε. Then it is natural to ask what happens when ε is negative. The case of boundary data depending on ε is also considered. The aim is to study real analytic families of harmonic functions which are not necessarily solutions of a particular boundary value problem.
KW - Harmonic functions
KW - Real analytic continuation in Banach space
KW - Singularly perturbed perforated planar domains
UR - http://www.scopus.com/inward/record.url?scp=85027921783&partnerID=8YFLogxK
U2 - 10.1016/j.jmaa.2014.08.037
DO - 10.1016/j.jmaa.2014.08.037
M3 - Article
AN - SCOPUS:85027921783
SN - 0022-247X
VL - 422
SP - 37
EP - 55
JO - Journal of Mathematical Analysis and Applications
JF - Journal of Mathematical Analysis and Applications
IS - 1
ER -