Abstract
In this paper we characterize the equilibrium measure for a nonlocal and anisotropic weighted energy describing the interaction of positive dislocations in the plane. We prove that the minimum value of the energy is attained by a measure supported on the vertical axis and distributed according to the semicircle law, a well-known measure that also arises as the minimizer of purely logarithmic interactions in one dimension. In this way we give a positive answer to the conjecture that positive dislocations tend to form vertical walls. This result is one of the few examples where the minimizer of a nonlocal energy is explicitly computed and the only one in the case of anisotropic kernels.
| Original language | English |
|---|---|
| Pages (from-to) | 136-158 |
| Number of pages | 23 |
| Journal | Communications on Pure and Applied Mathematics |
| Volume | 72 |
| Issue number | 1 |
| Early online date | 17 Jul 2018 |
| DOIs | |
| Publication status | Published - 1 Jan 2019 |
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ASJC Scopus subject areas
- Mathematics(all)
- Applied Mathematics
Cite this
The equilibrium measure for a nonlocal dislocation energy. / Mora, Maria Giovanna; Rondi, Luca; Scardia, Lucia.
In: Communications on Pure and Applied Mathematics, Vol. 72, No. 1, 01.01.2019, p. 136-158.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - The equilibrium measure for a nonlocal dislocation energy
AU - Mora, Maria Giovanna
AU - Rondi, Luca
AU - Scardia, Lucia
PY - 2019/1/1
Y1 - 2019/1/1
N2 - In this paper we characterize the equilibrium measure for a nonlocal and anisotropic weighted energy describing the interaction of positive dislocations in the plane. We prove that the minimum value of the energy is attained by a measure supported on the vertical axis and distributed according to the semicircle law, a well-known measure that also arises as the minimizer of purely logarithmic interactions in one dimension. In this way we give a positive answer to the conjecture that positive dislocations tend to form vertical walls. This result is one of the few examples where the minimizer of a nonlocal energy is explicitly computed and the only one in the case of anisotropic kernels.
AB - In this paper we characterize the equilibrium measure for a nonlocal and anisotropic weighted energy describing the interaction of positive dislocations in the plane. We prove that the minimum value of the energy is attained by a measure supported on the vertical axis and distributed according to the semicircle law, a well-known measure that also arises as the minimizer of purely logarithmic interactions in one dimension. In this way we give a positive answer to the conjecture that positive dislocations tend to form vertical walls. This result is one of the few examples where the minimizer of a nonlocal energy is explicitly computed and the only one in the case of anisotropic kernels.
UR - http://www.scopus.com/inward/record.url?scp=85050932748&partnerID=8YFLogxK
U2 - 10.1002/cpa.21762
DO - 10.1002/cpa.21762
M3 - Article
VL - 72
SP - 136
EP - 158
JO - Communications on Pure and Applied Mathematics
JF - Communications on Pure and Applied Mathematics
SN - 0010-3640
IS - 1
ER -