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.
|Number of pages||23|
|Journal||Communications on Pure and Applied Mathematics|
|Early online date||17 Jul 2018|
|Publication status||Published - Jan 2019|
ASJC Scopus subject areas
- Applied Mathematics
FingerprintDive into the research topics of 'The Equilibrium Measure for a Nonlocal Dislocation Energy'. Together they form a unique fingerprint.
- , Mathematics - Associate Professor
Person: Academic (Research & Teaching)