On the Regularity of the Minimizer of the Electrostatic Born–Infeld Energy

We consider the electrostatic Born–Infeld energy ∫ R N 1 - 1 - | ∇ u | 2 d x - ∫ R N ρ u d x , where ρ ∈ L m ( R N ) is an assigned charge density, m ∈ [ 1 , 2 ∗ ] , 2 ∗ : = 2 N N + 2 , N ≥ 3 . We prove that if ρ ∈ L q ( R N ) for q > 2 N , the unique minimizer u ρ is of class W loc 2 , 2 ( R N )...

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Bibliographic Details
Published in:Archive for rational mechanics and analysis 2019-05, Vol.232 (2), p.697-725
Main Authors: Bonheure, Denis, Iacopetti, Alessandro
Format: Article
Language:English
Subjects:
Boundary conditions
Charge density
Classical Mechanics
Complex Systems
Fluid- and Aerodynamics
Mathematical and Computational Physics
Physics
Physics and Astronomy
Theoretical
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Summary:We consider the electrostatic Born–Infeld energy ∫ R N 1 - 1 - | ∇ u | 2 d x - ∫ R N ρ u d x , where ρ ∈ L m ( R N ) is an assigned charge density, m ∈ [ 1 , 2 ∗ ] , 2 ∗ : = 2 N N + 2 , N ≥ 3 . We prove that if ρ ∈ L q ( R N ) for q > 2 N , the unique minimizer u ρ is of class W loc 2 , 2 ( R N ) . Moreover, if the norm of ρ is sufficiently small, the minimizer is a weak solution of the associated PDE - div ∇ u 1 - | ∇ u | 2 = ρ in R N , ( BI ) with the boundary condition lim | x | → ∞ u ( x ) = 0 , and it is of class C loc 1 , α ( R N ) for some α ∈ ( 0 , 1 ) .
ISSN:0003-9527
1432-0673
DOI:10.1007/s00205-018-1331-4