We consider the electrostatic Born–Infeld energy∫RN(1-1-|∇u|2)dx-∫RNρudx,where ρ∈ L m (R N ) is an assigned charge density, m∈ [1 , 2 ∗ ] , 2∗:=2NN+2, N≥ 3. We prove that if ρ∈ L q (R N ) for q> 2 N, the unique minimizer u ρ is of class Wloc2,2(RN). Moreover, if the norm of ρ is sufficiently small, the minimizer is a weak solution of the associated PDE-div(∇u1-|∇u|2)=ρinRN,(BI)with the boundary condition lim |x|→∞ u(x) = 0 , and it is of class Cloc1,α(RN) for some α∈ (0 , 1).
On the Regularity of the Minimizer of the Electrostatic Born–Infeld Energy
Bonheure D.
;Iacopetti A.
2019-01-01
Abstract
We consider the electrostatic Born–Infeld energy∫RN(1-1-|∇u|2)dx-∫RNρudx,where ρ∈ L m (R N ) is an assigned charge density, m∈ [1 , 2 ∗ ] , 2∗:=2NN+2, N≥ 3. We prove that if ρ∈ L q (R N ) for q> 2 N, the unique minimizer u ρ is of class Wloc2,2(RN). Moreover, if the norm of ρ is sufficiently small, the minimizer is a weak solution of the associated PDE-div(∇u1-|∇u|2)=ρinRN,(BI)with the boundary condition lim |x|→∞ u(x) = 0 , and it is of class Cloc1,α(RN) for some α∈ (0 , 1).
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