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

We consider the electrostatic Born-Infeld energy integral(RN) (1-root 1-vertical bar del u vertical bar(2)) dx - integral(RN) rho u dx, where rho epsilon L-m(R-N) is an assigned charge density, m epsilon [1, 2(*)], 2(*) := 2N/N+2, N >= 3. We prove that if rho epsilon L-q (R-N) for q > 2N, the unique minimizer u(rho) is of class W-loc(2,2)(R-N). Moreover, if the norm of. is sufficiently small, theminimizer is a weak solution of the associated PDE - div (del u/root 1-vertical bar del u vertical bar(2)) - rho in R-N, (BI) with the boundary condition lim(vertical bar x vertical bar ->infinity) u(x) = 0, and it is of class C-loc(1,alpha) (R-N) for some alpha epsilon (0, 1).