DBI Hamiltonian bound for an electric spike (${\mathcal{H}}^2/T_3^2=(1+\mathbf{d}\cdot \nabla X{)}^2+|\mathbf{d}-\nabla X{|}^2$)

For a static D3-brane with one scalar and no magnetic field, taking the Legendre transform of the DBI action gives the displayed identity. It bounds energy by the electric scalar charge and is saturated at $\mathbf{d}=\nabla X$. The opposite orientation uses the analogous identity with reversed signs. On saturation, the constitutive relation gives $\mathbf{E}=\mathbf{d}$ and the Gauss law constraint in gauge theory reduces to the Laplace equation for $X$.