Existence and Regularity for Prescribed Lorentzian Mean Curvature Hypersurfaces, and the Born–Infeld Model

Abstract

Given a measure $\rho$ on a domain $\Omega \subset {\mathbb {R}}^m$, we study spacelike graphs over $\Omega$ in Minkowski space with Lorentzian mean curvature $\rho$ and Dirichlet boundary condition on $\partial \Omega$, which solve

The graph function also represents the electric potential generated by a charge $\rho$ in electrostatic Born-Infeld’s theory. Even though there exists a unique minimizer $u_\rho$ of the associated action

$$\begin{aligned} I_\rho (\psi ) \doteq \int _{\Omega } \Big ( 1 - \sqrt{1-|D\psi |^2} \Big ) \textrm{d}x - \langle \rho , \psi \rangle \end{aligned}$$

among functions $\psi$ satisfying $|D\psi | \le 1$, by the lack of smoothness of the Lagrangian density for $|D\psi | = 1$ one cannot guarantee that $u_\rho$ satisfies the Euler-Lagrange equation ($\mathcal{B}\mathcal{I}$). A chief difficulty comes from the possible presence of light segments in the graph of $u_\rho$. In this paper, we investigate the existence of a solution for general $\rho$. In particular, we give sufficient conditions to guarantee that $u_\rho$ solves ($\mathcal{B}\mathcal{I}$) and enjoys $\log$-improved energy and $W^{2,2}_\textrm{loc}$ estimate. Furthermore, we construct examples which suggest a sharp threshold for the regularity of $\rho$ to ensure the solvability of ($\mathcal{B}\mathcal{I}$).

The Born-Infeld Model

We here recall the Born-Infeld model of electromagnetism [16, 17]. Concise but informative introductions can be found in [12, 13], see also [8, 35, 36, 58] for a thorough account of the physical literature. We remark that the Born-Infeld model also proved to be relevant in the theory of superstrings and membranes, see [32, 58] and the references therein.

As outlined in the Introduction, one of the main concerns of the theory was to overcome the failure of the principle of finite energy occurring in Maxwell’s model, which we now describe. In a spacetime $(N^4,g)$ with metric $g = g_{ab}\textrm{d}y^a \otimes \textrm{d}y^b$ of signature $(-,+,+,+)$ ($g_{00}<0$), the electromagnetic field is described as a closed 2-form $F = \frac{1}{2} F_{ab} \textrm{d}y^a \wedge \textrm{d}y^b$ which, according to the Lagrangian formulation of Maxwell’s theory due to Schwarzschild (cf. [50] and [46, p.88]), in the absence of charges and currents, is required to be stationary for the action

$$\begin{aligned} \mathscr {L}_{\textrm{MS}} \doteq \int _{N^4} \textsf {L}_{\textrm{MS}} \sqrt{-|g|} \textrm{d}y \qquad \text {with} \quad \textsf {L}_\textrm{MS}\doteq -\frac{F^{ab}F_{ab}}{4}, \end{aligned}$$

where |g| is the determinant of g and $F^{ab} \doteq g^{ac}g^{bd} F_{cd}$. The presence of a vector field J describing charges and currents is taken into account by adding the action

$$\begin{aligned} \mathscr {L}_{J} \doteq \int _{N^4} \textsf {L}_J \sqrt{-|g|} \textrm{d}y, \qquad \textsf {L}_J = J^a\Phi _a, \end{aligned}$$

where we assumed that F is globally exact and we set $F = \textrm{d}\Phi$. By its very definition, the energy-impulse tensor T associated to $\mathscr {L}_\textrm{MS}+ \mathscr {L}_J$ has components

$$\begin{aligned} T_{ab} = \frac{-2}{\sqrt{-|g|}} \frac{\partial ((\textsf {L}_\textrm{MS}+ \textsf {L}_J) \sqrt{-|g|})}{\partial g^{ab}} = F_{ac}F_{bp}g^{cp} - \frac{1}{4}F^{cp}F_{cp}g_{ab} + J^c \Phi _c g_{ab} \end{aligned}$$

and in particular $T_{00}$ describes the energy density. In Minkowski space $\mathbb {L}^4$, by writing in Cartesian coordinates $\{x^a\}$ the electromagnetic tensor in terms of the electric and magnetic fields $\textbf{E} = E_j \textrm{d}x^j$ and $\textbf{B} = B_j \textrm{d}x^j$ as

$$\begin{aligned} F = \sum _{j =1}^3 E_j \textrm{d}x^j \wedge \textrm{d}x^0 + B_1 \textrm{d}x^2 \wedge \textrm{d}x^3 + B_2 \textrm{d}x^3 \wedge \textrm{d}x^1 + B_3 \textrm{d}x^1 \wedge \textrm{d}x^2, \end{aligned}$$

the vector potential as $\Phi = -\varphi \textrm{d}x^0 + \textbf{A} = -\varphi \textrm{d}x^0 + A_j \textrm{d}x^j$ and $J = \rho \partial _{x^0} + \textbf{J} = \rho \partial _{x^0} + J^j \partial _{x^j}$, the Maxwell Lagrangian and energy densities become

$$\begin{aligned} \textsf {L}_\textrm{MS}+ \textsf {L}_J = \frac{1}{2}\left( |\textbf{E}|^2 - |\textbf{B}|^2 \right) - \rho \varphi + \textbf{A}(\textbf{J}), \qquad T_{00} = \frac{1}{2} \left( |\textbf{E}|^2 + |\textbf{B}|^2 \right) + \rho \varphi - \textbf{A}(\textbf{J}). \end{aligned}$$

Restricting to the electrostatic case with no current density ($\textbf{B} = 0$, $\textbf{E}$ independent of $x^0$, $\textbf{J} = 0$), from $\textbf{E} = -\textrm{d}\varphi$ the potential $\varphi$ turns out to be stationary for the reduced action

$$\begin{aligned} J_\rho (v) \doteq \frac{1}{2} \int _{{\mathbb {R}}^3} |Dv|^2 \textrm{d}x - \langle \rho , v \rangle , \end{aligned}$$

where $\langle \rho , v \rangle$ is the duality pairing given, for smooth $\rho$, by integration. However, for $\rho = \delta _{x_0}$ the Dirac delta centered at a point $x_0$, the Newtonian potential ${\bar{u}}_\rho = \textrm{const} \cdot |x-x_0|^{2-m}$ solving the Euler-Lagrange equation $-\Delta {\bar{u}}_\rho = \rho$ for $J_\rho$ has infinite energy on punctured balls centered at $x_0$:

$$\begin{aligned} \int _{B_R\backslash B_\varepsilon } T_{00} \textrm{d}x = \frac{1}{2} \int _{B_R\backslash B_\varepsilon } |D {\bar{u}}_\rho |^2 \textrm{d}x \rightarrow \infty \qquad \text {as } \, \varepsilon \rightarrow 0, \end{aligned}$$

a fact of serious physical concern (cf. [17]). The problem also persists for certain sources $\rho \in L^1({\mathbb {R}}^m)$, see [12, 27]. To avoid it, Born and Infeld in [16, 17] proposed an alternative Lagrangian density $\textsf {L}_{\textrm{BI}}$, defined by

$$\begin{aligned} \displaystyle \textsf {L}_{\textrm{BI}}\sqrt{-|g|} \doteq \sqrt{ - |g|} - \sqrt{ - |g + F|} \end{aligned}$$

(we follow the convention in [58], which changes signs in $\textsf {L}_\textrm{BI}$ with respect to [17]). Indeed, the authors also indicated a further Lagrangian, already suggested in [15], see [17, (2.27) and (2.28)]. However, later works in [9, 47] pointed out a distinctive feature of $\textsf {L}_\textrm{BI}$, that of generating an electrodynamics free from the phenomenon of birefringence (cf. [8, 36]). Computing the determinants in the expression of $\textsf {L}_\textrm{BI}$, Born and Infeld obtained

$$\begin{aligned} \displaystyle \textsf {L}_{\textrm{BI}} = 1- \sqrt{1 + \mathbb {F} - \mathbb {G}^2}, \qquad \text {where } \ \ \mathbb {F} \doteq \frac{F^{ab}F_{ab}}{2}, \ \ \mathbb {G} \doteq \frac{F_{ab}(\star F)^{ab}}{4} \end{aligned}$$

and $\star$ is the Hodge dual, so $\textsf {L}_\textrm{BI}$ is asymptotic to $\textsf {L}_\textrm{MS}$ for small F and flat g. In Minkowski space and Cartesian coordinates $\{x^a\}$, $\textsf {L}_\textrm{BI}$ becomes

$$\begin{aligned} \textsf {L}_{\textrm{BI}} = 1- \sqrt{ 1 - |\textbf{E}|^2 + |\textbf{B}|^2 - (\textbf{B}\cdot \textbf{E})^2}. \end{aligned}$$

(A.1)

Here, we set the maximal field strength to be 1 for convenience. The energy-impulse tensor associated to $\mathscr {L}_\textrm{BI}+ \mathscr {L}_J$, and its component $T_{00}$ in Cartesian coordinates, are thus

$$\begin{aligned} T_{ab}= & {} \displaystyle \textsf {L}_{\textrm{BI}} g_{ab} + \frac{F_{ac}F_{bp}g^{cp} - \mathbb {G}^2g_{ab}}{\sqrt{1 + \mathbb {F} - \mathbb {G}^2}} + J^c \Phi _c g_{ab}, \\ T_{00}= & {} \displaystyle \frac{1 + |\textbf{B}|^2}{\sqrt{ 1 - |\textbf{E}|^2 + |\textbf{B}|^2 - (\textbf{B}\cdot \textbf{E})^2}} -1 + \rho \varphi - \textbf{A}(\textbf{J}). \end{aligned}$$

In the electrostatic case, the potential $u_\rho$ generated by $\rho$ is therefore required to minimize the action $I_\rho$ in (1.2) on $\Omega = {\mathbb {R}}^3$ among weakly spacelike functions with a suitable decay at infinity, and its energy density is given by

$$\begin{aligned} T_{00} = \displaystyle \frac{1}{\sqrt{ 1 - |Du_\rho |^2}} -1 + \rho u_\rho = w_\rho -1 + \rho u_\rho , \end{aligned}$$

making apparent the link between $w_\rho$ and $T_{00}$ mentioned in the Introduction. Whence, the integrability (1.11) proved in [12, Proposition 2.7] can be rephrased as the remarkable property

$$\begin{aligned} T_{00}- \rho u_\rho \in L^1({\mathbb {R}}^3) \end{aligned}$$

(A.2)

for $\rho$ in a large class of distributions including any finite measure on ${\mathbb {R}}^3$. The case $\rho = \delta _{x_0}$ was previously considered by Born and Infeld in [17] to support the consistency of their theory. For solutions in bounded domains, Proposition 3.13 guarantees the same desirable property: $T_{00}- \rho u_\rho \in L^1_\textrm{loc}(\Omega )$, provided that the boundary datum $\phi$ is not too degenerate.