Maximal hypersurfaces with prescribed light-like cones in Lorentz-Minkowski space

1. Problem Statement

The paper investigates the existence and properties of maximal hypersurfaces in the $(N+1)$-dimensional Lorentz-Minkowski space $\mathbb{L}^{N+1}$. Specifically, the authors study hypersurfaces defined as graphs of functions $u: \mathbb{R}^N \to \mathbb{R}$ that satisfy the mean curvature equation with prescribed Dirac mass sources.

The core equation is:
$$-\nabla \cdot \left( \frac{\nabla u}{\sqrt{1 - |\nabla u|^2}} \right) = \sum_{j=1}^{m_0} \alpha_j \delta_{p_j} \quad \text{in } \mathcal{D}'(\mathbb{R}^N)$$
where:

• $N \ge 2$.
• $p_j \in \mathbb{R}^N$ are distinct points representing the vertices of light-cone singularities.
• $\alpha_j$ are coefficients (masses) which can be positive, negative, or mixed.
• The condition $|\nabla u| < 1$ (spacelike condition) holds everywhere except at the singular points $p_j$, where $|\nabla u| \to 1$ as $x \to p_j$.

The paper addresses three main scenarios:

1. $N \ge 3$ with positive masses: Existence of solutions decaying to 0 at infinity.
2. $N = 2$ with positive masses: Existence of solutions diverging to $-\infty$ at infinity (due to the logarithmic nature of the fundamental solution in 2D).
3. Mixed signs and infinite masses: Solutions with both positive and negative Dirac masses, and extensions to infinitely many singularities.

2. Methodology

The authors employ a rigorous approximation procedure combined with variational methods and geometric analysis.

• Approximation of Dirac Masses:
  The singular source terms $\sum \alpha_j \delta_{p_j}$ are approximated by a sequence of smooth, non-negative functions $g_n$ (using mollifiers $\eta_n$). The authors first solve the regularized problem:
  $$M_0 u_n = g_n \quad \text{in } B_R(0) \quad (\text{with } u_n = 0 \text{ on } \partial B_R(0))$$
  or in the whole space with appropriate boundary conditions at infinity.

• Variational Characterization:
  The solutions are constructed as maximizers of the associated energy functional:
  $$J(w) = \int \left( \sqrt{1 - |\nabla w|^2} - 1 \right) dx - \sum \alpha_j w(p_j)$$
  The authors establish that the weak solutions correspond to critical points of this functional within the space of spacelike functions ($|\nabla w| \le 1$).

• Comparison Principles and Rearrangement:

  • Symmetric Decreasing Rearrangement: Used to derive uniform bounds on the solutions by comparing them to radial fundamental solutions $\Phi_{N, \alpha}$.
  • Comparison Principle: Used to prove monotonicity with respect to the domain size ($R$) and the source strength ($\alpha_j$), ensuring the existence of limits as $R \to \infty$ and $n \to \infty$.

• Handling Dimensional Differences:

  • Case $N \ge 3$: The energy functional is well-defined. The authors pass to the limit $R \to \infty$ directly to obtain a solution decaying at infinity.
  • Case $N = 2$: The energy functional diverges for the standard fundamental solution. The authors cannot use the direct variational limit. Instead, they employ a normalization technique: they subtract the maximum value of the solution on the ball ($u_{R} - \max u_R$) to keep the sequence locally uniformly bounded, allowing them to extract a convergent subsequence.

• Regularity Analysis:
  The authors prove that the limiting solutions are classical solutions ($C^2$) in $\mathbb{R}^N \setminus \{p_j\}$. They utilize the classification of isolated singularities and Hopf's Lemma to rule out "lightlike segments" (regions where $|\nabla u| = 1$) connecting singular points, ensuring the solution is strictly spacelike away from the vertices.

3. Key Contributions and Results

A. Resolution of Open Questions (for $N \ge 3$)

The paper resolves two open questions raised in previous literature (specifically Bonheure et al. [7]):

1. Variational vs. Weak Solution: It is proven that the maximizer of the energy functional $J_N$ coincides with the unique weak solution of the Euler-Lagrange equation involving Dirac masses.
2. Regularity: The authors establish that the solution is classical ($C^2$) in the regular domain without requiring the singular points to be "well-separated" or the coefficients to be small. The only geometric constraint is that no lightlike segment connects any pair of singular points (which is naturally satisfied for positive masses).

B. Main Theorems

• Theorem 1.1 ($N \ge 3$, Positive Masses):

  • Existence and uniqueness of a weak solution $u_{N, \alpha_0} \in C^2(\mathbb{R}^N \setminus P_{m_0}) \cap C^{0,1}(\mathbb{R}^N)$.
  • Asymptotics: $u(x) \sim c_N \alpha_0 |x|^{2-N}$ as $|x| \to \infty$.
  • Singularity: $|\nabla u| \to 1$ as $x \to p_j$.
  • Variational Property: The solution is the unique maximizer of the energy functional $J_\infty$.

• Theorem 1.2 ($N = 2$, Positive Masses):

  • Existence of a weak solution $u_{2, \alpha_0}$ with asymptotic behavior $u(x) \sim -\frac{\alpha_0}{2\pi} \ln|x| + c$ as $|x| \to \infty$.
  • The solution is unique under the constraint of this specific logarithmic growth.
  • The variational characterization fails due to divergence, but the solution is constructed via the normalization method.

• Theorem 1.3 (Mixed Signs, $N \ge 3$):

  • Considers $M_0 u = \sum \alpha_j \delta_{p_j} - \sum \beta_j \delta_{q_j}$.
  • Establishes a sharper sufficient condition for $C^2$ regularity: $\Phi_{N, \alpha_0}(0) + \Phi_{N, \beta_0}(0) < \text{dist}(P_+, P_-)$. This condition ensures that the "upward" and "downward" cones do not intersect in a way that creates a lightlike segment between a positive and negative source.

• Theorem 5.2 (Infinite Singularities):

  • Extends the construction to infinitely many light-cone vertices $P_\infty = \{p_j\}_{j=1}^\infty$ provided $\sum \alpha_j < \infty$ and the points have a single cluster point.
  • Proves the existence of a minimal positive solution with the same asymptotic decay as the finite case.

4. Significance

• Mathematical Physics: The results provide a rigorous foundation for the Born-Infeld model in electrostatics, where the mean curvature equation describes the potential of a charged particle. The solutions represent the electromagnetic field generated by point charges in a nonlinear theory.
• Geometric Analysis: The paper advances the understanding of maximal hypersurfaces in Lorentzian manifolds, particularly regarding the structure of singularities. It clarifies the relationship between the variational formulation and the distributional equation, a subtle issue in nonlinear PDEs with singular sources.
• Methodological Innovation: The handling of the $N=2$ case via normalization and the derivation of sharp regularity conditions for mixed-sign sources represent significant technical advances over previous works that relied on smallness assumptions or separation conditions.
• Completeness: The paper provides a comprehensive classification of solutions for arbitrary numbers of singularities (finite and infinite) and arbitrary signs of masses, filling gaps in the existing literature regarding the global behavior of these maximal surfaces.

In summary, this work establishes the existence, uniqueness, and regularity of maximal hypersurfaces with prescribed light-cone singularities, resolving long-standing conjectures about the variational nature of these solutions and extending the theory to more complex configurations involving mixed signs and infinite sources.