A Local Linking Theorem for Relativistic Action Functionals

Technical Summary: A Local Linking Theorem for Relativistic Action Functionals

Problem Statement
The paper addresses the multiplicity of non-constant solutions for variational problems arising from relativistic action principles. Specifically, it targets a class of non-smooth functionals of Szulkin-type, defined on a Hilbert space $H$ but possessing an effective domain $K \subset H$ where the functional is finite. A prototypical example is the relativistic action functional:
$$I(q) = \int_{\Omega} \left( 1 - \sqrt{1 - |\nabla q|^2} + F(x, q, \nabla q) \right) dx,$$
where the domain $K$ is constrained by $|\nabla q| \leq 1$.

The central challenge is that these functionals are not $C^1$ (they are non-smooth) and, crucially, do not satisfy the classical strong Palais–Smale (PS) condition in the topology of $H$. Instead, compactness for Palais–Smale sequences is only available with respect to a weaker topology $\tau$ induced by the effective domain $K$. The authors seek to extend the classical Brezis–Nirenberg local linking theorem—which guarantees at least two non-zero critical points under a specific geometric condition and the strong PS condition—to this non-smooth, $\tau$-compact setting.

Methodology
The authors develop an abstract multiplicity theorem (Theorem 2.1) for functionals $I = \Psi + F$ in the class $X(K)$, where $\Psi$ is a proper, convex, lower semicontinuous functional with domain $K$, and $F$ is a $C^1$ perturbation. The methodology involves several key technical innovations:

1. Topology and Compactness: The framework utilizes a topology $\tau$ on $K$ such that bounded sets in an auxiliary Banach space $E$ (where $H$ is continuously embedded) are $\tau$-compact. The compactness assumption is replaced by a $(PS)^*_\tau$ condition: every Palais–Smale sequence at a non-zero level admits a subsequence converging in $\tau$ to a critical point.
2. Ekeland–Lasry Regularization: To handle the lack of smoothness and the absence of a strong PS condition, the authors employ the Ekeland–Lasry regularization technique. They define a regularized functional $I_\varepsilon$ which is $C^1$ and shares the same critical points as $I$.
3. Perturbative Construction of Min-Max Geometry: A major difficulty arises because the negative gradient flow for $I_\varepsilon$ is naturally associated with the strong topology of $H$, whereas the compactness of the original functional $I$ is only guaranteed in $\tau$. The authors overcome this by constructing a continuous map $\gamma^*$ using a specific perturbation $\omega_\varepsilon$ of the gradient flow trajectory. This map exploits the regularity of $I_\varepsilon$ while ensuring the image remains within the effective domain $K$ and respects the $\tau$-compactness properties of $I$.
4. Local Linking Condition: The theorem assumes a local linking geometry where the functional is non-positive on a finite-dimensional subspace $H_1$ and strictly positive (bounded below by a function of the norm) on the complementary subspace $H_2$ near the origin. An additional convexity assumption ($I + \mu\|\cdot\|_H^2$ is convex) is required to facilitate the regularization.

Key Contributions and Results
The primary contribution is Theorem 2.1, which establishes that if a functional $I \in X(K)$ is bounded from below with a negative global minimum, satisfies the $(PS)^*_\tau$ condition, and exhibits a local linking geometry, then $I$ possesses at least two non-zero critical points.

The paper applies this abstract result to two specific relativistic models:

1. The Lorentz Force Equation:

   • Context: The dynamics of a charged particle in an electromagnetic field with vanishing scalar potential, described by the equation $\frac{d}{dt}\left(\frac{\dot{q}}{\sqrt{1-|\dot{q}|^2}}\right) = E(t,q) + \dot{q} \times B(t,q)$.
   • Result: Under assumptions of isolated equilibria for the electromagnetic field and specific decay/boundedness conditions on the vector potential, the authors prove the existence of at least two non-constant $T$-periodic solutions. This extends previous results (e.g., [24, Theorem 2.1]) which established the existence of only one global minimizer.

2. The Prescribed Mean Curvature Equation in Minkowski Space:

   • Context: The Dirichlet problem $\text{div}\left(\frac{\nabla q}{\sqrt{1-|\nabla q|^2}}\right) = f(x, q)$, where the operator corresponds to the mean curvature in Minkowski space.
   • Result: For nonlinearities of the form $f(x, s) = a(x)s - g(s)$, where $a(x)$ generates negative eigenvalues for the associated linear operator and $g$ satisfies specific growth conditions at zero, the paper proves the existence of at least two non-trivial solutions. This is noted as the first application of a local linking theorem to this specific operator.

Significance and Claims
The paper claims to provide the first extension of the Brezis–Nirenberg local linking theorem to the framework of Szulkin-type functionals arising in relativistic mechanics, where the strong Palais–Smale condition fails.

• Theoretical Advancement: The work resolves the incompatibility between the strong topology required for Ekeland–Lasry regularization and the weak topology ($\tau$) where compactness holds for relativistic functionals. The authors explicitly state that previous extensions of the local linking theorem (e.g., [34]) relied on positivity assumptions near the boundary of the effective domain or strong compactness, which are incompatible with the relativistic models studied here.
• Methodological Novelty: The construction of the map $\gamma^*$ via the perturbation $\omega_\varepsilon$ is presented as the core technical achievement, allowing the recovery of the necessary min-max geometry without requiring the flow to stay within the interior of the domain or satisfy strong compactness.
• Application Scope: The results demonstrate that multiplicity of solutions can be guaranteed for relativistic systems under natural physical assumptions (such as isolated equilibria or specific spectral properties of the linear part), even when the action functional is non-smooth and lacks strong compactness.

The authors maintain a modest tone regarding the scope, noting that their results apply to specific subclasses of functionals and that the convexity assumption (15) is necessary for the regularization procedure. They do not claim to solve the general multiplicity problem for all relativistic variational problems but rather provide a robust framework for a significant class of them.