A Lavrentiev phenomenon in the neo-Hookean model

1. Problem Statement and Context

The paper addresses a fundamental issue in the mathematical theory of nonlinear elasticity: the existence of minimizers for the neo-Hookean energy in three dimensions.

• The Energy Functional: The energy is defined as:
  $$E(u) = \int_{\Omega} \left( |Du|^2 + H(|\det Du|) \right) dx$$
  where $u: \Omega \to \mathbb{R}^3$ is a deformation, $|Du|^2$ represents the elastic stretching, and $H$ is a convex function penalizing volume changes (compression and expansion). The function $H$ satisfies superlinear growth at infinity and a singularity at zero ($H(t) \to \infty$ as $t \to 0$), preventing material interpenetration.
• The Admissible Class: Physical admissibility requires deformations to be orientation-preserving ($\det Du > 0$) and injective (one-to-one almost everywhere) to prevent self-intersection. The authors define a "regular" class $\mathcal{A}_r$ consisting of maps $u \in H^1$ that satisfy these conditions, agree with a boundary datum $b$, and crucially, have an inverse $u^{-1}$ that belongs to $W^{1,1}$ (Sobolev regularity). This class ensures the Condition INV (invertibility) and the divergence identities hold, ruling out pathological cavitation.
• The Core Difficulty: The direct method of the calculus of variations fails for this energy because the class $\mathcal{A}_r$ is not sequentially closed under weak $H^1$ convergence. A sequence of regular maps can converge to a limit that violates the invertibility conditions or develops singularities (cavities) that are not captured by the energy functional $E$ itself.
• The Goal: The authors aim to demonstrate a Lavrentiev phenomenon. This occurs when the infimum of the energy over the regular class $\mathcal{A}_r$ is strictly larger than the infimum over its weak $H^1$-closure $\overline{\mathcal{A}_r}$. This implies that minimizing sequences in the regular class cannot converge to a minimizer within that same class; the "true" minimizer lies in a larger, less regular space.

2. Methodology and Strategy

The proof relies on constructing a specific boundary condition that forces the energy to concentrate on a specific type of singularity known as a dipole.

• The Conti–De Lellis Dipole Map: The authors utilize a singular map $v$ (originally constructed by Conti and De Lellis) defined on a ball $B(0,4)$. This map creates a "dipole" singularity where two distinct regions of the reference configuration (half-balls $a$ and $e$) are brought into contact across a spherical interface $\Gamma$ (a "bubble").
  • In the deformed configuration, matter from region $a$ passes through the origin to fill a cavity nucleated at a different point $0'$.
  • The inverse map $v^{-1}$ has a jump discontinuity across $\Gamma$. Consequently, $v^{-1}$ is in the space of Special Functions of Bounded Variation (SBV) but not in $W^{1,1}$. Thus, $v \notin \mathcal{A}_r$.
• Relaxed Energy: The authors use a relaxed energy functional $\mathcal{F}$ defined on a larger space $\mathcal{B}$ (where $u^{-1} \in BV$).
  $$\mathcal{F}(u) = E(u) + 2\|D_s u^{-1}\|$$
  Here, $\|D_s u^{-1}\|$ is the norm of the singular part of the gradient of the inverse. For the dipole map $v$, the singular part corresponds to the jump across $\Gamma$, contributing an energy cost of exactly $2\pi$.
• Approximation: It is known that $v$ can be approximated by a sequence of regular maps $u_\varepsilon \in \mathcal{A}_r$ such that $u_\varepsilon \rightharpoonup v$ in $H^1$ and $E(u_\varepsilon) \to \mathcal{F}(v) = E(v) + 2\pi$.
• The "Patching" Construction: The map $v$ itself is not a valid boundary condition because it is not a homeomorphism (it collapses segments and disks to the origin). To fix this, the authors construct a bi-Lipschitz homeomorphism $b_\delta$ that:
  1. Agrees with the singular map $v$ outside a small neighborhood $U_\delta$ of the singular set.
  2. Is "patched" (regularized) inside $U_\delta$ to be a smooth, invertible map.
  3. Is constructed using the regularized approximations $u_\varepsilon$ inside the neighborhood, interpolated smoothly to $v$ at the boundary of the neighborhood.

3. Key Contributions and Results

Main Theorem (Theorem 1.1)

The authors prove that for the neo-Hookean energy with specific growth conditions on $H$ (specifically $H(t) \leq c t^{-\alpha}$ near 0 with $\alpha < 1/3$), there exist a domain $\Omega$, a subdomain $\tilde{\Omega}$, and a bi-Lipschitz boundary datum $b$ such that:
$$\inf \{ E(u) : u \in \mathcal{A}_r \} > \inf \{ E(u) : u \in \overline{\mathcal{A}_r} \}$$
where $\overline{\mathcal{A}_r}$ is the weak $H^1$ closure of $\mathcal{A}_r$.

Mechanism of the Gap

• Energy Concentration: The gap is driven by the dipole singularity. Any map in the weak closure $\overline{\mathcal{A}_r}$ that satisfies the boundary condition must essentially replicate the dipole structure of $v$ to minimize energy.
• The Cost: While the relaxed energy $\mathcal{F}$ accounts for the $2\pi$ cost of the singular inverse, the original energy $E$ does not "see" this cost if the map is not in $\mathcal{A}_r$. However, any map in $\mathcal{A}_r$ attempting to approximate this configuration must pay a high energy price to smooth out the singularity, or it simply cannot achieve the low energy of the singular limit.
• Contrast with Previous Work: Unlike previous Lavrentiev examples in elasticity (which relied on $p > d$ coercivity or self-contact mechanisms), this example works in the physically natural regime $p=2 < d=3$. The gap arises not from the forward map's regularity but from the failure of the inverse map's regularity (Sobolev vs. BV) to be preserved under weak convergence.

Variants (Section 4)

The authors show that the phenomenon persists even if one restricts the search to:

1. Axisymmetric maps: The gap remains in the class of axisymmetric deformations.
2. Sobolev Homeomorphisms: The gap exists even when considering the weak closure of the class of Sobolev homeomorphisms satisfying Lusin's condition (N). In these variants, the infimum over the larger class is actually attained, but the minimizer possesses a singularity (dipole) that excludes it from the regular class.

4. Technical Significance

• Failure of Direct Methods: The paper provides a rigorous counterexample showing that the standard strategy of the calculus of variations (minimize in a large space, then prove regularity) fails for the neo-Hookean model in 3D. The minimizer in the relaxed space is inherently singular.
• Role of Invertibility: It highlights that the condition of invertibility (specifically the regularity of the inverse) is a fragile property in 3D elasticity. The "Condition INV" and divergence identities, while physically motivated, are not stable under weak $H^1$ convergence for the critical exponent $p=2$.
• Dipole Singularities: The work establishes the dipole singularity as a fundamental mechanism for energy concentration in nonlinear elasticity, analogous to dipole singularities in harmonic maps into spheres.
• Implications for Modeling: The existence of a Lavrentiev gap suggests that the classical neo-Hookean energy functional is ill-posed for certain boundary data. To obtain well-posed problems, one must either:
  1. Use a relaxed energy (like $\mathcal{F}$) that explicitly penalizes the formation of singularities (cavities/dipoles).
  2. Impose stronger regularity conditions on the admissible class that prevent the formation of such singularities, potentially at the cost of physical realism.

5. Conclusion

Barchiesi et al. demonstrate that the neo-Hookean model in three dimensions exhibits a Lavrentiev gap phenomenon driven by dipole-type singularities. This result proves that the infimum of the energy over physically regular deformations is strictly higher than the infimum over the weak closure of that class. The gap is a direct consequence of the lack of weak compactness of the invertibility condition, specifically the regularity of the inverse map, in the critical Sobolev space $H^1$. This necessitates the use of relaxed energy functionals for a mathematically consistent treatment of nonlinear elasticity in 3D.