Partial regularity for minimizing constraint maps for the Alt-Phillips energy

1. Problem Statement and Context

The Core Problem:
The paper investigates the regularity of minimizing constraint maps for a specific class of energy functionals known as the Alt-Phillips energy.

• Domain: A bounded domain $\Omega \subset \mathbb{R}^n$.
• Target: A smooth domain $M \subset \mathbb{R}^m$ ($m \ge 2$) with a compact, connected complement.
• Functional: The energy functional is defined as:
  $$E_{\lambda, \gamma}[u] = \int_{\Omega} \left( \frac{1}{2}|Du|^2 + \lambda(\rho \circ u)^\gamma \right) dx$$
  where:
  • $u \in W^{1,2}(\Omega; M)$ is the map constrained to take values in $M$.
  • $\rho$ is the signed distance function to the boundary $\partial M$ (positive inside $M$).
  • $\lambda > 0$ is a parameter.
  • $\gamma \in (0, 2)$ is an exponent.

Motivation:
Previous studies have addressed:

1. Dirichlet Energy ($\lambda=0$): Regularity of harmonic maps with constraints (e.g., [5]).
2. Alt-Caffarelli Energy ($\gamma=0$): Free boundary problems with constraints (e.g., [4]).
3. Alt-Phillips Energy (Unconstrained): Regularity without the constraint $u \in M$ (e.g., [6, 7]).

This work bridges the gap by analyzing the Alt-Phillips energy under the constraint $u(x) \in M$. The presence of both the geometric constraint and the source term $\lambda(\rho \circ u)^\gamma$ creates unique analytical challenges not present in the individual cases.

2. Methodology

The author employs a combination of variational methods, elliptic regularity theory, and blow-up techniques. The methodology is structured around overcoming two primary difficulties:

1. The Source Term: The term $\lambda(\rho \circ u)^\gamma$ introduces a component in the Euler-Lagrange equation that is difficult to control for $\gamma \in (0, 1)$ but is crucial for regularity.
2. The Constraint: Existing results for the Alt-Phillips term (without constraints) cannot be directly applied because the constraint $u \in M$ fundamentally alters the comparison maps and the structure of the Euler-Lagrange system.

Key Technical Tools:

• Comparison Maps: Instead of using harmonic functions as competitors (standard in unconstrained problems), the author uses $E_0$-minimizing constraint maps (minimizers of the Dirichlet energy with the same constraint). This ensures competitors remain within the target manifold $M$.
• Maximum Principle for $E_0$-minimizers: A new lemma (Lemma 3.1) is established to control the oscillation of $E_0$-minimizers, which is essential for the $\epsilon$-regularity argument.
• Euler-Lagrange Equations: The author derives specific equations for the distance map $\rho \circ u$ and the projection map $\Pi \circ u$ (Lemmas 2.7 and 2.8).
  • For $\rho \circ u$: $\Delta(\rho \circ u) = \text{Hess}\,\rho(Du, Du) + \lambda\gamma(\rho \circ u)^{\gamma-1}$.
  • For $\Pi \circ u$: A system involving the second fundamental form of $\partial M$.
• Bootstrap Argument: The proof starts with a suboptimal regularity estimate ($C^{1, \alpha}$) and iteratively improves the exponent to reach the optimal regularity.
• Monotonicity Formula: An "almost monotonicity formula" is derived (Lemma 2.2) to handle the extra source term, adapting techniques from Luckhaus [9] and Alt-Caffarelli [4].

3. Key Contributions and Results

A. $\epsilon$-Regularity Theorem (Theorem 1.1)

The central result is an $\epsilon$-regularity theorem stating that if the energy is sufficiently small in a ball, the minimizer is smooth.

• Condition: If the scaled energy $(2R)^{2-n}\int_{B_{2R}} |Du|^2$ and the scaled source term $R^2 \lambda d^\gamma$ are both smaller than a small constant $\epsilon$, then $u$ is regular in a smaller ball $B_R$.
• Regularity Class:
  • For $\gamma \in (0, 2)$: $u \in C^{1, \alpha}$ with $\alpha = \min\left\{ \frac{\gamma}{2-\gamma}, \sigma \right\}$.
  • For $\gamma \in [1, 2)$: If the map stays close to the boundary (within a tubular neighborhood), $u \in C^{1,1}$.
• Significance: This establishes that small energy implies optimal smoothness, a prerequisite for analyzing the singular set.

B. Optimal Partial Regularity (Corollary 1.2)

Using Theorem 1.1 and existing partial regularity theory for stationary maps (Luckhaus [9]), the author proves:

• There exists a relatively closed singular set $\Sigma \subset \Omega$ such that $u$ is $C^{1, \alpha}$ on $\Omega \setminus \Sigma$.
• Dimension of Singular Set: The Hausdorff dimension of the singular set satisfies $\dim_H(\Sigma) \le n - 3$.
  • In 3D ($n=3$), the singular set is discrete (isolated points).
• Boundary Regularity: If $\gamma \ge 1$, the map is $C^{1,1}$ in a neighborhood of the free boundary (where $u$ touches $\partial M$), excluding the singular set.

C. Technical Lemmas

• Lemma 3.1: A maximum principle for $E_0$-minimizing constraint maps, crucial for the iteration scheme.
• Lemma 3.3: Establishes $C^{1,1}$ regularity for the case $\gamma \in [1, 2)$ by combining elliptic estimates for the distance and projection maps.

4. Significance and Impact

1. Extension of Theory: This work extends the theory of constraint maps from the classical Dirichlet energy and the Alt-Caffarelli functional to the broader class of Alt-Phillips functionals with source terms.
2. Handling Constraints with Sources: It resolves the difficulty of controlling the interaction between the geometric constraint ($u \in M$) and the non-linear source term ($\lambda(\rho \circ u)^\gamma$). The author demonstrates that standard techniques for one cannot simply be applied to the other; a new framework using $E_0$-minimizers as competitors is required.
3. Optimal Regularity: The paper provides the optimal Hölder exponent for the gradient ($C^{1, \alpha}$) and establishes $C^{1,1}$ regularity for higher $\gamma$, significantly improving upon previous results (specifically Luckhaus' result [9] which only guaranteed $C^{0, \alpha}$).
4. Foundation for Free Boundary Analysis: By establishing the smoothness of the map away from a small singular set, this paper lays the groundwork for future studies on the structure of the free boundary (the interface where $u$ transitions from the interior of $M$ to $\partial M$) in the presence of source terms.

In summary, Ziganshina's paper successfully generalizes partial regularity theory to a complex variational problem involving both geometric constraints and non-homogeneous source terms, providing sharp regularity estimates and a precise characterization of the singular set.