$C^{\infty}$ regularity of the Alt-Phillips Functional for negative powers

Abstract: In this paper, we study the regularity of the free boundary for minimizers of the Alt-Phillips functional with negative exponent [\mathcal{E}γ(u)=\intΩ\frac{1}{2}|\nabla u|^2+\frac{1}γu^{-γ}χ_{{u>0}}dx,\quadγ\in(0,2).] We proved that the free boundaries are $C<sup>{\infty}$ at regular points.

• The paper establishes C∞ regularity at every regular free boundary point for the Alt-Phillips functional with negative power singularities.
• It employs iterative bootstrap methods, linearization, and refined boundary Harnack techniques to overcome singular behavior near u = 0.
• Implications include improved interface models and an extension of Schauder theory to degenerate elliptic problems with negative singular drift.

$C^{\infty}$ Regularity of the Alt-Phillips Functional for Negative Powers

Introduction and Background

The analysis of minimizers for variational energy functionals with free boundaries represents a central theme in the calculus of variations. Classical models such as the obstacle problem and the Bernoulli free boundary problem have been extensively studied, leading to sharp regularity results for both solutions and free boundaries. The Alt-Phillips functional represents a significant family of such variational problems, embodying a range of singular potentials parameterized by a power $\gamma$. This paper investigates the case where the potential is a negative power, i.e., the energy functional takes the form:

$$\mathcal{E}_{\gamma}(u) = \int_{\Omega} \frac{1}{2}\lvert \nabla u \rvert^2 + \frac{1}{\gamma}u^{-\gamma}\chi_{\{u>0\}}\; dx, \quad \gamma \in (0,2).$$

The study is motivated by applications to models for cohesive forces in low-density liquid regimes and by connections to phase transition theory. For negative exponents, the singular behavior as $u \to 0$ dramatically affects the structure and regularity of minimizers, giving rise to challenging free boundary problems.

Historically, for nonnegative powers, regularity theory for the free boundary in such problems has achieved sharp results, including $C^\infty$ regularity at regular points. The difficult negative power regime poses new technical obstacles because the singularity enhances as $u$ vanishes. Prior work established $C^{1,\delta_0}$ regularity for the free boundary (with a singular set of codimension at least three). The primary contribution of this paper is to obtain full $C^\infty$ regularity at all regular points of the free boundary for these negative power functionals.

Precise Problem Formulation

The minimizers of $\mathcal{E}_\gamma$ are considered under nonnegative boundary data. The associated Euler-Lagrange equation in the positivity set $\{u>0\}$ reads:

$\gamma$0

At the free boundary $\gamma$1, a precise boundary expansion holds:

$\gamma$2

Here, $\gamma$3 is a regular free boundary point, and $\gamma$4 is a constant explicitly determined by the singularity exponent. The overarching aim is to establish the $\gamma$5 regularity of $\gamma$6 near regular points and the related smoothness of scaled versions of $\gamma$7 near the free boundary.

Methodology

The core strategy is to iteratively refine the regularity of the free boundary and associated quantities by analyzing suitable linearized operators.

Key Innovations and Tools:

• Distance Function Techniques: The regularized distance function $\gamma$8 plays a central role, allowing for precise expansions of $\gamma$9 near the free boundary. The analysis relies on delicate estimates for $$\mathcal{E}_{\gamma}(u) = \int_{\Omega} \frac{1}{2}\lvert \nabla u \rvert^2 + \frac{1}{\gamma}u^{-\gamma}\chi_{\{u>0\}}\; dx, \quad \gamma \in (0,2).$$0 and its derivatives up to arbitrary order in $$\mathcal{E}_{\gamma}(u) = \int_{\Omega} \frac{1}{2}\lvert \nabla u \rvert^2 + \frac{1}{\gamma}u^{-\gamma}\chi_{\{u>0\}}\; dx, \quad \gamma \in (0,2).$$1 domains.
• Linearization and Characteristic Exponents: Near the free boundary, $$\mathcal{E}_{\gamma}(u) = \int_{\Omega} \frac{1}{2}\lvert \nabla u \rvert^2 + \frac{1}{\gamma}u^{-\gamma}\chi_{\{u>0\}}\; dx, \quad \gamma \in (0,2).$$2 behaves like $$\mathcal{E}_{\gamma}(u) = \int_{\Omega} \frac{1}{2}\lvert \nabla u \rvert^2 + \frac{1}{\gamma}u^{-\gamma}\chi_{\{u>0\}}\; dx, \quad \gamma \in (0,2).$$3. Differentiation and linearization yield that each $$\mathcal{E}_{\gamma}(u) = \int_{\Omega} \frac{1}{2}\lvert \nabla u \rvert^2 + \frac{1}{\gamma}u^{-\gamma}\chi_{\{u>0\}}\; dx, \quad \gamma \in (0,2).$$4 approximately solves

$$\mathcal{E}_{\gamma}(u) = \int_{\Omega} \frac{1}{2}\lvert \nabla u \rvert^2 + \frac{1}{\gamma}u^{-\gamma}\chi_{\{u>0\}}\; dx, \quad \gamma \in (0,2).$$5

for explicit constants $$\mathcal{E}_{\gamma}(u) = \int_{\Omega} \frac{1}{2}\lvert \nabla u \rvert^2 + \frac{1}{\gamma}u^{-\gamma}\chi_{\{u>0\}}\; dx, \quad \gamma \in (0,2).$$6, $$\mathcal{E}_{\gamma}(u) = \int_{\Omega} \frac{1}{2}\lvert \nabla u \rvert^2 + \frac{1}{\gamma}u^{-\gamma}\chi_{\{u>0\}}\; dx, \quad \gamma \in (0,2).$$7, and $$\mathcal{E}_{\gamma}(u) = \int_{\Omega} \frac{1}{2}\lvert \nabla u \rvert^2 + \frac{1}{\gamma}u^{-\gamma}\chi_{\{u>0\}}\; dx, \quad \gamma \in (0,2).$$8 smooth. The characteristic exponents for the associated one-dimensional problem are

$$\mathcal{E}_{\gamma}(u) = \int_{\Omega} \frac{1}{2}\lvert \nabla u \rvert^2 + \frac{1}{\gamma}u^{-\gamma}\chi_{\{u>0\}}\; dx, \quad \gamma \in (0,2).$$9

with $u \to 0$0 for $u \to 0$1, which is critical for the method's success.

• Adaptation of Higher Regularity Techniques: Building upon and adapting the method from [Restrepo and Ros-Oton, 2025], originally for positive powers, the authors manage the distinct exponent structure in the negative power case. The technical innovation is the ability to propagate regularity for $u \to 0$2 but above the threshold $u \to 0$3.
• Boundary Harnack and Schauder Theory for Degenerate Elliptic Equations: Regularity transfer for the quotients of partial derivatives (tangential to the free boundary) is achieved by analyzing

$u \to 0$4

for $u \to 0$5, employing higher-order boundary Harnack results [TTV, 2024].

Iterative Bootstrap:

The analysis proceeds by induction on regularity: if the free boundary is $u \to 0$6, one can show $u \to 0$7 and $u \to 0$8 are $u \to 0$9 up to the boundary, which in turn enables one, via boundary Harnack technology, to upgrade the regularity of normal derivatives and hence the boundary itself to $C^\infty$0. This process iterates to $C^\infty$1.

Main Results and Numerical Statements

The central theorem is:

At every regular free boundary point $C^\infty$2, - The free boundary $C^\infty$3 is $C^\infty$4 in a neighborhood of $C^\infty$5. - The functions $C^\infty$6 and $C^\infty$7 are $C^\infty$8 up to the free boundary near $C^\infty$9.

This result removes all possible singularities for the regular part of the boundary, matching the best known results for the positive power Alt-Phillips functional and extending the scope of highly regular free boundary theory to the negative exponent regime.

Implications and Theoretical Consequences

The $u$0 regularity established here implies a complete compatibility between the asymptotic expansions for $u$1 near the free boundary and the smoothness of the geometric free boundary surface itself. This has several consequences:

• Justification of Interface Asymptotic Models: The result validates the formal expansions and sharp interface limits assumed in homogenization, $u$2-convergence, and matched asymptotic analyses for singular perturbations in free boundary problems involving strongly singular potentials.
• Stability of Regular Points: The result underscores the rigidity and robustness of the structure of regular points for these variational models, leaving singularities confined to a negligible set (of codimension at least three).
• Technical Advances in Degenerate Elliptic Theory: The adaptation and extension of Schauder and boundary Harnack techniques to operators with negative singular drift (but $u$3) is a notable technical achievement, with potential cross-application to other free boundary and obstacle-type problems governed by highly singular potentials.

Prospects for Future Research

• Classification of Singular Points and Global Minimizers: Although regular points are now completely understood, the classification of possible singularities and global minimizers—particularly in high dimensions and for the endpoint exponents—is open.
• Quantitative Estimates and Geometric Measure Theory: Finer quantitative understanding of the singular set and its possible structure remains a natural extension.
• Extension to Systems and Anisotropic Functionals: Whether the methods generalize to systems, vector-valued minimizers, or non-isotropic settings is open.
• Applications to Stochastic and Mean Field Models: The mathematical structure here is closely related to models in statistical physics and stochastic interfaces, suggesting further application to mean field games and probabilistic free boundary problems.

Conclusion

This work achieves sharp $u$4 regularity of free boundaries for the Alt-Phillips functional with negative power singularities at all regular points. The analysis demonstrates that the highly singular nature of the potential does not preclude the emergence of smooth free boundaries, provided the regularity bootstrapping can be initiated. These findings place the regularity theory for negative power variational problems on par with their positive power counterparts, setting a firm foundation for subsequent developments in the analytic and geometric theory of free boundary problems.

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Reference: "$u$5 regularity of the Alt-Phillips Functional for negative powers" (2604.15863)