Two-Phase Free Interface Problems

• Two-phase free interface problems are defined by regions of differing phases separated by an unknown free boundary with discontinuities in physical and geometric quantities.
• Advanced variational methods and PDE techniques reveal sharp regularity results, including C¹,α bounds for 0 < γ < 1 and Log-Lipschitz continuity for γ = 0.
• The analysis has practical implications in cavitation, fluid vortex dynamics, and reaction-diffusion, employing innovative comparison methods and stability estimates.

A two-phase free interface problem involves regions of different physical or mathematical phases separated by an a priori unknown interface—referred to as the "free boundary"—across which physical or geometric quantities may have discontinuities or singular behaviors. The interface's evolution and regularity are governed simultaneously by the bulk equations in each phase and specific coupling (jump or transmission) conditions prescribed on the interface. These problems arise in nonlinear elliptic and parabolic PDEs, with singular or degenerate structures arising from non-differentiable potentials, discontinuous coefficients, or degenerate absorption terms. Prominent applications include cavitation, two-phase fluid mechanics, phase transitions, and combustion. The analysis of two-phase free interface problems requires advanced PDE theory to address singularities at the interface and to obtain sharp regularity estimates for both solutions and the free boundary itself.

1. Variational Formulation and Singular Energies

A general two-phase free interface problem is formulated as the minimization of variational energies with singular, non-differentiable terms. A canonical example is the minimization of: $$J_\gamma(v) = \int_\Omega \left( |\nabla v(x)|^p + F_\gamma(v(x)) + f(x) v(x) \right) dx$$ where $v$ is sought in an appropriate Sobolev space with boundary data, $p > 1$, and $F_\gamma$ is a non-smooth potential that encodes the two-phase nature: $$F_\gamma(v) = \lambda_+ (v^+)^{1+\gamma} + \lambda_- (v^-)^{1+\gamma}, \qquad (0 < \gamma \leq 1)$$ with $v^+ = \max\{v, 0\}$, $v^- = \min\{v, 0\}$, and in the limiting case $\gamma = 0$, $$F_0(v) = \lambda_+ \chi_{\{v>0\}} + \lambda_- \chi_{\{v<0\}}$$ where $\chi$ denotes the characteristic function. The parameters $\lambda_\pm$ and $\gamma$ modulate the strength and discontinuity of absorption/jump effects between phases.

The associated Euler–Lagrange equations become singular along the free interface $\{u = 0\}$. For $u$ minimizer: $$\Delta_p u = \lambda_+ (u^+)^{\gamma-1} \chi_{\{u>0\}} - \lambda_- |u^-|^{\gamma-1} \chi_{\{u<0\}} + f(x)$$ where $$\Delta_p u = \nabla \cdot (|\nabla u|^{p-2} \nabla u)$$ is the $p$-Laplacian. For $\gamma = 0$, the right-hand side becomes distributional, and the interface condition emerges as a flux jump: $$|\nabla u^+|^{p-2} (\nabla u^+ \cdot \nu) - |\nabla u^-|^{p-2} (\nabla u^- \cdot \nu) = \lambda_+ - \lambda_-$$ with $\nu$ the unit normal to the interface.

This framework captures a wide range of problems, including degenerate absorption, obstacle-type systems, and models with non-smooth double-well potentials.

2. Degenerate Operators and Interface Singularities

Degenerate operators, most notably the $p$-Laplacian with $p \neq 2$, induce geometric and analytic challenges distinct from linear cases. Regularity theory for $p$-harmonic functions provides only $C^{1,\alpha}$ continuity, with the exponent $\alpha$ dependent on $p$ and the spatial dimension. The presence of a degenerate absorption term $u^{\gamma-1}$, especially as $\gamma \to 0$, causes the right-hand side of the Euler–Lagrange equation to blow up as the interface is approached, and standard tools, such as monotonicity formulas, are unavailable. The interface is not prescribed and must be found as part of the solution—often an implicit, highly non-smooth set along which the differential operator degenerates or becomes singular.

The case $\gamma = 0$ corresponds to the sharpest singular regime, typical of cavitation, jet, or vortex-core problems, where the model enforces the absorption/discontinuity directly via the indicator functions. Such constructions appear in heterogeneous Prandtl–Batchelor-type models in fluid mechanics (vortex regions with prescribed vorticity surrounded by irrotational flow), in chemical diffusion, and in other free boundary phenomena.

3. Sharp Regularity Theory for Minimizers

The regularity of minimizers in these problems is highly sensitive to the degeneracy and the absorption singularity. The principal findings are:

• For $0 < \gamma < 1$, every local minimizer is $C^{1,\alpha}$ in the interior, with the sharp Hölder exponent $\alpha$ dependent on dimension $n$, $p$, and $\gamma$:

$$\alpha = \min \left\{ \alpha_p,\ \frac{\gamma(q-n)}{p-\gamma(p-1)q} \right\}$$

where $\alpha_p$ is the optimal exponent for $p$-harmonic functions and $q > n$ describes the integrability of $f$.

• For $\gamma = 0$, minimizers are universally Log-Lipschitz continuous:

$|u(x) - u(y)| \le C |x - y| \cdot \log|x - y|$

at the borderline integrability $f \in L^n(\Omega)$.

These regularity results are sharp—that is, optimal with respect to the structure of the coefficients, operator degeneracy, and singularity strength. The approach leverages comparison with $p$-harmonic functions, Campanato-type arguments, and the fine analysis of singular absorption, extending beyond purely variational minimality to encompass viscosity/weak formulations for the interface conditions.

The distinction between the $C^{1,\alpha}$ and Log-Lipschitz regimes reflects a phase transition in the interface regularity as the strength of the singular potential changes ($\gamma \searrow 0$).

4. Physical Models, Applications, and Interface Geometry

Such two-phase free interface problems, as formulated above, are central to multiple physical phenomena, including:

• Heterogeneous cavitation and fluid vortex dynamics: The limiting case ($\gamma=0,\,p=2$) yields Prandtl–Batchelor-type models where constant vorticity regions (jets or cavities) are embedded within irrotational flow, and the interface is determined as a sharp vortex boundary.
• Reaction–diffusion in porous media: The absorption term models concentrated sources or sinks where chemical density undergoes sudden disappearance or ignition at the interface.
• Obstacle-type systems with degenerate nonlinearities: The potential encodes non-smooth phase constraints, typical of contact or phase separation problems.

The improved regularity of both solutions and interfaces is pivotal for analyzing interface stability, shape optimization, and geometric evolution. For instance, in cavitation and vortex core detection, the sharp $C^{1,\alpha}$ or Log-Lipschitz continuity ensures geometric control of the interface, excluding fractal or pathological behaviors in relevant parameter regimes.

5. Novel Analytical Techniques and Stability

The derivation of these sharp regularity results introduces several innovations:

• Quantitative $C^{1,\alpha}$ regularity via effective comparison functions: Using suitable $p$-harmonic comparison and Campanato's method, the regularity up to the free boundary is achieved, showing that singular absorption does not destroy interior gradient bounds for positive $\gamma$.
• Asymptotically optimal Log-Lipschitz bounds at the degenerate endpoint ($\gamma = 0$), not attainable via classical energy estimates.
• Stability and convergence of minimizers as $\gamma \to 0$: Minimizers of $J_\gamma$ converge to minimizers of the limiting cavitation energy $J_0$, demonstrating the robustness of the approach across the parameter space.
• Lower gradient (non-degeneracy) bounds close to the free boundary, ensuring sharp transitions between phases.
• Novel comparison arguments and BMO-based tools that compensate for the lack of monotonicity formulas in the degenerate regime.

These methods are essential for extending regularity theory from variational to fully nonlinear, degenerate, or singular free boundary PDEs.

6. Broader Implications, Limitations, and Future Directions

The analytical framework and regularity theory established generalize and strengthen classical results, providing explicit dependence of regularity exponents on dimension, operator degeneracy, and absorption singularity. Notably, the results are novel even for linear ($p=2$) cases—a context where the interplay between non-differentiable potentials and interface geometry had not been fully resolved.

Limitations stem from the intrinsic singularity: for example, in the strongest singular regime ($\gamma=0$), one can at best expect Log-Lipschitz continuity; full Lipschitz or higher regularity is out of reach without further structural assumptions.

This body of work underpins subsequent advances in the paper of more complex or higher-dimensional two-phase degenerate free boundary models, including those with variable exponent or nonlocal (fractional) operators, furnishing a unifying analytic foundation for the geometric and functional analysis of such interfaces.

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Key Regularity Table

Parameter regime	Regularity of minimizer	Dependence of exponent
$0 < \gamma < 1$	$C^{1,\alpha}$	$\alpha$ explicit in $n$, $p$, $\gamma$
$\gamma = 0$	Log-Lipschitz	Universal

This classification highlights the singular limit's effect on regularity and encapsulates the main analytical insight provided by (Leitão et al., 2012).