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Lipschitz regularity of solutions to two-phase $p$ -Laplacian free boundary problems with right hand side

This paper establishes the optimal local Lipschitz continuity of viscosity solutions for two-phase free boundary problems involving the $p$ -Laplacian with a non-zero right-hand side, while also proving local Hölder continuity for a broader class of such problems.

Original authors: Fausto Ferrari, Claudia Lederman

Published 2026-03-17

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Fausto Ferrari, Claudia Lederman

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a landscape architect tasked with designing a terrain that separates two distinct regions: a "positive" valley (where the ground is above sea level) and a "negative" valley (where the ground is below sea level). The line where these two meet is the Free Boundary.

Your job is to shape this terrain so that it follows specific physical laws, but there's a catch: the rules for how the ground slopes change depending on where you are, and there's a constant "wind" or "force" pushing on the ground (the Right Hand Side).

This paper by Ferrari and Lederman is about proving that this terrain, no matter how complex the rules or the wind, will always be smooth enough to walk on without tripping over a sudden, jagged cliff. In mathematical terms, they prove the solution is Lipschitz continuous (it has a bounded slope).

Here is the breakdown of their work using everyday analogies:

1. The Problem: The Shifting Sand and the Wind

The Terrain ( $u$ ): Imagine a flexible sheet of rubber representing the ground.
The Two Phases: One side of the sheet is pushed up (positive), the other is pushed down (negative).
The Free Boundary: The exact line where the sheet crosses sea level. This line isn't fixed; it moves and shapes itself based on the physics.
The $p$ -Laplacian: This is the rulebook for how the rubber sheet bends.
- If $p=2$ , the rubber is standard and behaves like a normal trampoline (linear).
- If $p \neq 2$ , the rubber is "weird." It gets super stiff when you try to bend it sharply (degenerate) or super floppy when the slope is gentle (singular). It's like trying to walk on mud that hardens when you step hard, or on ice that melts when you step lightly.
The Right Hand Side ( $f$ ): This is the "wind" or external force. In many previous studies, scientists assumed the wind was zero (calm day). This paper tackles the much harder problem where the wind is blowing constantly and unevenly.

2. The Boundary Rule: The "Handshake"

At the Free Boundary (the sea level line), the two sides of the rubber sheet must "shake hands."

The slope of the positive side ( $u^+_\nu$ ) must match a specific function of the slope of the negative side ( $u^-_\nu$ ).
Think of it like a dance: If the negative side leans at a 45-degree angle, the positive side must lean at a specific angle dictated by the function $G$ . If they don't match, the solution is invalid.

3. The Big Question: Is the Terrain Safe?

The central question mathematicians have been asking is: Is the slope of this terrain ever infinite?
Could there be a point where the ground suddenly drops off a vertical cliff?

The Goal: Prove that the slope is always finite. No matter how hard the wind blows or how weird the rubber is, you can always find a maximum steepness. This is called Lipschitz Regularity. It's the "optimal" (best possible) level of smoothness for this type of problem.

4. The Solution: The "Zoom-In" Strategy

The authors use a clever strategy to prove the slope is safe. Imagine you are looking at a map of the terrain.

Step 1: The "Flatness" Assumption
They start by assuming that if you zoom in really close to the Free Boundary, the terrain looks almost flat, like a calm lake.

Analogy: If you look at a beach from space, it looks like a flat line. If you zoom in, you see waves. If you zoom in really close, the sand looks flat again.

Step 2: The "Blow-Up" Technique
They take a tiny piece of the terrain and stretch it out (zoom in infinitely).

Because of the specific rules they set for the "dance" (the function $G$ ), when they zoom in on a "flat" boundary, the weird rubber sheet ( $p$ -Laplacian) and the wind ( $f$ ) eventually behave like a simple, straight line.
The complex, non-linear rules simplify into a basic, linear rule where the slopes on both sides are just equal ( $G(t) = t$ ).

Step 3: The "No-Jump" Discovery
In this zoomed-in, simplified world, they prove that the slopes on both sides of the boundary must match perfectly. There is no "jump" or cliff.

The Metaphor: It's like realizing that if two dancers are moving in perfect sync on a tiny stage, they can't suddenly jump apart. The continuity of the dance forces them to stay connected.

Step 4: Zooming Back Out
Once they prove the slope is safe in the "zoomed-in" world, they use a logical chain reaction to prove it must be safe in the "real" world.

If the slope were infinite in the real world, it would look like a vertical cliff even when zoomed in. But they just proved that zoomed-in cliffs are impossible. Therefore, the real-world cliff cannot exist.

5. Why This Matters

New Ground: Previous work only solved this for calm days (no wind) or for standard rubber ( $p=2$ ). This paper solves it for the "stormy" days (non-zero wind) and the "weird rubber" ( $p \neq 2$ ).
Real World Applications: This math isn't just abstract. It applies to:
- Fluid Dynamics: How oil and water separate in porous rock (like in oil wells).
- Material Science: How cracks form in materials under stress.
- Optimal Design: Designing structures that use the least amount of material while withstanding specific forces.

Summary

Ferrari and Lederman proved that even when you have a complex, non-linear material being pushed by a constant, uneven force, the boundary between its two states will never develop a jagged, vertical cliff. The terrain will always have a "safe" maximum slope. They did this by showing that if you look closely enough, the chaos simplifies into a smooth, predictable line.

1. Problem Statement

The paper investigates the regularity of viscosity solutions to a two-phase free boundary problem governed by the $p$ -Laplacian operator ( $\Delta_p u = \text{div}(|\nabla u|^{p-2}\nabla u)$ ) with a non-zero right-hand side ( $f$ ).

The specific problem is defined in a bounded domain $\Omega \subset \mathbb{R}^n$ as:
$\begin{cases} \Delta_p u = f, & \text{in } \Omega^+(u) \cup \Omega^-(u), \\ u^+_\nu = G(u^-_\nu, x), & \text{on } F(u) := \partial \Omega^+(u) \cap \Omega, \end{cases}$
where:

$\Omega^+(u) = \{x \in \Omega : u(x) > 0\}$ and $\Omega^-(u) = \{x \in \Omega : u(x) \leq 0\}^\circ$ .
$F(u)$ is the free boundary.
$u^+_\nu$ and $u^-_\nu$ are the inward normal derivatives on the positive and negative phases, respectively.
$f \in L^\infty(\Omega)$ is continuous in the phases.
$G(t, x)$ is a strictly increasing function satisfying specific asymptotic and regularity conditions (detailed below).

The primary goal is to establish the local Lipschitz continuity of the solution $u$ , which is considered the optimal regularity for this class of problems.

2. Methodology

The authors employ a strategy inspired by the work of De Silva, Ferrari, and Salsa (specifically the approach in [DS] for uniformly elliptic fully nonlinear operators), adapted to the singular/degenerate nature of the $p$ -Laplacian. The methodology involves several key steps:

Viscosity Solution Framework: The analysis is conducted entirely within the framework of viscosity solutions. This allows for broader applicability compared to classical or weak solutions, particularly when dealing with the singular ( $1 < p < 2$ ) or degenerate ( $p > 2$ ) behavior of the operator.
Scaling and Compactness: The proof relies heavily on blow-up arguments. By rescaling the solution $u$ and the data ( $f$ and $G$ ), the authors analyze the behavior of the solution at small scales.
Flatness Implies Regularity: A central pillar of the proof is the "flatness implies $C^{1,\gamma}$ " principle. If a solution is sufficiently close to a "two-plane" solution (a linear profile $U_\beta(x) = \alpha x^+ - \beta x^-$ ) in a ball, then the free boundary is $C^{1,\gamma}$ and the solution is $C^{1,\gamma}$ up to the boundary.
Refinement of Previous Results: The authors revisit and refine results from their previous work ([FL3]). They extend $C^{1,\gamma}$ regularity theorems for flat free boundaries to more general classes of boundary conditions $G$ and specific cases (homogeneous $f \equiv 0$ , linear case $p=2$ , and autonomous $G(t)$ ).
Alternative Lemma (Lemma 5.2): The core of the Lipschitz proof is an alternative lemma. For a solution in a ball, either:
- The solution is Lipschitz with a controlled gradient, OR
- The solution's $L^\infty$ norm decays significantly upon rescaling (implying the gradient is small).
  By iterating this alternative, the authors prove that the gradient cannot blow up, establishing global Lipschitz bounds.

3. Key Assumptions

The main theorem requires the function $G(t, x)$ to satisfy:

(P1) Strictly increasing and $G(t, x) \to \infty$ as $t \to \infty$ .
(P2) Asymptotic linearity: $\frac{\partial G}{\partial t} \to 1$ and $\frac{\partial^2 G}{\partial t^2} = O(1/t)$ as $t \to \infty$ .
(P3) Uniform Hölder continuity in $x$ .
(P4) (Specifically for the inhomogeneous case $f \not\equiv 0$ and $p \neq 2$ ): $G(t, x) \equiv G(t)$ with $t^{-1}G(t)$ strictly decreasing for large $t$ . This condition handles the technical difficulties arising from the non-zero right-hand side combined with the nonlinearity of the $p$ -Laplacian.

4. Main Results

Theorem 1.1: Optimal Regularity (Lipschitz Continuity)

The primary result states that if $u$ is a viscosity solution to the problem (1.1) in $B_1$ satisfying the assumptions on $f$ and $G$ , then $u$ is locally Lipschitz continuous. Specifically:
$\|\nabla u\|_{L^\infty(B_{1/2})} \leq C(\|u\|_{L^\infty(B_{3/4})} + 1)$
where $C$ depends on dimension $n$ , $p$ , $\|f\|_\infty$ , and $G$ .

Significance: This is the first proof of Lipschitz continuity for the general two-phase $p$ -Laplacian problem with a non-zero right-hand side. It is also new even in the homogeneous case ( $f \equiv 0$ ) for general $p \neq 2$ , as previous literature mostly covered the $p=2$ (linear) case or specific variational minimizers.

Theorem 1.2: Local Hölder Continuity

Before proving Lipschitz continuity, the authors establish local Hölder continuity ( $u \in C^{0,\alpha}$ ) for a broader class of problems where $G$ satisfies a linear growth condition ( $\sigma t \leq G(t, x) \leq \sigma^{-1} t$ ). This result is of independent interest and serves as a stepping stone for the Lipschitz proof.

Theorems 3.5, 3.7, and 3.9: Refined Flatness Results

The authors prove that "flatness implies $C^{1,\gamma}$ " under relaxed assumptions:

Theorem 3.5: Applies when $f \equiv 0$ with a more general $G$ (satisfying condition H3').
Theorem 3.7: Applies when $p=2$ (Laplace operator) with a general $G$ .
Theorem 3.9: Applies when $G$ is autonomous ( $G(t, x) \equiv G(t)$ ) with $f \not\equiv 0$ .
These refinements are crucial technical tools that allow the main theorem to handle the specific structure of the $p$ -Laplacian with non-zero source terms.

5. Significance and Contributions

Optimal Regularity for Nonlinear Operators: The paper resolves the question of optimal regularity for two-phase free boundary problems involving the $p$ -Laplacian with non-zero sources. Prior to this, monotonicity formulas (used for $p=2$ in [CJK]) were not available for $p \neq 2$ , making the proof significantly more difficult.
Handling Singular/Degenerate Operators: The authors successfully adapt techniques from uniformly elliptic theory to the $p$ -Laplacian, which is singular when $1 < p < 2$ and degenerate when $p > 2$ . The presence of a non-zero right-hand side ( $f \not\equiv 0$ ) makes the factor $|\nabla u|^{p-2}$ in the operator's structure non-negligible, a difficulty the authors overcome through careful scaling and the use of the companion paper [FL3].
Viscosity Solution Generality: By working strictly with viscosity solutions, the results apply to a wider class of solutions than those obtained via variational methods (minimizers of functionals), increasing the applicability of the theory.
New Even in Homogeneous Case: The authors note that their results are new even when $f \equiv 0$ for general $p$ , as existing literature on Lipschitz continuity across the free boundary was largely restricted to the linear case or specific variational settings.

In summary, this paper provides a comprehensive and rigorous proof of the Lipschitz continuity of solutions to a challenging class of nonlinear free boundary problems, bridging the gap between linear theory and fully nonlinear degenerate/singular operators.