Partial regularity for variational integrals with Morrey-Hölder zero-order terms, and the limit exponent in Massari's regularity theorem

This paper revisits the partial $C^{1,\alpha}$ regularity theory for minimizers of variational integrals with Morrey-Hölder zero-order terms to establish the sharp dependence of the Hölder exponent on structural assumptions, thereby confirming optimal regularity up to the limit exponent in Massari's theorem for prescribed-mean-curvature hypersurfaces.

The Gist

Imagine you are trying to smooth out a crumpled piece of fabric or a wrinkled sheet of metal. In mathematics, this "smoothing" process is described by equations that model how things settle into their most efficient, lowest-energy shape. These shapes are called minimizers.

This paper is about figuring out just how smooth these shapes can be. Specifically, the authors are asking: If we have a slightly messy or "rough" force acting on our fabric, how smooth will the final result still be?

Here is a breakdown of the paper's concepts using everyday analogies:

1. The Setup: The Fabric and the Wind

Think of the mathematical integral (the formula they are studying) as a piece of fabric stretched over a frame.

• The Fabric ($f$): This part of the equation represents the fabric's natural tendency to stay flat and smooth. The authors assume this fabric is "well-behaved" (mathematically, it's strictly convex and has quadratic growth). It wants to be smooth.
• The Wind ($g$): This is the "zero-order term." Imagine a gust of wind blowing on the fabric, or perhaps a patch of sticky mud on it. This force is messy. It might change strength depending on where you are, and it might not be perfectly smooth itself. The authors call this a Morrey-Hölder condition.
  • Analogy: Think of the wind as a storm. Sometimes it's a gentle breeze (smooth), sometimes it's a chaotic gale (rough). The paper asks: "If the wind is this rough, how much will it ruin the smoothness of the fabric?"

2. The Problem: The "Roughness" Limit

In the past, mathematicians knew that if the wind was very rough, the fabric would end up with wrinkles (singularities). If the wind was very smooth, the fabric would be perfectly smooth.

But there was a "gray area" in the middle. If the wind had a specific type of roughness (measured by something called a Hölder exponent, let's call it $\alpha$), mathematicians could prove the fabric was smooth most of the time, but they weren't sure exactly how smooth it could get. They had a "best guess" for the smoothness, but it wasn't the absolute limit.

3. The Breakthrough: Finding the Perfect Limit

The authors, Thomas Schmidt and Jule Helena Schütt, have done the math to find the exact limit.

• The Discovery: They proved that even with a very specific, somewhat rough wind, the fabric can be smoothed out to a degree of perfection that was previously thought impossible to guarantee.
• The "Sharp Dependence": They created a precise formula. If you tell them exactly how rough the wind is (the parameters $\beta$, $q$, and the function $\Gamma$), they can tell you the exact maximum smoothness ($\alpha$) the fabric will achieve.
  • Analogy: It's like a chef who can tell you exactly how much salt you can add to a soup before it becomes unpalatable. Before, they knew "a little salt is fine, a lot is bad." Now, they know the exact gram limit.

4. The Special Case: Massari's Theorem (The Soap Bubble)

The paper has a very famous application involving soap bubbles (or more technically, surfaces with a specific curvature).

• The Scenario: Imagine a soap bubble trying to form a shape while being pushed by a variable wind (representing a "prescribed mean curvature").
• The Old Result: A mathematician named Massari had a theorem that said, "If the wind isn't too crazy, the bubble is smooth." But his theorem stopped short of the absolute limit.
• The New Result: The authors took their new "perfect limit" formula and applied it to the soap bubble. They proved that the bubble is smooth right up to the very edge of what is physically possible. They closed the gap that had existed for decades.
  • Analogy: Imagine a bridge engineer who knew a bridge could hold up to 10 tons. The authors proved it can actually hold 10.00001 tons, right up to the theoretical breaking point, without collapsing.

5. Why Does This Matter?

You might ask, "Who cares about the exact smoothness of a mathematical fabric?"

• Predictability: In engineering and physics, knowing the exact limit of smoothness helps us predict when a material will fail or when a structure will become unstable.
• Efficiency: It tells us that we don't need to assume the "wind" is perfectly smooth to get a smooth result. We can tolerate more chaos in the input and still get a high-quality output.
• Completing the Puzzle: For decades, mathematicians had a puzzle piece that was slightly too big or too small. This paper carved that piece to fit perfectly, completing the picture of how these complex shapes behave.

Summary

In simple terms, this paper is a masterclass in precision. The authors took a complex problem about how rough forces affect smooth shapes, analyzed the exact relationship between the two, and found the absolute maximum smoothness possible. They then used this new knowledge to fix a famous old theorem about soap bubbles, proving that these natural shapes are even more perfect than we previously thought.

They didn't just say "it's smooth"; they said, "It is smooth exactly this much, and no more, and here is the mathematical proof that you can't do better."

Technical Summary

1. Problem Statement

The paper addresses the partial $C^{1,\alpha}$ regularity of minimizers for non-parametric variational integrals of the form:
$$\mathcal{F}[w] = \int_{\Omega} \left[ f(Dw) + g(x, w) \right] dx$$
where:

• $f: \mathbb{R}^{N \times n} \to \mathbb{R}$ is the first-order integrand (depending on the gradient $Dw$).
• $g: \Omega \times \mathbb{R}^N \to \mathbb{R}$ is the zero-order integrand (depending on the function value $w$ and position $x$).

Key Challenges:

1. General Zero-Order Terms: The authors focus on very general hypotheses for $g$, specifically allowing for non-differentiability in the $y$-variable (the function value). This prevents the use of the standard Euler-Lagrange equation.
2. Sharp Dependence of $\alpha$: The primary goal is to determine the optimal Hölder exponent $\alpha$ for the gradient $Du$ and establish its sharp dependence on the structural assumptions of $g$.
3. Massari's Theorem Limit: A specific application targets Massari's regularity theorem for hypersurfaces with prescribed mean curvature in $L^p$. Previous work established regularity for exponents $\alpha < \alpha_{opt}$, but the limit case $\alpha = \alpha_{opt}$ remained open.

2. Mathematical Framework and Assumptions

The Zero-Order Term ($g$)

The function $g$ satisfies a Morrey-Hölder condition (Equation 1.2):
$$|g(x, y) - g(x, \tilde{y})| \leq \Gamma(x) (1 + |y| + |\tilde{y}|)^{q-\beta} |y - \tilde{y}|^\beta$$

• Exponents: $0 < \beta \leq q < \infty$.
• Coefficient: $\Gamma$ is a non-negative function controlled by Morrey spaces.
• Significance: This condition allows $g$ to be non-smooth (even non-differentiable) in $y$, provided the singularity is controlled by $\Gamma$.

The First-Order Term ($f$)

• $f$ is a $C^2$ integrand.
• It satisfies 2-strict quasiconvexity (a generalization of convexity crucial for vectorial problems).
• It has at most quadratic growth.

Main Theorems (Theorems 1.2, 1.3, 1.4)

The paper establishes partial regularity ($u \in C^{1,\alpha}_{loc}$ on an open set $\Omega_{reg}$ with $|\Omega \setminus \Omega_{reg}| = 0$) under three distinct settings:

1. General Case (Theorem 1.2): Requires specific Morrey conditions on $\Gamma$ (Equations 1.4 and 1.5).
2. A Priori $L^\infty$ Case (Theorem 1.3): If the minimizer $u$ is known to be bounded, the stricter Morrey condition (1.5) can be dropped, and the growth exponent $q$ can be arbitrarily large.
3. A Priori $W^{1,\infty}$ Case (Theorem 1.4): If the minimizer has a bounded gradient, the assumption of uniform ellipticity on $f$ can be relaxed to local uniform ellipticity.

3. Methodology

The proof relies on the $A$-harmonic approximation method, a powerful technique in partial regularity theory introduced by Duzaar and Steffen and further developed by Schmidt and others.

Key Steps in the Proof:

1. Caccioppoli Inequality (Section 4):

   • Derives an energy estimate for the difference between the minimizer $u$ and an affine function $u_{\xi, \zeta}$.
   • Crucially, handles the zero-order term $g$ using the Morrey-Hölder condition and Young's inequality to separate the gradient terms from the $\Gamma$ terms.
   • Distinguishes cases based on dimension ($n \geq 3$ vs. $n \in \{1, 2\}$) and the growth exponent $q$.

2. Approximate $A$-harmonicity (Section 5):

   • Shows that the minimizer is "close" to being a solution of a linear system with constant coefficients $A = D^2f(\xi)$.
   • Since $g$ is not differentiable, the standard Euler equation is unavailable. Instead, the authors use the minimality of $u$ to estimate the defect from $A$-harmonicity, bounding it by terms involving the excess and the Morrey norm of $\Gamma$.

3. Excess Estimates and Decay (Section 6):

   • Uses the $A$-harmonic approximation lemma (Lemma 2.9) to compare the minimizer with a smooth $A$-harmonic function.
   • Establishes an excess decay estimate (Lemma 6.2): The quadratic excess $\Phi(x_0, \rho)$ decays at a rate determined by $\alpha$ and the Morrey parameters.
   • This decay is iterated (Lemma 6.3) to prove Hölder continuity of the gradient.

4. Refinement of the Exponent (Section 7.2):

   • Initially, the decay yields $C^{1, \min\{\alpha, \delta\}}$ regularity.
   • By utilizing the a priori boundedness of the minimizer (in Settings 2 and 3) or by bootstrapping (Setting 1), the authors remove the $\delta$ restriction and achieve the optimal exponent $\alpha$.

4. Key Contributions and Results

A. Sharp Dependence of $\alpha$ on Structural Parameters

The paper provides the precise formula for the Hölder exponent $\alpha$ based on the parameters $\beta$ (Hölder exponent of $g$), $q$ (growth), and the integrability of $\Gamma$.

• Case (A): If $\Gamma \in L^\infty$, the exponent is $\alpha = \frac{\beta}{2-\beta}$. This recovers known optimal results.
• Case (B): If $\Gamma \in L^p$, the exponent is $\alpha = \frac{\beta - n/p}{2-\beta}$.
• Significance: Previous works often yielded suboptimal exponents or required $q=\beta$. This paper handles the general case $q \geq \beta$ and identifies the exact threshold for regularity.

B. Resolution of Massari's Regularity Theorem (Theorem 1.5)

The paper applies the non-parametric results to the parametric setting of prescribed mean curvature.

• Context: Massari's theorem concerns sets $E$ minimizing $P(E, U) - \int_{E} H dx$ with $H \in L^p(U)$.
• Previous State: Regularity was known for $\alpha < \frac{p-(n+1)}{p+1}$.
• New Result: The authors prove that the boundary $\partial^* E$ is a $C^{1, \alpha_{opt}}$-submanifold with the limit exponent:
  $$\alpha_{opt} = \frac{p - (n+1)}{p+1}$$
• Singular Set: The singular set has Hausdorff dimension at most $n-7$ (for $n \geq 7$) and is empty for $n \leq 6$.
• Method: They transfer the parametric minimality to a non-parametric minimization problem (via local graph representation), apply the new non-parametric regularity theorem (Theorem 1.4), and transfer the result back.

C. Handling Non-Differentiable Zero-Order Terms

The methodology successfully bypasses the need for the Euler-Lagrange equation by working directly with the minimality condition and the Morrey-Hölder bound on $g$. This extends the scope of partial regularity theory to a much broader class of integrands where $g$ may be rough in the $y$-variable.

5. Significance

1. Optimality: The paper closes the gap in Massari's regularity theory, proving that the previously conjectured optimal exponent is indeed achievable. This confirms that the regularity threshold is sharp.
2. Generality: By treating general Morrey-Hölder conditions on $g$ without requiring differentiability, the results unify and extend numerous previous partial regularity theorems (e.g., those by [40], [56], [77], [79]).
3. Technical Innovation: The refined analysis of the Morrey conditions (specifically the interplay between the exponents $\beta, q$ and the Morrey space indices) provides a precise blueprint for determining regularity in variational problems with lower-order terms.
4. Applications: The results are immediately applicable to problems in geometric measure theory (prescribed mean curvature), scalar variational problems with non-smooth potentials, and systems with non-standard growth.

In summary, Schmidt and Schütt provide a definitive answer to the regularity of minimizers with rough zero-order terms, culminating in the proof of the optimal $C^{1,\alpha}$ regularity for Massari's theorem, a long-standing open problem in geometric analysis.