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Non-convexity of level sets for solutions to $k$ -Hessian equations in exterior domains

This paper demonstrates that solutions to $k$ -Hessian equations with quadratic growth in exterior domains are not necessarily quasiconvex for $1 \leq k \leq n/2$ , while simultaneously providing a new proof for the quasiconvexity of harmonic functions decaying to zero in the same setting.

Original authors: Wang Bo, Wang Cong, Wang Zhizhang

Published 2026-03-31

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Wang Bo, Wang Cong, Wang Zhizhang

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

The Big Picture: The Shape of Solutions

Imagine you are standing in a vast, flat field (this is our Exterior Domain). In the middle of the field, there is a perfectly round, smooth hill (this is the Strictly Convex Domain, or $\Omega_0$ ).

The mathematicians in this paper are studying a specific type of "force" or "pressure" that exists everywhere in the field outside that hill. They are asking a simple question: If the hill in the middle is perfectly round and smooth, will the "pressure lines" (or level sets) surrounding it also be perfectly round and smooth as they stretch out to infinity?

In math terms, they are asking if the solutions to these equations are quasiconvex.

Quasiconvex means: If you draw a line between any two points on a specific "pressure line," that line stays inside the area defined by that line. Visually, it means the shapes don't have "dents" or "pinches." They are nice, bulging shapes like eggs or spheres.

The Main Discovery: "Not Always!"

For a long time, mathematicians suspected that if the inner hill is convex (bulging out), the outer shapes should also be convex. It seemed logical: a smooth rock in a pond creates smooth ripples.

This paper says: "Not necessarily."

The authors (Wang, Wang, and Wang) constructed a specific scenario where:

The inner hill is perfectly smooth and convex.
The "pressure" at the edge of the hill is fixed.
Far away, the pressure grows in a predictable, quadratic way (like a bowl shape).

The Result: Even with all these perfect conditions, the "ripples" (the level sets) can develop weird, non-convex shapes. They might pinch in the middle or develop a dumbbell shape. The smoothness of the inner hill does not guarantee the smoothness of the outer shapes for these specific types of equations (called k-Hessian equations).

The Analogy: The Rubber Sheet and the Rock

Imagine a giant, stretchy rubber sheet covering the whole world.

The Hill ( $\Omega_0$ ): You push a smooth, round rock up from underneath the sheet.
The Equation ( $S_k(D^2u)=1$ ): This is a rule about how the rubber sheet wants to stretch. It's a very specific, complex rule about how the sheet curves in different directions.
The Level Sets: These are the contour lines you would draw on the sheet if you were looking at it from above (like a topographic map).

The Intuition: You'd think, "If I push a round rock up, the contour lines around it should be round circles."
The Reality: For these specific, complex rules of stretching, the rubber sheet can get "confused." Even though the rock is round, the tension in the sheet might cause the contour lines to wiggle, pinch, or become non-convex further out, depending on how the sheet is anchored far away.

The Two Main Parts of the Paper

Part 1: Breaking the Rule (The Counterexamples)

The authors proved that for a wide range of these complex equations (where $1 \le k \le n/2$ ), you can find a setup where the outer shapes are not convex.

How they did it: They didn't just guess; they built a mathematical "machine" (a specific domain $\Omega_0$ and specific boundary conditions) that forces the solution to behave badly.
The Surprise: They showed that the solution is extremely sensitive. If you move the inner rock just a tiny bit or change its shape slightly, the outer ripples can suddenly lose their convexity. It's like a house of cards; a tiny nudge makes the whole structure collapse into a weird shape.

Part 2: Saving the Day for Simple Cases (Harmonic Functions)

The paper also revisits a classic, simpler case: Harmonic functions (which describe things like heat distribution or electrostatic potential).

The Old Result: It was already known that for these simple equations, if the inner hill is convex, the outer ripples are convex.
The New Proof: The authors provided a new, microscopic proof for this. Instead of looking at the whole picture (macroscopic), they looked at the tiny details of the curvature at the very edge of infinity.
The Metaphor: Imagine watching a wave travel out to the horizon. They proved that as the wave gets infinitely far away, its curvature behaves in a very specific, "super-harmonic" way that forces it to stay smooth and convex all the way back to the source.

Why Does This Matter?

It Shatters a Myth: It shows that in the complex world of non-linear physics (like certain models of fluid flow or material stress), "smooth input does not guarantee smooth output."
Sensitivity: It highlights that these systems are incredibly sensitive to their environment. Small changes in the boundary can lead to dramatic changes in the global shape.
New Tools: The methods they used to prove the negative results (constructing counterexamples) and the positive results (analyzing curvature at infinity) give other mathematicians new tools to solve similar problems.

Summary in One Sentence

"Just because the obstacle in the middle is perfectly round doesn't mean the ripples spreading out from it will stay round; for complex physical laws, those ripples can pinch and warp, though for simple heat-like laws, they stay perfectly smooth."

1. Problem Statement

The paper investigates the quasiconvexity of solutions to the exterior Dirichlet problem for $k$ -Hessian equations. Specifically, it addresses whether the sublevel sets of a solution $u$ remain convex when the domain is the exterior of a strictly convex bounded domain $\Omega_0$ .

The problem is formulated as:
$\begin{cases} S_k(D^2 u) = 1 & \text{in } \mathbb{R}^n \setminus \bar{\Omega}_0, \\ u = 0 & \text{on } \partial \Omega_0, \\ u(x) - \left(\frac{1}{2}x^T A x + b \cdot x + c\right) \to 0 & \text{as } |x| \to \infty, \end{cases}$
where:

$S_k(D^2 u)$ is the $k$ -Hessian operator (the $k$ -th elementary symmetric function of the eigenvalues of the Hessian matrix $D^2 u$ ).
$1 \le k \le n/2$ .
$A$ is a real symmetric positive definite matrix with $S_k(A)=1$ .
The solution is required to be $k$ -convex (its eigenvalues lie in the Gårding cone $\Gamma_k$ ).

The Core Question: While the inner boundary $\Omega_0$ is convex and the asymptotic behavior at infinity is quadratic (implying convex sublevel sets at infinity), does this convexity propagate globally? In other words, is every $k$ -convex solution to this problem quasiconvex (i.e., are all its sublevel sets convex)?

2. Methodology

The authors employ a combination of approximation techniques, Hessian measure theory, and asymptotic analysis to construct counterexamples and prove new results.

A. Construction of Counterexamples (Theorems 1.3 & 1.4)

To disprove the global quasiconvexity, the authors construct explicit counterexamples where the domain $\Omega_0$ is a small ball $B_\varepsilon(x_0)$ centered at a point $x_0$ away from the origin.

Approximating Problem: They first solve the problem in a bounded non-concentric convex ring $\Omega_R^\varepsilon = E_R(0) \setminus \bar{B}_\varepsilon(x_0)$ , where $E_R(0)$ is an ellipsoid.
Subsolution Construction: Using generalized symmetric functions, they construct a subsolution to ensure the existence of a unique $k$ -convex solution $u_{\varepsilon, R}$ in the ring.
Uniform Estimates: They derive uniform $C^0$ , $C^1$ , and $C^2$ estimates for $u_{\varepsilon, R}$ that are independent of the outer radius $R$ and the inner radius $\varepsilon$ . This allows them to take the limits $R \to \infty$ and $\varepsilon \to 0$ .
Hessian Measures: To handle the singularity at $x_0$ as $\varepsilon \to 0$ , they utilize the theory of $k$ -Hessian measures (developed by Trudinger and Wang). They prove that the sequence of solutions converges to a function $v$ in the sense of $k$ -Hessian measures, and that $v$ coincides with the prescribed quadratic polynomial $\psi(x) = \frac{1}{2}x^T A x + c$ everywhere.
Contradiction Argument: They assume the solution is quasiconvex. By analyzing the geometry of the sublevel sets near the singularity $x_0$ and comparing them with the quadratic asymptotic profile, they derive a contradiction. Specifically, the convexity of sublevel sets would force the quadratic function $\psi$ to satisfy an inequality ( $\psi(x_0) \le \psi(0)$ ) that contradicts the strict positivity of the quadratic form at $x_0$ .

B. New Proof for Harmonic Functions (Theorem 1.6)

For the case $k=1$ (Laplace equation) with decay to zero at infinity, the paper provides a new microscopic proof of the known result that solutions are quasiconcave.

Deformation Method: They use a continuous deformation of the domain $\Omega_t$ from a small ball to the target domain.
Gaussian Curvature Analysis: They establish the asymptotic behavior of the Gaussian curvature $K(x)$ of the level sets at infinity, showing $K(x) \sim |x|^{1-n}$ .
Superharmonicity: Leveraging a result by Ma and Zhang, they show that a specific curvature function $\psi_t = (|Du|^{n-3}K)^{1/(n-1)}$ is superharmonic.
Maximum Principle: By applying the strong maximum principle and Hopf lemma to the curvature function during the deformation, they prove that the Gaussian curvature cannot vanish, ensuring the level sets remain strictly convex throughout the deformation.

3. Key Contributions and Results

Negative Result: Non-Convexity for $k \ge 2$

Theorem 1.3: For $2 \le k \le n/2$ , there exist smooth strictly convex domains $\Omega_0$ and constants $c$ such that the unique $k$ -convex solution to the exterior problem is not quasiconvex.
Theorem 1.4: For the Laplace case ( $k=1$ ) with a specific condition on the matrix $A$ ( $\lambda_{\max}(A) < 1/2$ ), similar non-quasiconvex solutions exist if the solution grows quadratically at infinity.
Significance: This resolves Question 1.1 in the negative. It demonstrates that quasiconvexity is not an inherent property of $k$ -Hessian solutions in exterior domains, even when the boundary and asymptotic data are convex. The convexity of level sets is highly sensitive to the position of the inner domain relative to the origin and the specific quadratic growth rate.

Positive Result: Harmonic Functions with Decay

Theorem 1.6: Re-proves that for the Laplace equation with $u=1$ on $\partial \Omega_0$ and $u \to 0$ at infinity, the solution is quasiconcave (superlevel sets are convex).
Novelty: Unlike previous macroscopic proofs (e.g., Lewis, Kawohl), this proof uses a microscopic approach involving the evolution of Gaussian curvature and superharmonicity of curvature functions.

4. Significance and Implications

Resolution of Open Problems: The paper settles the question of whether convexity of the domain and asymptotic behavior guarantees global quasiconvexity for $k$ -Hessian equations. The answer is no for $1 \le k \le n/2$ under quadratic growth conditions.
Extension of Counterexamples: While previous work (Wang and Xiao) showed non-convexity in bounded convex rings, this paper extends the phenomenon to unbounded exterior domains, which are physically more relevant for potential theory and exterior problems.
Methodological Advancement: The successful application of $k$ -Hessian measure theory to handle the singular limit of shrinking inner domains in exterior problems provides a robust framework for future studies on fully nonlinear equations in unbounded domains.
Contrast in Behavior: The results highlight a sharp distinction between:
- Quadratic Growth ( $u \sim |x|^2$ ): Solutions can fail to be quasiconvex.
- Decay to Zero ( $u \to 0$ ): For harmonic functions, solutions remain quasiconcave.
  This suggests that the nature of the asymptotic behavior (growth vs. decay) plays a critical role in the geometric properties of the level sets.

In summary, the paper provides a rigorous disproof of the global quasiconvexity conjecture for $k$ -Hessian equations in exterior domains via explicit counterexamples, while simultaneously offering a refined geometric proof for the classical harmonic case.