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Strictly stable solutions in uniformly convex planar domains may have nonconvex superlevel sets

This paper constructs smooth, uniformly convex planar domains supporting strictly stable solutions to semilinear Dirichlet problems with prototypical nonlinearities (such as $e^u$ and $(a+u)^p$ ) that possess nonconvex superlevel sets, thereby providing a negative answer to a question posed by Brezis regarding whether stability necessarily implies quasiconcavity.

Original authors: Yi Ru-Ya Zhang

Published 2026-07-08

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Yi Ru-Ya Zhang

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 Question: Does a Round Bowl Make Round Soup?

Imagine you have a perfectly smooth, round bowl (a convex domain). You pour a special kind of soup into it (a mathematical solution to an equation). In the world of physics and math, there was a long-held belief that if your bowl is perfectly round, the "surface" of the soup should also be perfectly round.

In mathematical terms, this meant: If the shape of the container is convex, the "level sets" (the shapes you get if you slice the solution at a specific height) must also be convex.

For decades, mathematicians believed this was true, especially for "stable" solutions—solutions that don't wobble or collapse easily. A famous mathematician named Brezis asked: "If the solution is stable and the bowl is convex, must the soup level always be convex?"

This paper says: No.

The author, Yi Ru-Ya Zhang, has built a specific, smooth, round bowl where the soup level is actually bumpy or non-convex, even though the solution is perfectly stable.

The Analogy: The "Saddle-Node" Trap

To understand how this happens, imagine a very specific type of hill.

The One-Dimensional Hill:
Imagine a 1D hill (a line) where you can adjust the height of the water. As you pour more water (increasing the "amplitude"), the water spreads out.
- At first, adding water makes the water spread wider smoothly.
- But then, you hit a "cliff" or a fold. At a specific point, adding just a tiny bit more water causes the water level to behave strangely. It's like a saddle-node bifurcation: the system is stable right up until the very edge of the fold, where it suddenly wants to snap into a different shape.
The paper proves that for certain types of "soup" (nonlinearities like $e^u$ or $(a+u)^p$ ), this fold exists. Near this fold, the relationship between the width of the container and the height of the water becomes extremely sensitive. It's not a smooth curve anymore; it's a sharp, square-root turn.
The 2D "Slow Channel":
Now, imagine taking that 1D line and stretching it into a long, thin, slightly curved tunnel (a slow channel).
- The tunnel is shaped like a very long, slightly curved rectangle.
- The walls of the tunnel are slightly curved inward (concave) in the middle, but the ends are capped off with perfect curves so the entire tunnel remains a smooth, convex shape (like a long, slightly bent sausage).
The author designs this tunnel so that the width of the tunnel changes very slowly along its length.
- In the middle of the tunnel, the width is just barely wide enough to be on the "stable side" of that tricky fold we mentioned earlier.
- Because of the square-root sensitivity at the fold, a tiny change in the tunnel's width causes a surprisingly large, non-smooth change in the height of the water level.

The Magic Trick: How the Bump Happens

Here is the core mechanism, visualized:

The Setup: You have a long, convex tunnel. The width of the tunnel varies slightly from left to right.
The Reaction: Because the system is operating right near that "fold" (the cliff edge), the water level doesn't just follow the width smoothly. Instead, the water level reacts with a square-root distortion.
The Result: Even though the tunnel walls are smooth and convex, the water level in the middle dips down or bulges up in a way that creates a "dent" in the shape of the water's surface.

Think of it like a trampoline. If you have a perfectly round trampoline (the domain), you expect the fabric to sag in a perfect circle. But if the springs in the middle are tuned to a very specific, sensitive tension (the fold), and you step in the middle, the fabric might sag in a weird, non-circular shape, even though the frame is perfectly round.

Why This Matters (According to the Paper)

It Breaks a Rule: It proves that "Convex Container + Stable Solution = Convex Level Set" is false. You can have a perfectly round bowl and a very stable solution, yet the "slice" of the solution is not round.
It's Not a Fluke: The author didn't just find a weird, broken bowl. The bowl is smooth, uniformly convex (curved everywhere), and the solution is minimal (the simplest possible solution) and strictly stable (it won't collapse).
The Cause: The non-convexity isn't caused by the bowl having holes, or the soup being weird, or the solution having multiple peaks. It is caused purely by the mathematical geometry of the "fold" near the stability limit. The "square-root" behavior at the fold overrides the convexity of the container.

The Specific "Soups" Tested

The paper specifically tested two famous types of "soup" (nonlinearities) that mathematicians had been wondering about:

The Exponential Soup ( $e^u$ ): This is the famous "Gelfand problem."
The Power Soup ( $(a+u)^p$ ): A shifted power function.

The paper shows that for both of these, you can build a convex bowl where the stable solution has a non-convex level set.

Summary

The author built a mathematical "trap." They created a perfectly round, smooth container and filled it with a stable solution. However, by tuning the container's width to be just right near a mathematical "cliff" (a fold), they forced the solution to develop a non-convex shape.

The takeaway: Just because a container is perfectly round and the solution is stable, it doesn't guarantee that the solution's shape will be round. Sometimes, the math of the "edge of stability" creates a bump that the container's shape cannot smooth out.

Technical Summary: Strictly Stable Solutions in Uniformly Convex Planar Domains May Have Nonconvex Superlevel Sets

Problem Statement
The paper addresses a fundamental question in the qualitative theory of semilinear elliptic equations: whether the convexity of a domain $\Omega \subset \mathbb{R}^2$ forces the convexity of the level sets (or superlevel sets) of positive solutions to the Dirichlet problem:
$\begin{cases} -\Delta u = f(u) & \text{in } \Omega, \\ u > 0 & \text{in } \Omega, \\ u = 0 & \text{on } \partial\Omega. \end{cases}$
Specifically, the paper investigates whether stability of a solution restores convexity of level sets for prototypical nonlinearities, such as the Gelfand nonlinearity $f(u) = e^u$ and shifted power-type nonlinearities $f(u) = (a+u)^p$ ( $a>0, p>1$ ). While classical results (e.g., by Makar-Limanov, Brascamp-Lieb, and Caffarelli-Friedman) established quasiconcavity for specific cases (torsion functions, first eigenfunctions, or specific concave nonlinearities), and Hamel, Nadirashvili, and Sire [22] showed that convexity fails for general nonlinearities, the status of minimal strictly stable solutions in uniformly convex domains remained an open question posed by Brezis [7].

Methodology
The author constructs a counterexample using a geometric mechanism driven by a one-dimensional saddle-node (fold) bifurcation. The methodology proceeds in three main stages:

One-Dimensional Fold Analysis:
The paper analyzes the 1D problem $-U'' = f(U)$ on an interval $(-h, h)$ with $U(\pm h)=0$ . For a "fold-admissible" nonlinearity (where the map from amplitude $A$ to half-width $h(A)$ has a nondegenerate fold), the author establishes that near the critical amplitude $A_c$ (where $h'(A_c)=0$ ), the relationship between the half-width $h$ and the height $Y_t(h)$ of a fixed level set $t$ exhibits a square-root singularity. Specifically, as $h$ approaches the critical width $h_c$ from below (along the stable lower branch), the height behaves as:
$Y_t(h) \approx Y_c - c_t \sqrt{h_c - h}$
This expansion is derived using the implicit function theorem and the properties of the linearized operator at the fold.
Geometric Construction of the Domain:
The author constructs a "slow channel" domain $\Omega_\epsilon$ that is smooth, bounded, and uniformly convex. The domain is designed to be long and thin, with a slowly varying width $H(X)$ in the horizontal direction. The function $H(X)$ is chosen to be concave and to vary slightly below the critical width $h_c$ . Crucially, the author selects a specific profile $H(X)$ such that the term $\sqrt{h_c - H(X)}$ (which dictates the level set height) becomes strictly midpoint-convex (violating the concavity required for a convex superlevel set) at a specific midpoint, despite $H(X)$ itself being concave. This is achieved by exploiting the square-root singularity: if $D(X) = h_c - H(X)$ is convex, then $-\sqrt{D(X)}$ can be strictly midpoint-convex near a point where $D(X)$ is small and positive.
Analytical Approximation and Stability:
Using a barrier argument and the maximum principle, the paper proves that the actual solution $u_\epsilon$ in the constructed domain $\Omega_\epsilon$ remains uniformly close (in $L^\infty$ ) to the formal 1D profile $V(X, \eta) = U_{H(X)}(\eta)$ as the domain aspect ratio $\epsilon \to 0$ .
- Existence and Minimality: A minimal solution is constructed via monotone iteration between the zero subsolution and a global supersolution derived from the 1D profile at the maximum width.
- Strict Stability: The strict stability of the minimal solution is established by comparing the 2D linearized operator to the 1D linearized operator on the lower branch. Since the 1D operator has a positive spectral gap on the stable branch, the 2D solution inherits strict stability.
- Nonconvexity: The $L^\infty$ approximation error is shown to be $O(\epsilon^2)$ , which is insufficient to correct the finite "midpoint violation" in the level set height created by the square-root singularity. Thus, the actual superlevel set of the minimal strictly stable solution is nonconvex.

Key Contributions and Results

Theorem 1.2: For any fold-admissible nonlinearity $f$ , there exists a smooth, uniformly convex planar domain $\Omega$ and a minimal, strictly stable solution $u$ such that a superlevel set $\{u > t_0\}$ is not convex.
Corollary 1.3: This result applies specifically to the Gelfand nonlinearity ( $f(u)=e^u$ ) and the shifted power nonlinearity ( $f(u)=(a+u)^p$ with $a>0, p>1$ ), providing a negative answer to Brezis' question regarding these prototypical cases.
Corollary 3.8: The result extends to the parameterized problems $-\Delta u = \lambda f(u)$ , showing that the lack of convexity persists for minimal strictly stable solutions across the parameter range.
Verification of Fold-Admissibility: The paper explicitly verifies that $e^u$ and $(a+u)^p$ satisfy the structural conditions required for the fold mechanism (Propositions 2.10 and 2.11).

Significance and Claims
The paper claims to provide a definitive negative answer to the question of whether stability restores level set convexity in uniformly convex domains for standard nonlinearities.

Distinction from Previous Counterexamples: Unlike previous counterexamples by Hamel, Nadirashvili, and Sire [22] (which relied on specially constructed, non-convex nonlinearities) or Monneau and Shahgholian [37] (which involved free-boundary limits), this construction uses positive, increasing, and convex nonlinearities on smooth, uniformly convex, simply connected domains.
Mechanism: The nonconvexity is not attributed to topology, multiple critical points, or non-monotonic reaction terms. Instead, it arises from the square-root response of the level set height near a stable one-dimensional fold. The paper argues that the "effective" one-dimensional fold dominates and overrides the convexity properties of the ambient domain.
Minimal Solutions: The result is particularly strong because it applies to minimal solutions, which are typically the most "regular" and stable configurations in the theory, and these solutions are shown to be strictly stable.

In summary, the paper demonstrates that even in the most favorable geometric settings (uniformly convex domains) and for the most robust solution types (minimal, strictly stable), the convexity of the domain does not guarantee the convexity of the superlevel sets for standard nonlinearities like the Gelfand and power laws.