🔢 mathematics

Very weak solutions of quadratic Hessian equations

This paper extends existing methods to investigate very weak solutions of the complex $\sigma_2$ equation in domains of $\mathbb{C}^n$ ( $n\geq 2$ ) and, as a consequence, refines the regularity threshold for counterexamples in the real case.

Original authors: Sławomir Dinew, Szymon Myga

Published 2026-07-22

📖 8 min read🧠 Deep dive

CC BY 4.0

Original authors: Sławomir Dinew, Szymon Myga

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 trying to solve a giant, complex puzzle where the pieces are shapes and the rules are written in a secret code called mathematics. Specifically, this paper lives in the world of Partial Differential Equations (PDEs). Think of PDEs as the ultimate instruction manuals for how things change and move in the universe, from the way heat spreads across a pan to how light bends through a lens. Usually, to solve these puzzles, mathematicians demand that the shapes they are working with be perfectly smooth and well-behaved, like a polished marble statue. They call these "strong" solutions.

But what if the shape is a bit rough, or wiggly, or even a little bit broken? What if it's more like a crumpled piece of paper than a statue? For a long time, mathematicians thought you couldn't solve the puzzle with these messy shapes. However, a clever trick called convex integration changed the game. Imagine you are trying to fit a square peg into a round hole. Instead of forcing it, you wiggle the peg so fast and so finely that, to the naked eye, it looks like it fits perfectly, even though up close, it's vibrating wildly. This paper explores this "wiggling" method for a specific type of equation involving Hessian matrices. You can think of a Hessian matrix as a map that tells you how a surface curves in every direction. The authors are asking: "If we allow our surface to wiggle and vibrate (but not too wildly), can we still solve the equation?"

This question matters because it challenges our understanding of rigidity. In the real world, we often assume that if a structure looks a certain way, it must be built in a specific, rigid way. This research suggests that if you look closely enough, there might be a hidden layer of chaos and flexibility underneath that we didn't know existed. It's like realizing that a seemingly solid wall might actually be made of billions of tiny, vibrating springs that can rearrange themselves to fit any shape you want, as long as you don't look too closely at the individual springs.

The Wiggle Room in Math: A New Kind of Solution

In this paper, the authors, Sławomir Dinew and Szymon Myga, take a bold step into the realm of "very weak solutions." To understand this, imagine you are trying to describe the shape of a bumpy road. A "strong" solution would require you to know the exact height of every single pebble. A "weak" solution might just ask for the average height over a small area. But a "very weak" solution is even more relaxed: it asks, "If I squint my eyes and look at the road from far away, does it feel like it fits the description?"

The authors focus on a specific equation called the complex $\sigma_2$ equation (or the complex 2-Hessian equation). In plain English, this equation is a rule about how a surface curves in a multi-dimensional space (specifically, in complex space, which is a fancy mathematical playground with dimensions that behave a bit differently than our everyday 3D world). Usually, to solve this, you need the surface to be "2-subharmonic," which is a strict rule ensuring the surface curves in a specific, positive way everywhere.

The authors ask a daring question: What if we drop the strict rule? What if we allow the surface to be a bit messy, as long as it satisfies the equation in this "squinting" (very weak) sense?

The Big Discovery: Chaos is Everywhere

The paper proves that if you are willing to accept these "very weak" solutions, the answer is a resounding yes, and not just a few, but lots of them. In fact, the authors show that these wiggly, vibrating solutions are dense in the space of all continuous functions.

To use an analogy: Imagine you have a smooth, flat sheet of clay. You want to mold it into a specific shape. Usually, you have to follow strict rules to keep it from collapsing. But the authors show that if you allow the clay to vibrate and wiggle at a microscopic level, you can mold it into almost any shape you want, and it will still satisfy the mathematical rules of the equation. They prove that for any function $f$ (which acts like the target shape) that is "smooth enough" (specifically, in the class $L^p$ where $p > n + \frac{1}{2}$ ), you can find a solution that is $C^{1,\beta}$ regular.

What does $C^{1,\beta}$ mean? It means the solution is smooth enough to have a defined slope everywhere, but it can wiggle a bit. The authors calculate exactly how much it can wiggle. They find that the "wiggleness" (the exponent $\beta$ ) can be anything less than $\frac{1}{1+2n}$ , where $n$ is the number of dimensions in the space.

For example, if you are working in a space with 2 dimensions ( $n=2$ ), the solution can be as smooth as $\frac{1}{5}$ . If you have 3 dimensions, the limit drops to $\frac{1}{7}$ . The higher the dimension, the more the solution has to wiggle to fit the rules.

Why This Matters (and What It Doesn't)

The authors are very careful to point out what they are not doing. They are not saying that these wiggly solutions are the "real" physical shapes we see in nature. In fact, they admit that these solutions are usually not "2-subharmonic." In the real world, if you tried to build a structure based on these solutions, it might look like a vibrating mess rather than a solid object.

However, the paper is a massive breakthrough in mathematical flexibility. It shows that the "rules" of the equation are much looser than we thought. If you are willing to accept a solution that vibrates wildly at a tiny scale, the equation becomes incredibly flexible, allowing for a vast array of solutions that were previously thought impossible.

They also sharpen the results for the real version of this equation (the kind that deals with ordinary numbers instead of complex ones). Before this paper, the best-known limit for how smooth these real solutions could be was $\frac{1}{1+n+n^2}$ . The authors improved this to $\frac{1}{1+2n}$ . This is a significant upgrade, making the solutions smoother and the math more precise. For $n=2$ , this recovers a famous threshold of $\frac{1}{5}$ found by other mathematicians, but now they have a unified way to reach it for any dimension.

The Secret Weapon: Diagonalization

How did they do it? The paper relies on a clever trick called diagonalization. Imagine you have a messy pile of tangled wires (the equation). The authors found a way to untangle them just enough so that they can be rearranged into a neat, diagonal stack. They proved a specific mathematical lemma (Proposition 5.1) that shows you can take a matrix that is "close enough" to a perfect identity matrix and break it down into a diagonal part (the neat stack) and a "kernel" part (the wiggly, vibrating mess that doesn't break the rules).

This diagonalization is the key that unlocks the door to the "very weak" solutions. It allows them to use an iteration procedure. Think of it like sculpting:

Start with a rough guess.
Add a tiny, high-frequency wiggle to fix the errors.
Repeat this process over and over.
With each step, the wiggle gets smaller and faster, and the shape gets closer to the target.

Because they have this diagonalization trick, they can control the wiggles perfectly, ensuring that the final result is a valid solution, even though it's vibrating like crazy under the microscope.

The Bottom Line

This paper doesn't solve a physical engineering problem or predict the weather. Instead, it solves a deep theoretical puzzle: How flexible are the rules of curvature?

The answer is: More flexible than we thought. As long as you are okay with solutions that are "very weak" (meaning they satisfy the equation in a loose, averaged sense and wiggle at a microscopic scale), you can find solutions for almost any target shape. The authors have mapped out exactly how much wiggle room exists, proving that in the world of complex Hessian equations, chaos and order can coexist in a very specific, mathematically beautiful way.

They leave us with a lingering question for the future: Can this same "wiggling" trick work for even more complex equations (like $\sigma_k$ for $k \ge 3$ )? The authors don't know yet, but they've opened the door, and the path looks promising for anyone brave enough to follow the vibrations.

Technical Summary: Very Weak Solutions of Quadratic Hessian Equations

Problem Statement
The paper addresses the existence and regularity of "very weak" (distributional) solutions to the complex $\sigma_2$ (2-Hessian) equation, denoted as $S^C_2(u) = f$ , in domains $\Omega \subset \mathbb{C}^n$ for $n \ge 2$ . Specifically, the authors investigate solutions with low regularity, specifically in the class $C^{1,\beta}$ , where the equation is not necessarily elliptic in the classical sense, and the solutions are not necessarily 2-subharmonic.

This work extends the theory of convex integration, previously applied to the Monge-Ampère equation in dimension 2 and real $\sigma_2$ equations in higher dimensions, to the complex setting. A central difficulty in defining very weak solutions for Hessian operators in dimensions $n \ge 3$ is the inability to define the operator without utilizing Hessian terms, a challenge shared with the domain of definition for the complex Monge-Ampère operator on plurisubharmonic functions.

Methodology
The authors employ the method of convex integration, following the iterative schemes developed by Lewicka, Pakzad, Cao, and Székelyhidi. The core of the methodology involves constructing highly oscillating solutions through an iteration procedure that introduces small-amplitude, high-frequency perturbations to an approximate solution.

Key technical components include:

Very Weak Formulation: The authors define the very weak solution to the complex 2-Hessian equation using differential forms. For $u \in W^{1,2}_{loc}(\Omega)$ , the equation is interpreted as:
$-\frac{1}{2}\text{tr}(\text{curl}_L \text{curl}_R (\partial u \otimes \bar{\partial} u)) = f$
where $\text{curl}_L$ and $\text{curl}_R$ are left and right curl operators acting on tensor fields, and $\text{tr}$ is a trace operator defined via the Hodge star. An equivalent formulation using the operator $d d^c$ is also provided.
Diagonalization Proposition (Proposition 5.1): A crucial step in the iteration scheme is the ability to decompose a matrix field close to the identity into a sum of a diagonal matrix (squares of functions) and an element belonging to the kernel of the $\text{curl}\text{curl}$ operator. The authors prove a diagonalization result for Hermitian matrices that differs from the real case counterpart found in previous literature. This involves solving a system of PDEs for auxiliary functions $\psi_{j\bar{k}}$ to construct the kernel element.
Iteration Scheme: The proof constructs a sequence of approximations $(v_q, w_q, k_q)$ where $v_q$ and $w_q$ are vector fields and $k_q$ is a matrix field in the weak kernel. At each step, the "deficit tensor" (the error in satisfying the equation) is reduced by mollifying the data and adding perturbations involving specific oscillating functions $\Gamma_1$ and $\Gamma_2$ .
Real Case Extension: The authors adapt their diagonalization technique to the real 2-Hessian equation, improving upon the regularity thresholds established by Li and Qiu.

Key Contributions and Results

Theorem 1.1 (Complex Case): Let $\Omega$ be a bounded simply-connected domain in $\mathbb{C}^n$ . For any $f \in L^p(\Omega)$ with $p > n + \frac{1}{2}$ and any $\beta < \frac{1}{1+2n}$ , there exist $C^{1,\beta}$ very weak solutions to $S^C_2(u) = f$ . Furthermore, these solutions are dense in $C^0(\Omega)$ .
Corollary 3.9 (Real Case): Let $\Omega$ be a bounded simply-connected domain in $\mathbb{R}^n$ . For any $f \in L^p(\Omega)$ with $p > \frac{2n+1}{4}$ and any $\beta < \frac{1}{1+2n}$ , the $C^{1,\beta}$ very weak solutions to the real 2-Hessian equation $S_2(u) = f$ are dense in $C^0(\Omega)$ .
Regularity Threshold Improvement: By refining the diagonalization result (Proposition 5.1) compared to the work of Cao and Székelyhidi, the authors improve the regularity exponent threshold for oscillating very weak solutions in the real case from $\alpha < \frac{1}{1+n+n^2}$ (Li and Qiu) to $\alpha < \frac{1}{1+2n}$ . For $n=2$ , this recovers the $\frac{1}{5}$ threshold established by Cao and Székelyhidi.

Significance and Claims
The paper claims to extend the circle of ideas regarding convex integration to specialists dealing with complex Partial Differential Equations (PDEs). While convex integration has been used in complex analysis for constructing nontrivial inner functions, this work represents the first application of these methods to complex PDEs of Hessian type.

The authors highlight several motivations for this generalization:

Rigidity vs. Flexibility: The existence of highly oscillating solutions that are far from being 2-subharmonic contrasts with rigidity results (such as Sverak's theorem) which require higher regularity or specific conditions to enforce convexity. The paper suggests that understanding these very weak solutions is necessary to formulate questions about rigidity (e.g., whether $W^{2,p}$ solutions close to a convex quadratic must be plurisubharmonic).
Overcoming Admissibility Obstructions: In Hessian-type equations, the "admissibility" requirement (e.g., 2-subharmonicity) often obstructs the application of implicit function theorem arguments in Sobolev spaces. Constructing a theory of solutions that generalizes past this requirement is presented as an important step for building a broader theory.
Viscosity Solutions: The authors note that the theory of $L^p$ -viscosity solutions for complex Hessian equations is still developing, particularly regarding supersolutions. They suggest that very weak solutions of suitable Sobolev regularity could serve as test functions within such a theory.

The paper concludes by noting that whether similar results can be obtained for $S^C_k$ or $S^R_k$ operators for $n \ge k \ge 3$ remains an open problem.

The Wiggle Room in Math: A New Kind of Solution

The Big Discovery: Chaos is Everywhere

Why This Matters (and What It Doesn't)

The Secret Weapon: Diagonalization

The Bottom Line

More like this