Very weak solutions of quadratic Hessian equations

Q: What does this research mean for the field?

The notion of very weak solutions can be extended to the complex $\sigma_2$ equation in domains in $\mathbb{C}^n$ ($n \geq 2$), which yields a sharpened regularity threshold for counterexamples in the real case. Novelty: ClaimNovelty.INCREMENTAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

Sławomir Dinew; Szymon Myga

doi:10.1016/j.na.2026.114236

July 24, 2026Nonlinear AnalysisOpen Access

Very weak solutions of quadratic Hessian equations

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Key Points

The aim is to extend existing methods to study very weak solutions of the complex σ2 equation in multiple domains.
Methods of Lewicka - Pakzad and others were extended to address very weak solutions.
Focus is on domains in complex n-dimensional space (C^n) where n is at least 2.
Regularity thresholds from previous work were analyzed and sharpened.
Regularity thresholds for counterexamples were improved as compared to prior findings.
Insights into very weak solutions enhance understanding of the σ2 equation framework.

Implication

Investigates very weak solutions of quadratic Hessian equations, suggesting improved understanding.

Abstract

We extend the methods of Lewicka - Pakzad, Székelyhidi - Cao and Li - Qiu to study the notion of very weak solutions to the complex σ₂ equation in domains in Cⁿ,\ n≥ 2. As a by-product we sharpen the regularity threshold of the counterexamples obtained by Li and Qiu in the real case.

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Cite This Study

Dinew et al. (2026) studied this question.

synapsesocial.com/papers/6a67004840bca442e0d49f44 https://doi.org/10.1016/j.na.2026.114236

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Very weak solutions of quadratic Hessian equations

Key Points

Implication

Abstract

Discussion

Cite This Study

Very weak solutions of quadratic Hessian equations

Key Points

Implication

Abstract

Discussion

Cite This Study

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