A Priori Estimates for Nonlinear Hessian Equations

Definitions and mathematical statement

For a real symmetric $n\times n$ matrix $H$ with eigenvalues $\lambda_1,\ldots,\lambda_n$, define \[\sigma_2(H)=\sum_{i<j}\lambda_i\lambda_j=\frac12\left((\operatorname{tr}H)^2-\operatorname{tr}(H^2)\right).\] To read the equation, let $\Omega\subseteq{\mathbb{R}}^n$ be open, $n\geq2$, let $u$ be twice differentiable on $\Omega$, and let $f$ be defined on the relevant triples $(x,z,p)\in\Omega\times{\mathbb{R}}\times{\mathbb{R}}^n$. The nonlinear Hessian equation is \[\sigma_2(D^2u(x))=f(x,u(x),Du(x)),\qquad x\in\Omega.\] The source asks: what conditions on $f$ ensure estimates analogous to those obtained for $\sigma_2(D^2u)=1$? The choice of $\Omega$, dimension, solution regularity, and the estimates being generalized are not specified in the supplied fragment; the notation above only makes its displayed equation meaningful. Elliptic-branch assumptions are also missing. A standard branch is the cone $\Gamma_2=\{\lambda\in{\mathbb{R}}^n:\sigma_1(\lambda)>0,\ \sigma_2(\lambda)>0\}$, with $\sigma_1(\lambda)=\sum_i\lambda_i$, but the draft does not silently impose $\Gamma_2$-admissibility, convexity, positivity of $f$, or any particular structural condition as a recovered source hypothesis. A precise estimate would have to state the controlled norm, whether it is interior or up to the boundary, and the quantities on which its constant may depend. None of these is supplied. The 2026-08-21 literature triage labels the record open and explicitly describes the extracted line as incomplete, with no general classification of the requested conditions established by its sources.

Short English statement

Ask which structural assumptions on the right-hand side of the second Hessian equation preserve the estimates known for constant right-hand side one. The supplied fragment does not identify the estimate, dimension, domain, boundary data, or elliptic solution class.

Sources

• Local UnsolvedMath registry, permanent ID 30000800, source problem catalogue entry, A Priori Estimates for Nonlinear Hessian Equations, proposed year 2007; no individual proposer supplied.
• Supplied citation: Partielle Differentialgleichungen (2008), Oberwolfach Reports, DOI 10.4171/owr/2007/35, pp. 2035--2078: https://doi.org/10.4171/owr/2007/35. Supplied literature label: original Oberwolfach report.