Existence and Nonexistence for Hessian Exterior Dirichlet Problems with \(k\)-Admissible Asymptotic Matrices

Abstract: We study exterior Dirichlet problems for (k)-Hessian equations with prescribed quadratic asymptotics, allowing the asymptotic matrix to be merely (k)-admissible and not necessarily positive definite. The key point is that the correct metric at infinity is not determined by the asymptotic matrix itself, but by the coefficient matrix obtained by linearizing the (k)-Hessian operator at this matrix. This gives the exterior barriers and subsolutions needed to solve the Dirichlet problem, both in viscosity and smooth settings, for all sufficiently large asymptotic constants. In the case of smooth, strictly star-shaped domains with strictly ((k-1))-convex boundary, we complete the characterization of existence and nonexistence through a linearized capacitary comparison and a tangential-trace contradiction on the inner boundary.

• The paper demonstrates a sharp threshold between existence and nonexistence for k-admissible asymptotic matrices in the exterior Dirichlet problem.
• The authors develop innovative barrier constructions that separate the asymptotic metric from the linearized metric to handle degeneracies at both the boundary and infinity.
• Explicit asymptotic estimates and a robust framework for fully nonlinear Hessian equations offer practical insights for applications in affine geometry and nonlinear potential theory.

Existence and Nonexistence for Hessian Exterior Dirichlet Problems with $k$-Admissible Asymptotic Matrices

Introduction and Background

The paper "Existence and Nonexistence for Hessian Exterior Dirichlet Problems with $k$-Admissible Asymptotic Matrices" (2606.28703) addresses the solvability of exterior Dirichlet problems for the $k$-Hessian equation on unbounded domains, particularly focusing on the existence and sharp nonexistence criteria when the prescribed asymptotic matrix is merely $k$-admissible and not necessarily positive definite. The study generalizes classical results on Monge–Ampère-type equations to the broader class of $k$-Hessian PDEs, providing a comprehensive analysis of boundary regularity, asymptotic behavior, and the intricate role of the linearized operator in characterizing metrics at infinity.

Let $D$ be a bounded domain in $\mathbb{R}^n$, $n\geq3$, and $\Omega = \mathbb{R}^n \setminus \overline{D}$. The primary object of study is the exterior Dirichlet problem for the $k$-Hessian equation: $k$0 with prescribed quadratic asymptotics

$k$1

where $k$2 is the $k$3th elementary symmetric polynomial, $k$4, and $k$5 denotes the set of symmetric matrices with spectrum in the Gårding cone $k$6 ($k$7-admissible).

Classically, exterior problems for fully nonlinear elliptic PDEs relied on strong convexity of the asymptotic matrix $k$8 (i.e., $k$9), but for intermediate $k$0-Hessian operators ($k$1), $k$2-admissibility is substantially more general. The technical barrier arises because the ansatz based directly on $k$3, such as the generalized symmetric function of Bao–Li–Li, fails when $k$4 is degenerate or indefinite; new constructions are required.

Main Technical Contributions

Separation of Asymptotic Metric and Linearized Metric

A key conceptual advance is the separation between the matrix $k$5 prescribing quadratic growth at infinity and the metric at infinity, which is governed by the coefficient matrix $k$6 of the linearized $k$7-Hessian operator at $k$8: $k$9 Given $k$0, $k$1 is positive definite. The natural "exterior norm" is then

$k$2

so barrier and subsolution constructions rely on $k$3 instead of the standard $k$4, leading to robust arguments even in the absence of strict convexity or positivity of $k$5.

Barrier Construction and Gluing

The development of both far-field and near-boundary barriers is central. Far-field subsolutions adopt the form

$k$6

with explicit control of the error and subsolution property via the linearized operator. Near the boundary, quadratic supports are crafted as

$k$7

where $k$8, circumventing the invertibility and positivity constraints previously needed for classical supports.

Employing viscosity solutions and Perron's method, the authors construct global subsolutions and supersolutions, subsequently proving existence and uniqueness in the appropriate function spaces.

Sharp Threshold and Nonexistence

A major result is the sharp solvability threshold for the additive constant $k$9 in the asymptotic profile. There exists a critical value $k$0 (depending on data), such that:

• For any $k$1, the problem admits a unique $k$2-admissible smooth solution.
• For $k$3, no such smooth solution exists.

The nonexistence argument leverages a capacitary comparison in the linearized metric: subsolutions must be subharmonic with respect to $k$4, and at boundary extremal points, a contradiction arises with the nonnegativity of the tangential trace for any $k$5-admissible Hessian.

Asymptotic Expansions

The existence theory is sharpened by explicit asymptotic estimates. For large $k$6, solutions satisfy

$k$7

with further higher-order expansion $k$8 at infinity, uniform in the full admissible range.

Regularity and Domain Geometry

Solvability is analyzed in both viscosity and smooth settings. Smooth existence theory is established under strict star-shapedness and strict $k$9-convexity of the boundary. However, the theory highlights that strict $D$0-convexity alone is insufficient for star-shapedness, as detailed in the appendix via geometric counterexamples.

Implications and Broader Context

This work substantially advances understanding of fully nonlinear elliptic PDEs of Hessian type on unbounded domains with degenerate or indefinite asymptotic configurations. The separation of asymptotic and linearized metrics provides both a novel technical tool and a general guiding principle applicable to other fully nonlinear, elliptic spectral equations, such as Hessian quotient equations and special Lagrangian equations, whenever the linearized operator admits a definite metric.

Practically, the approach enables precise classification of solvability regimes for a broad class of geometric PDEs, with implications for affine geometry, geometric flow, and nonlinear potential theory. The sharp threshold phenomenon, with explicit nonexistence below the critical parameter, is especially noteworthy, as such dichotomies are rarely accessible in nonlinear, unbounded settings.

The techniques developed here—barrier construction using the linearized metric, regularized maxima, and capacitary comparison—establish a robust framework that is flexible enough to handle degeneracies and geometric intricacies near the boundary and at infinity. Further developments may explore extensions to more general right-hand sides (e.g., decaying or compactly supported), anisotropic problems, or lower regularity boundary data.

Conclusion

The paper (2606.28703) delivers a rigorous and detailed analysis of exterior Dirichlet problems for $D$1-Hessian equations with merely $D$2-admissible asymptotic matrices. By separating the roles of the asymptotic and linearized operators, the authors overcome significant technical obstacles and achieve a complete characterization of existence and nonexistence, including explicit asymptotic estimates and a sharp threshold phenomenon for the asymptotic constant. This work both consolidates and extends the theoretical understanding of fully nonlinear elliptic PDEs on exterior domains, providing potent methods and results for further advances in the field.