Research Archive

Luminary Research ArchiveIndependent scholarship

Luminary Portal

The collection

All Research

Explore the Archive by subject, subfield, and paper type.

2 papers · Page 1 of 1

Differential Equations

Edition 2

Single-time boundary heat-flux rigidity for smooth domains

Scott Schubr

Primary subject
Mathematics
Subject subfields
Differential Equations
Paper type
Research Article
Manuscript stage
Preprint

For a bounded connected smooth Euclidean domain in dimension at least two, consider Dirichlet heat evolution with initial temperature identically one. If the signed outward normal derivative is spatially constant on the entire boundary at one positive time, the domain is a ball; connected boundary is not assumed. The proof uses moving planes, an L2 initial-corner comparison, component reconstruction and a parabolic corner barrier. We also give a local near-ball quantitative stability estimate in strong boundary norms, with volume and centroid normalization, at times in a fixed compact positive interval. In contrast, one fixed time observed on a moving proper boundary cap admits an infinite-dimensional family of nearby normalized nonballs with exactly the ball flux on that cap. Global stability and reconstruction from a fixed physical patch are not asserted.

    Read paper

    Analysis and Spectral Theory

    Edition 2

    Positive Affine Eigenfunctions Above the First Eigenvalue on Nonconvex Pentagons

    Scott Schubr

    Primary subject
    Mathematics
    Subject subfields
    Analysis and Spectral Theory · Differential Equations
    Paper type
    Research Article
    Manuscript stage
    Preprint

    We construct bounded connected nonconvex planar pentagons for which the affine 2-Laplacian has a strictly positive Dirichlet eigenfunction at an eigenvalue strictly above its first variational eigenvalue. The same domain therefore supports positive affine eigenfunctions at two distinct values. The construction gives a negative answer, for n = p = 2, to the general-domain positive-eigenfunction question of Haddad, Jiménez and Montenegro. The mechanism is an affine-isotropic saddle formed by pinching a pentagon into two right isosceles triangles. A symmetry-preserving diagonal normalization has positive curvature, while a shear couples the ground state to a nearly coincident odd mode. We prove the sharp limit (λ₂ − λ₁)∂²λ₁/∂q² → −32768/81 at the isotropic positions. An area-preserving apex slide has the corresponding limit −50π⁴. The saddles persist under small vertex perturbations, including examples with nonparallel convex-hull sides. Consequences include distinct isotropic positions on one linear orbit and failure of positive-definite-deformation log-convexity outside convexity. We further preserve the saddle under smooth inner exhaustion and use a symmetry-balanced product construction to obtain positive affine 2-eigenfunctions at two distinct eigenvalues on smooth connected nonconvex domains in every dimension n ≥ 2.

      Read paper