Area-preserving curvature flow for open curves with general contact angles on skew lines

Fuya Hiroi

Home/Archive/arXiv

arXiv · 2610.05909

Area-preserving curvature flow for open curves with general contact angles on skew lines

Fuya Hiroi

View original source ↗Read source document ↗Export metadata ↓

Abstract

We study an area-preserving curvature flow for planar open curves whose end points lie on skew lines and meet the lines at general contact angles. In previous studies of moving boundary problems for higher-order geometric gradient flows, the right-angle condition has often been imposed to ensure that the evolving curves naturally satisfy the boundary conditions. However, under general contact-angle conditions, the evolution must be described by a free boundary problem. Under suitable assumptions on the contact angles $θ_\pm$, the angle $θ$ between the skew lines, and the initial data $γ_0$, we prove global-in-time existence and uniqueness up to reparametrization. We further prove that, after choosing a constant-speed parametrization at each time, the solution converges exponentially to a circular arc with the same signed area as the initial curve. Our analysis is based on a formulation of the evolution as a free boundary problem together with the derivation of uniform a priori estimates.

Explore related subjects

math.AP

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Fuya Hiroi. 2026-10-05. Area-preserving curvature flow for open curves with general contact angles on skew lines. https://arxiv.org/abs/2610.05909

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A nonlocal-to-local transfer principle for parabolic Calderón problems

We prove a nonlocal-to-local transfer principle for fractional powers of variable coefficient parabolic operators. Let $H_σ=\partial_t-\nabla\cdotσ\nabla$, where $σ$ is a uniformly elliptic $C^2$ matrix-valued coefficient. For any $s\in(0,1)$ and any nonempty bounded Lipschitz exterior observation set $W$ with $\overline W\cap\overlineΩ=\emptyset$, we show that equality of the partial exterior DN maps for $H_{σ_j}^s$, $j=1,2$, together with the exterior agreement $σ_1=σ_2$ in $Ω_e$, implies equality of the full lateral local Cauchy data sets for $H_{σ_j}$ on $(\partialΩ)_T$. The proof uses the parabolic Caffarelli--Silvestre extension and the averaging transform \[ v(t,x)=\int_0^\infty y^{1-2s}\widetilde u(t,x,y)\,dy. \] This transform sends fractional parabolic solutions to local parabolic energy solutions. The main analytic ingredient is a boundary density theorem: the lateral traces generated by these averaged exterior solutions are dense in the corresponding closed local energy trace space. Unlike the elliptic reduction, no exterior normalization to a constant is imposed; the common exterior coefficient may be variable. The result holds for all spatial dimensions $n\geq2$. Consequently, local uniqueness results for parabolic Calderón problems transfer to the corresponding nonlocal problems. In particular, scalar leading coefficients are uniquely determined in $Ω$ from partial exterior nonlocal measurements, relative to their exterior values.

math.AP↗

arXiv2026

A proof of Onsager's Conjecture for the SQG equation

We construct solutions to the SQG equation that fail to conserve the Hamiltonian while having the maximal allowable regularity for this property to hold. This result solves the generalized Onsager conjecture on the threshold regularity for Hamiltonian conservation for SQG. We also prove nonuniqueness for arbitrary smooth data and an h-principle result.

math.AP↗

arXiv2026

An inverse problem for the Monge-Ampère equation

We prove uniqueness for an inverse source problem for the Monge--Ampère equation $\det D^2u=F$ on a bounded uniformly convex planar domain. The source is smooth and strictly positive, and its boundary derivatives through order two are prescribed. We show that the Dirichlet-to-Neumann map on the convex solution branch, restricted to a neighborhood of zero boundary data, determines the source throughout the domain. The first linearization gives an anisotropic equation whose principal coefficient is the inverse Hessian of an unknown background solution. We determine its conformal class in the original Euclidean coordinates and use the second linearized equation to determine the remaining scalar factor. We construct complex geometric optics solutions and prove arbitrary order expansions of their correction terms in $W^{2,p}$ for phases without critical points. A termwise stationary phase analysis of the second linearized identity gives a differential equation with a Cauchy integral term. We prove the required unique continuation property by incorporating that integral as an additional unknown in a first order system and applying a Carleman estimate.

math.AP↗