Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.13592

The conformal Poisson embedding method for the anisotropic Calderón problem with a potential

Cătălin I. Cârstea·Tony Liimatainen·Yi-Hsuan Lin·William Trad
View original source ↗Read source document ↗Export metadata ↓

Abstract

We prove that the local Dirichlet-to-Neumann map of $Δ_g+q$ determines a compact connected real-analytic Riemannian manifold of dimension at least three and a real-analytic real-valued potential, up to a diffeomorphism fixing the observed boundary and the conformal gauge of the equation. The result is new even when \(q\) is constant, i.e. for the fixed-frequency (Helmholtz) equation \(Δ_g u+λu=0\). The measurements are restricted to an arbitrary nonempty open boundary set $Γ$. The proof develops conformal Poisson embeddings method and uses coordinates given by quotients of solutions.

Explore related subjects

math.AP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Cătălin I. Cârstea, Tony Liimatainen, Yi-Hsuan Lin, William Trad. 2026-10-09. The conformal Poisson embedding method for the anisotropic Calderón problem with a potential. https://arxiv.org/abs/2610.13592

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

KEEP EXPLORING

Related papers

arXiv2026

Global higher integrability and Hardy inequalities for double-phase functionals under a capacity density condition

We prove global higher integrability for functionals of double-phase type under a uniform local capacity density condition on the complement of the considered domain $Ω\subset \mathbb{R}^n$. In this context, we investigate a new natural notion of variational capacity associated to the double-phase integrand. Under the related fatness condition for the complement of $Ω$, we establish an integral Hardy inequality. Further, we show that fatness of $\mathbb{R}^n \setminus Ω$ is equivalent to a boundary Poincaré inequality, a pointwise Hardy inequality and to the local uniform $p$-fatness of $\mathbb{R}^n \setminus Ω$. We provide a counterexample that shows that the expected Maz'ya type inequality - a key intermediate step toward global higher integrability - does not hold with the notion of capacity involving the double-phase functional itself.

math.AP↗
arXiv2026

Modulus of continuity for solutions of non-local heat equations

We extend the two-point modulus-of-continuity method for parabolic equations---as used, for instance, in the proof of the fundamental gap conjecture---to non-local heat equations on $\R^n$. An initial modulus of continuity is shown to be preserved provided that its odd extension is a supersolution of an associated one-dimensional non-local equation. The result applies to both integrable and singular kernels, including the fractional Laplacian, and the method further applies to a class of nonlinear non-local heat equations on $\R^n$. We also examine the corresponding regional problem on bounded convex sets. The natural distance-only modulus comparison fails: a smooth counterexample is constructed in one dimension, and the same obstruction is demonstrated to persist in higher dimensions. For the regional fractional Laplacian, we define convex domains of fixed diameter for which the first non-trivial eigenvalue tends to zero. There is consequently no positive Payne--Weinberger-type lower bound for this eigenvalue depending only on the diameter of the domain.

math.AP↗
arXiv2026

Long time behavior of discrete velocity kinetic equations

We study long time behavior of some nonlinear discrete velocity kinetic equations in the one and three dimensions with periodic boundary conditions. We prove the exponential time decay of solutions towards the global equilibrium in the $L^2$ space. Our result holds for a wide class of interaction rates including the Goldstein-Taylor and Carleman equations, and the estimates on the rate of convergence are explicit and constructive. The technique is based on the construction of suitable Lyapunov functionals by modifying Boltzmann's entropy.

math.AP↗