Related papers: Heat content asymptotics for sub-Riemannian manifo…

In this paper we use the heat equation in a group of Heisenberg type $\mathbb{G}$ to provide a unified treatment of the two very different extension problems for the time independent pseudo-differential operators $\mathscr L^s$ and…

We consider the Hodge Laplacian on manifolds with incomplete edge singularities, with infinite dimensional von Neumann spaces and intricate elliptic boundary value theory. We single out a class of its algebraic self-adjoint extensions. Our…

In this paper, we observe how the heat equation in a non-cylindrical domain can arise as the asymptotic limit of a parabolic problem in a cylindrical domain, by adding a potential that vanishes outside the limit domain. This can be seen as…

Heat-invariants are a class of spectral invariants of Laplace-type operators on compact Riemannian manifolds that contain information about the geometry of the manifold, e.g., the metric and connection. Since Brownian motion solves the heat…

We show that there is non-uniqueness for the Calder{\'o}n problem with partial data for Riemannian metrics with H{\"o}lder continuous coefficients in dimension greater or equal than three. We provide simple counterexamples in the case of…

We study the heat trace asymptotics associated with the Steklov eigenvalue problem on a Riemannian manifold with boundary. In particular, we describe the structure of the Steklov heat invariants and compute the first few of them explicitly…

This is the second in a series of two articles where we study various aspects of the spectral theory associated to families of hyperbolic Riemann surfaces obtained through elliptic degeneration. In the first article, we investigate the…

We develop a method to study the structure of the common part of the boundaries of disjoint and possibly non-complementary time-varying domains in $\mathbb{R}^{n+1}$, $n \geq 2$, at the points of mutual absolute continuity of their…

We obtain a sub-Riemannian version of the classical Gauss-Bonnet theorem. We consider subsurfaces of a three dimensional contact sub-Riemannian manifolds, and using a family of taming Riemannian metric, we obtain a pure sub-Riemannian…

On a large class of Riemannian manifolds with boundary, some dimension-free Harnack inequalities for the Neumann semigroup is proved to be equivalent to the convexity of the boundary and a curvature condition. In particular, for $p_t(x,y)$…

We consider the asymptotic behavior of solutions to the convection-diffusion equation: \[ \partial_t u - \mathrm{div}\left(a(x)\nabla u\right) = d\cdot\nabla \left(\left\lvert u\right\rvert ^{q-1}u\right),\ \ x\in\mathbb{R}^n, \ t>0 \] with…

Supersymmetric domain-wall spacetimes that lift to Ricci-flat solutions of M-theory admit generalized Heisenberg (2-step nilpotent) isometry groups. These metrics may be obtained from known cohomogeneity one metrics of special holonomy by…

This is the second part of a two parts work on the analysis of heat-type equations on manifolds with fibered boundary equipped with a $\Phi$-metric. This setting generalizes the asymptotically conical (scattering) spaces and includes…

We derive a detailed asymptotic expansion of the heat trace for the Laplace-Beltrami operator on functions on manifolds with conic singularities, using the Singular Asymptotics Lemma of Jochen Bruening and Robert T. Seeley [BS]. In the…

We consider a quantum graph where the operator contains a potential. We show that this operator admits a heat kernel. Under some assumptions on the potential, this heat kernel admits an asymptotic expansion at t=0 with coefficients that…

The main goal of this work is to study the $L^p$-asymptotic behavior of solutions to the heat equation on arbitrary rank Riemannian symmetric spaces of non-compact type $G/K$ for non-bi-$K$ invariant initial data. For initial data $u_0$…

We develop a new method for the calculation of the heat trace asymptotics of the Laplacian on symmetric spaces that is based on a representation of the heat semigroup in form of an average over the Lie group of isometries and obtain a…

We use a Riemannnian approximation scheme to define a notion of $\textit{sub-Riemannian Gaussian curvature}$ for a Euclidean $C^{2}$-smooth surface in the Heisenberg group $\mathbb{H}$ away from characteristic points, and a notion of…

In this work, we establish the uniform heat kernel asymptotics as well as sharp bounds for its derivatives on the free step-two Carnot group with $3$ generators. As a by-product, on this highly non-trivial toy model, we completely solve the…

Let $\{e_j\}$ be an orthonormal basis of Laplace eigenfunctions of a compact Riemannian manifold $(M,g)$. Let $H \subset M$ be a submanifold and let $\{\psi_k\}$ be an orthonormal basis of Laplace eigenfunctions of $H$ with the induced…