arXiv subjects
Publications and source records attributed to Laurent Kayser.
We adapt in the present note the perturbation method introduced in [3] to get a Gaussian lower bound for the Neumann heat kernel of the Laplace-Beltrami operator on an open subset of a compact Riemannian manifold.
Given a domain $Ω$ of a complete Riemannian manifold $\mathcal{M}$ and define $\mathcal{A}$ to be the Laplacian with Neumann boundary condition on $Ω$. We prove that, under appropriate conditions, the corresponding heat kernel satisfies the Gaussian upper bound $$ h(t,x,y)\leq \frac{C}{\left[V\_Ω(x,\sqrt{t})V\_Ω(y,\sqrt{t})\right]^{1/2}}\left( 1+\frac{d^2(x,y)}{4t}\right)^δe^{-\frac{d^2(x,y)}{4t}},\;\; t\textgreater{}0,\; x,y\in Ω. $$ Here $d$ is the geodesic distance on $\mathcal{M}$, $V\_Ω(x,r)$ is the Riemannian volume of $B(x,r)\cap Ω$, where $B(x,r)$ is the geodesic ball of center $x$ and radius $r$, and $δ$ is a constant related to the doubling property of $Ω$. As a consequence we obtain analyticity of the semigroup $e^{-t {\mathcal A}}$ on $L^p(Ω)$ for all $p \in [1, \infty)$ as well as a spectral multiplier result.
Based on the fact that the Neumann Green function can be constructed as a perturbation of the fundamental solution by a single-layer potential, we establish gaussian two-sided bounds for the Neumann Green function for a general parabolic operator. We build our analysis on classical tools coming from the construction of a fundamental solution of a general parabolic operator by means of the so-called parametrix method. At the same time we provide a simple proof for the gaussian two-sided bounds for the fundamental solution. We also indicate how our method can be adapted to get a gaussian lower bound for the Neumann heat kernel of a compact Riemannian manifold with boundary having non negative Ricci curvature.
Let $Ω$ be a $C^\infty$-smooth bounded domain of $\mathbb{R}^n$, $n \geq 1$, and let the matrix ${\bf a} \in C^\infty (\overlineΩ;\R^{n^2})$ be symmetric and uniformly elliptic. We consider the $L^2(Ω)$-realization $A$ of the operator $-\mydiv ( {\bf a} \nabla \cdot)$ with Dirichlet boundary conditions. We perturb $A$ by some real valued potential $V \in C_0^\infty (Ω)$ and note $A_V=A+V$. We compute the asymptotic expansion of $\mbox{tr}\left( e^{-t A_V}-e^{-t A}\right)$ as $t \downarrow 0$ for any matrix ${\bf a}$ whose coefficients are homogeneous of degree $0$. In the particular case where $A$ is the Dirichlet Laplacian in $Ω$, that is when ${\bf a}$ is the identity of $\R^{n^2}$, we make the four main terms appearing in the asymptotic expansion formula explicit and prove that $L^\infty$-bounded sets of isospectral potentials of $A$ are $H^s$-compact for $s <2$.