Related papers: Lowest eigenvalues and formally self-adjoint fourt…

Let $D\subset R^d$ be a bounded domain and let \[ L=\frac12\nabla\cdot a\nabla +b\cdot\nabla \] %\[ %L=\frac12\sum_{i,j=1}^da_{i,j}\frac{\partial^2}{\partial x_i\partial x_j}+\sum_{i=1}^db_i\frac{\partial}{\partial x_i}, %\] be a second…

In $L_2({\mathbb R}^d;{\mathbb C}^n)$, we study a selfadjoint strongly elliptic operator $A_\varepsilon$ of order $2p$ given by the expression $b({\mathbf D})^* g({\mathbf x}/\varepsilon) b({\mathbf D})$, $\varepsilon >0$. Here $g({\mathbf…

This paper investigates links between the eigenvalues and eigenfunctions of the Laplace-Beltrami operator, and the higher Cheeger constants of smooth Riemannian manifolds, possibly weighted and/or with boundary. The higher Cheeger constants…

Let M be a compact manifold without boundary. Associated to a metric g on M there are various Laplace operators, for example the de Rham Laplacian on forms and the conformal Laplacian on functions. For a general Laplace type operator we…

We characterize the action of isotropic pseudodifferential operators on functions in terms of their action on Hermite functions. We show that an operator $A : S(\mathbb{R}) \to S(\mathbb{R})$ is an isotropic pseudodifferential operator of…

Let $\Omega\subset\mathbb{R}^{n}$ be a smooth bounded domain and $m\in C(\overline{\Omega})$ be a sign-changing weight function. For $1<p<\infty$, consider the eigenvalue problem $$ \left\{ \begin{array} [c]{ll} -\Delta_{p}u=\lambda…

On a Riemannian manifold with a smooth function $f: M\to \mathbb{R}$, we consider the linearization of the Perelman scalar curvature $\mathcal{R}$ and its $L^2$-formal adjoint operator $\delta\mathcal{R}^*$. A manifold endowed with a metric…

We study closed extensions A of an elliptic differential operator on a manifold with conical singularities, acting as an unbounded operator on a weighted L_p-space. Under suitable conditions we show that the resolvent (\lambda-A)^{-1}…

We prove the existence of extremal domains for the first eigenvalue of the Laplace-Beltrami operator in some compact Riemannian manifolds of dimension $n \geq 2$, with volume close to the volume of the manifold. If the first (positive)…

We study the spectrum of the Laplace-Beltrami operator on ellipsoids. For ellipsoids that are close to the sphere, we use analytic perturbation theory to estimate the eigenvalues up to two orders. We show that for biaxial ellipsoids…

Let $M$ be a complete Riemannian manifold and $G$ a Lie subgroup of the isometry group of $M$ acting freely and properly on $M.$ We study the Dirichlet Problem \begin{align*} \operatorname{div}\left( \frac{a\left( \left\Vert \nabla…

Let $M$ be a compact Riemannian manifold with or without boundary, and let $-\Delta $ be its Laplace-Beltrami operator. For any bounded scalar potential $q$, we denote by $\lambda\_i(q)$ the $i$-th eigenvalue of the Schr\"{o}dinger type…

We study multiplicity of the eigenvalues of the Hodge Laplacian on smooth, compact Riemannian manifolds of dimension five for generic families of metrics. We prove that generically the Hodge Laplacian, restricted to the subspace of co-exact…

Let $M$ be an $n(>2)$-dimensional closed orientable submanifold in an $(n+p)$-dimensional space form $\mathbb{R}^{n+p}(c)$. We obtain an optimal upper bound for the second eigenvalue of a class of elliptic operators on $M$ defined by…

The spectral shift function of a pair of self-adjoint operators is expressed via an abstract operator valued Titchmarsh--Weyl $m$-function. This general result is applied to different self-adjoint realizations of second-order elliptic…

For a smooth, compact Riemannian manifold (M,g) of dimension $N \geg 3$, we are interested in the critical equation $$\Delta_g u+(N-2/4(N-1) S_g+\epsilon h)u=u^{N+2/N-2} in M, u>0 in M,$$ where \Delta_g is the Laplace--Beltrami operator,…

Based on a recent work of Mancini-Thizy [28], we obtain the nonexistence of extremals for an inequality of Adimurthi-Druet [1] on a closed Riemann surface $(\Sigma,g)$. Precisely, if $\lambda_1(\Sigma)$ is the first eigenvalue of the…

We derive various eigenvalue estimates for the Hodge Laplacian acting on differential forms on weighted Riemannian manifolds. Our estimates unify and extend various results from the literature and we provide a number of geometric…

We study the evolution equation $\partial_{t}u=-\Lambda_{t}u$ where $\Lambda_ {t}$ is the Dirichlet-Neumann operator of a decreasing family of Riemannian manifolds with boundary $\Sigma_{t}$. We derive a lower bound for the solution of such…

Given a complex manifold $X$ and a smooth positive function $\eta$ thereon, we perturb the standard differential operator $d=\partial + \bar\partial$ acting on differential forms to a first-order differential operator $D_\eta$ whose…