Related papers: Operator estimates for the crushed ice problem

For a family of elliptic operators with rapidly oscillating periodic coefficients, we study the convergence rates for Dirichlet eigenvalues and bounds of the normal derivatives of Dirichlet eigenfunctions. The results rely on an…

In this paper, the theory of Gelfand problems is adapted to the 1--Laplacian setting. Concretely, we deal with the following problem \begin{equation*} \left\{\begin{array}{cc} -\Delta_1u=\lambda f(u) &\hbox{in }\Omega\,;\\[2mm] u=0…

We investigate the limit properties of a family of Schr\"odinger operators of the form $H_\varepsilon= -\frac{\mathrm{d}^2}{\mathrm{d}x^2}+ \frac{\lambda(\varepsilon)}{\varepsilon^2}Q \big(\frac{x}{\varepsilon}\big)$ acting on $n$-edge star…

In this article we study an inverse problem for the space-time fractional parabolic operator $(\partial_t-\Delta)^s+Q$ with $0<s<1$ in any space dimension. We uniquely determine the unknown bounded potential $Q$ from infinitely many…

We establish dimension-independent estimates related to heat operators e^{tL} on manifolds. We first develop a very general contractivity result for Markov kernels which can be applied to diffusion semigroups. Second, we develop estimates…

We study the variational problem $$\inf \{\lambda_k(\Omega): \Omega\ \textup{open in}\ \R^m,\ |\Omega| < \infty, \ \h(\partial \Omega) \le 1 \},$$ where $\lambda_k(\Omega)$ is the $k$'th eigenvalue of the Dirichlet Laplacian acting in…

In this paper, we first define a discrete version of the fractional Laplace operator $(-\Delta)^{s}$ through the heat semigroup on a stochastically complete, connected, locally finite graph $G = (V, E, \mu, w)$. Secondly, we define the…

In the whole space $R^d$, $d\ge 2$, we study homogenization of a divergence form elliptic operator $A_\varepsilon$ of order $2m\ge 4$ with measurable $\varepsilon$-periodic coefficients, where $\varepsilon$ is a small parameter. For the…

We establish new connections between integral curvature bounds and the Euler characteristic of closed Riemannian manifolds through the perspective of Schr\"odinger-type operators. Central to our approach is the twisted Dirac operator…

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…

We consider Cheeger-like shape optimization problems of the form $$\min\big\{|\Omega|^\alpha J(\Omega) : \Omega\subset D\big\}$$ where $D$ is a given bounded domain and $\alpha$ is above the natural scaling. We show the existence of a…

We study the generator $G$ of the one-dimensional damped wave equation with unbounded damping. We show that the norm of the corresponding resolvent operator, $\| (G - \lambda)^{-1} \|$, is approximately constant as $|\lambda| \to +\infty$…

Let $\Omega$ be an open set in $\R^d$ $(d > 1)$ and $h(\Omega)$ the Fr\'echet space of harmonic functions on $\Omega$. Given a bounded linear operator $L :h(\Omega)\to h(\Omega)$, we show that its eigenvalues $\lambda_n$, arranged in…

In this paper, we consider the inverse boundary value problem of the elliptic operator $\Delta+q$ in a fixed region $\Omega\subset\mathbb{R}^3$ with unknown embedded obstacles $D$. In particular, we give a new and simple proof to uniquely…

We investigate bubbling solutions for the nonlocal equation \[ A_\Omega^s u =u^p,\ u >0 \quad \mbox{in } \Omega, \] under homogeneous Dirichlet conditions, where $\Omega$ is a bounded and smooth domain. The operator $A_\Omega^s$ stands for…

We consider a class of constant-coefficient partial differential operators on a finite-dimensional real vector space which exhibit a natural dilation invariance. Typically, these operators are anisotropic, allowing for different degrees in…

For a an arbitrary periodic Borel measure $\mu$, we prove order $O(\varepsilon)$ operator-norm resolvent estimates for the solutions to scalar elliptic problems in $L^2({\mathbb R}^d, d\mu^\varepsilon)$ with $\varepsilon$-periodic…

This paper presents a posteriori error estimates for conforming numerical approximations of eigenvalue clusters of second-order self-adjoint elliptic linear operators with compact resolvent. Given a cluster of eigenvalues, we estimate the…

In this paper the approximation of Dirac operators with general $\delta$-shell potentials supported on $C^2$-curves in $\mathbb{R}^2$ or $C^2$-surfaces in $\mathbb{R}^3$, which may be bounded or unbounded, is studied. It is shown under…

This study investigates the boundedness of the \( H^\infty \)-calculus for the discrete negative Laplace operator, subject to homogeneous Dirichlet boundary conditions. The discrete negative Laplace operator is implemented using the finite…