Related papers: On the M.Kac problem with augmented data

For a given bounded domain $\Omega\subset {\Bbb R}^n$ with smooth boundary, we explicitly calculate the first two coefficients of the asymptotic expansion of the heat trace associated with the Stokes operator as $t\to 0^+$. These…

This paper is dedicated to the spectral optimization problem \begin{equation*} \min \big\{ \lambda_1(\Omega)+\cdots+\lambda_k(\Omega) + \Lambda|\Omega| \ : \ \Omega \subset D \text{ quasi-open} \big\} \end{equation*} where…

For a compact Riemannian manifold $(M,g)$ with boundary $\partial M$, the Diri\-chl\-et-to-Neumann operator $\Lambda_g:C^\infty(\partial M)\longrightarrow C^\infty(\partial M)$ is defined by $\Lambda_gf=\left.\frac{\partial…

In this paper, we consider the solutions for elliptic equations involving regional fractional Laplacian \begin{equation}\label{0} \arraycolsep=1pt \begin{array}{lll} \displaystyle (-\Delta)^\alpha_\Omega u=f \qquad & {\rm in}\quad…

In this paper, we investigate the Dirichlet problem of Laplacian on complete Riemannian manifolds. By constructing new trial functions, we obtain a sharp upper bound of the gap of the consecutive eigenvalues in the sense of the order, which…

Let $0<\alpha<2$, $\beta>0$ and $\alpha/2<|s|\leq 1$. In a previous work, we obtained all possible values of the Lebesgue exponent $p=p(\gamma)$ for which the Fourier transform of $ E_{\alpha,\beta}(e^{\dot{\imath}\pi s} |\cdot|^{\gamma} )$…

We introduce an integrable Hamiltonian system which Lax deforms the Dirac operator D=d+d* on a finite simple graph or compact Riemannian manifold. We show that the nonlinear isospectral deformation always leads to an expansion of the…

In this paper, we are interested in the possible values taken by the pair $(\lambda_1(\Omega), \mu_1(\Omega))$ the first eigenvalues of the Laplace operator with Dirichlet and Neumann boundary conditions respectively of a bounded plane…

Using minimax methods and Lusternik-Schnirelmann theory, we study multiple positive solutions for the Schr\"{o}dinger - Kirchhoff equation $$ M\left(\dis\int_{\Omega_{\lambda}}|\nabla…

We study the approximation of functions that map a Euclidean domain $\Omega\subset \mathbb{R}^{d}$ into an $n$-dimensional Riemannian manifold $(M,g)$ minimizing an elliptic, semilinear energy in a function set $H\subset W^{1,2}(\Omega,M)$.…

We study Stark Hamiltonians with a $\delta$-interaction supported on a compact hypersurface in $\mathbb R^d$. Let $\Sigma$ be a compact Lipschitz hypersurface and let $\alpha\in L^\infty(\Sigma;\mathbb R)$. We define the operator…

We show that a bilinear estimate for biharmonic functions in a Lipschitz domain $\Omega$is equivalent to the solvability of the Dirichlet problem for the biharmonic equationin $\Omega$. As a result, we prove that for any given bounded…

We investigate the global solutions of the Dirac equation on the Anti-de-Sitter Universe. Since this space is not globally hyperbolic, the Cauchy problem is not, {\it a priori}, well-posed. Nevertheless we can prove that there exists…

Let $\Omega=\widetilde{\Omega}\setminus \overline{D}$ where $\widetilde{\Omega}$ is a bounded domain with connected complement in $\mathbb C^n$ (or more generally in a Stein manifold) and $D$ is relatively compact open subset of…

We study the Laplace operator subject to Dirichlet boundary conditions in a two-dimensional domain that is one-to-one mapped onto a cylinder (rectangle or infinite strip). As a result of this transformation the original eigenvalue problem…

We determine a fundamental solution for the differential operator (Delta - lambda_z)^n on the Riemannian symmetric space G/K, where G is any complex semi-simple Lie group, and K is a maximal compact subgroup. We develop a global zonal…

We study constant mean curvature graphs in the Riemannian 3-dimensional Heisenberg spaces ${\cal H}={\cal H}(\tau)$. Each such ${\cal H}$ is the total space of a Riemannian submersion onto the Euclidean plane $\mathbb{R}^2$ with geodesic…

We draw attention to the fact that a Hermitian matrix is always diagonalizable and has real discrete spectrum whereas the Hermitian Schr{\"o}dinger Hamiltonian: $H=p^2/2\mu+V(x)$, may not be so. For instance when $V(x)=x, x^3, -x^2$, $H$…

The existence of non-isomorphic graphs which share the same Laplace spectrum (to be referred to as isospectral graphs) leads naturally to the following question: What additional information is required in order to resolve isospectral…

We describe a method for constraining Laplacian and Dirac spectra of two dimensional compact orientable hyperbolic spin manifolds and orbifolds. The key ingredient is an infinite family of identities satisfied by the spectra. These spectral…