Related papers: Lieb-Thirring inequality on the four-dimensional s…
We provide a unified theoretical analysis of Linear Discriminant Analysis with simultaneous multilabel scatter matrix formulations and Stiefel orthogonality constraints. Our contributions span both algebraic structure and statistical…
We prove new inequalities of the Lieb-Thirring type on the eigenvalues of Schr\"odinger operators in wave guides with local perturbations. The estimates are optimal in the weak-coupling case. To illustrate their applications, we consider,…
We prove a new family of sharp $L^2(\mathbb S^{d-1})\to L^4(\mathbb R^d)$ Fourier extension inequalities from the unit sphere $\mathbb S^{d-1}\subset \mathbb R^d$, valid in arbitrary dimensions $d\geq 3$.
This paper is essentially derived from the observation that some results used for improving constants in the Lieb-Thirring inequalities for Schrodinger operators in L2(-\infty,\infty) can be translated to the discrete Schrodinger op-…
We consider the cubic Hyperbolic Schr\"odinger equation \eqref{eq:nls} on torus $\T^2$. We prove that sharp $L^4$ Strichartz estimate, which implies that \eqref{eq:nls} is analytic locally well-posed in in $H^s(\T^2)$ with $s>1/2$,…
We obtain a new solution of the TOV-equation for an anisotropic fluid distribution by imposing the Karmarkar condition. In order to close the system of equations we postulate an interesting form for the grr gravitational potential which…
We study the cubic non linear Schr\"odinger equation (NLS) on compact surfaces. On the sphere $\mathbb{S}^2$ and more generally on Zoll surfaces, we prove that, for $s>1/4$, NLS is uniformly well-posed in $H^s$, which is sharp on the…
We establish a fourth order sharp Sobolev trace inequality on three-balls, and its equivalence to a third order sharp Sobolev inequality on two-spheres.
The flat torus ${\mathbb T}={\mathbb S}^1\left (\frac{1}{2} \right ) \times {\mathbb S}^1\left (\frac{1}{2} \right )$ admits a proper biharmonic isometric immersion into the unit $4$-dimensional sphere ${\mathbb S}^4$ given by $\Phi=i \circ…
We consider the question of showing a log-Sobolev inequality for the Gibbs measure of the focusing Schr\"odinger equation built by Lebowitz-Rose-Speer (1988), formally given by $$ d\rho \propto \exp\big(\frac 1 p\int_{\mathbb T} |u|^p d x -…
In this paper, we consider the problem of prescribing the scalar curvature under minimal boundary conditions on the standard four dimensional half sphere. We provide an Euler-Hopf type criterion for a given function to be a scalar curvature…
We study the 2D Ising model on three different types of lattices that are topologically equivalent to spheres. The geometrical shapes are reminiscent of the surface of a pillow, a 3D cube and a sphere, respectively. Systems of volumes…
We consider the low regularity behavior of the fourth order cubic nonlinear Schr\"odinger equation (4NLS) \begin{align*} \begin{cases} i\partial_tu+\partial_x^4u=\pm \vert u \vert^2u, \quad(t,x)\in \mathbb{R}\times \mathbb{R}\\…
In this paper we derive Lieb-Thirring estimates for eigenvalues of Dirichlet Laplacians below the threshold of the essential spectrum on asymptotically Archimedean spiral-shaped regions.
We prove that an integral Cauchy-Riemann inequality holds for any pair of smooth functions $(f,h)$ on the 2-sphere $\mathbb{S}^2$, and equality holds iff $f$ and $h$ are related $\lambda_1$-eigenfunctions. We extend such inequality to…
Given an arbitrary $1$-Lipschitz function $f$ on the torus $\mathbb{T}^n $, we find a $k$-dimensional subtorus $M \subseteq \mathbb{T}^n$, parallel to the axes, such that the restriction of $f$ to the subtorus $M$ is nearly a constant…
We consider a Kirchhoff problem of Brezis-Nirenberg type in a smooth bounded domain of $\mathbb{R}^4$ with Dirichlet boundary conditions. Our approach, novel in this framework and based upon approximation arguments, allows us to cope with…
In this paper, we study the Liouville-type equation \[\Delta ^2 u-\lambda_1\kappa\Delta u+\lambda_2\kappa^2(1-\mathrm e^{4u})=0\] on a closed Riemannian manifold \((M^4,g)\) with \(\operatorname{Ric}\geqslant 3\kappa g\) and \(\kappa>0\).…
We estimate the lowest eigenvalue in the gap of the essential spectrum of a Dirac operator with mass in terms of a Lebesgue norm of the potential. Such a bound is the counterpart for Dirac operators of the Keller estimates for the…
In 1976 Lieb and Thirring established upper bounds on sums of powers of the negative eigenvalues of a Schr\"odinger operator in terms of semiclassical phase-space integrals. Over the last 45 years the optimal constants in these…