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We obtain a trace Hardy inequality for the Euclidean space with a bounded cut $Sigmasubsetmathbb R^d$, $d ge 2$. In this novel geometric setting, the Hardy-type inequality non-typically holds also for $d = 2$. The respective Hardy weight is given in terms of the geodesic distance to the boundary of $Sigma$. We provide its applications to the heat equation on $mathbb R^d$ with an insulating cut at $Sigma$ and to the Schrodinger operator with a $delta$-interaction supported on $Sigma$. We also obtain generalizations of this trace Hardy inequality for a class of unbounded cuts.
We investigate the large-distance asymptotics of optimal Hardy weights on $mathbb Z^d$, $dgeq 3$, via the super solution construction. For the free discrete Laplacian, the Hardy weight asymptotic is the familiar $frac{(d-2)^2}{4}|x|^{-2}$ as $|x|toin
We establish an analog Hardy inequality with sharp constant involving exponential weight function. The special case of this inequality (for n=2) leads to a direct proof of Onofri inequality on S^2.
We prove a local Faber-Krahn inequality for solutions $u$ to the Dirichlet problem for $Delta + V$ on an arbitrary domain $Omega$ in $mathbb{R}^n$. Suppose a solution $u$ assumes a global maximum at some point $x_0 in Omega$ and $u(x_0)>0$. Let $T(x_
There are at least two directions concerning the extension of classical sharp Hardy-Littlewood-Sobolev inequality: (1) Extending the sharp inequality on general manifolds; (2) Extending it for the negative exponent $lambda=n-alpha$ (that is for the c
We study the quadratic form associated to the kinetic energy operator in the presence of an external magnetic field in d = 3. We show that if the radial component of the magnetic field does not vanish identically, then the classical lower bound given