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For a general subcritical second-order elliptic operator $P$ in a domain $Omega subset mathbb{R}^n$ (or noncompact manifold), we construct Hardy-weight $W$ which is optimal in the following sense. The operator $P - lambda W$ is subcritical in $Omega$ for all $lambda < 1$, null-critical in $Omega$ for $lambda = 1$, and supercritical near any neighborhood of infinity in $Omega$ for any $lambda > 1$. Moreover, if $P$ is symmetric and $W>0$, then the spectrum and the essential spectrum of $W^{-1}P$ are equal to $[1,infty)$, and the corresponding Agmon metric is complete. Our method is based on the theory of positive solutions and applies to both symmetric and nonsymmetric operators. The constructed Hardy-weight is given by an explicit simple formula involving two distinct positive solutions of the equation $Pu=0$, the existence of which depends on the subcriticality of $P$ in $Omega$.
In this paper we prove stable determination of an inverse boundary value problem associated to a magnetic Schrodinger operator assuming that the magnetic and electric potentials are essentially bounded and the magnetic potentials admit a Holder-type modulus of continuity in the sense of $L^2$.
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
In this paper, we study an extension problem for the Ornstein-Uhlenbeck operator $L=-Delta+2xcdot abla +n$ and we obtain various characterisations of the solution of the same. We use a particular solution of that extension problem to prove a trace Ha
Generalizing results by Bryant and Griffiths [Duke Math. J., 1995, V.78, 531-676], we completely describe local conservation laws of second-order (1+1)-dimensional evolution equations up to contact equivalence. The possible dimensions of spaces of co
We study the inverse scattering problem of determining a magnetic field and electric potential from scattering measurements corresponding to finitely many plane waves. The main result shows that the coefficients are uniquely determined by $2n$ measur