arXiv subjects
Publications and source records attributed to Giorgio Metafune.
We prove uniqueness results and Harnack inequality for Bessel operators \begin{align*} %\label{def L transf alpha} D_t-\Delta_{x} -2a\cdot\nabla_xD_y- D_{yy}- \frac cy D_y % \nonumber \\[1ex]&=y^{\alpha}\sum_{i,j=1}^{N+1}a_{ij}D_{ij}+y^{\alpha-1}\left(v,\nabla\right)-by^{\alpha-2}. \end{align*} in the strip $[0,T]\times \mathbb{R}^{N+1}_+=\{0 \leq t \leq T, x \in \mathbb{R}^N, y>0\}$ under Neumann boundary conditions at $y=0$.
We characterize the domain of the Schrödinger operators $S=-Δ+c|x|^{-α}$ in $L^p(\mathbb{R}^N)$, with $0<α<2$ and $c\in\mathbb{R}$. When $αp< N$, the domain characterization is essentially known and can be proved using different tools, for instance kernel estimates and potentials in the Kato class or in the reverse Hölder class. However,the other cases seem not to be known, so far.In this paper, we give the explicit description of the domain of $S$ for all range of parameters $p,α$ and $c$.
We study elliptic and parabolic problems governed by the singular elliptic operators \begin{align*} \mathcal L=y^{α_1}\mbox{Tr }\left(QD^2_xu\right)+2y^{\frac{α_1+α_2}{2}}q\cdot \nabla_xD_y+γy^{α_2} D_{yy}+Cy^{α_2-1}D_y \end{align*} under Neumann boundary condition, in the half-space $\mathbb{R}^{N+1}_+=\{(x,y): x \in \mathbb{R}^N, y>0\}$. We prove elliptic and parabolic $L^p$-estimates and solvability for the associated problems. In the language of semigroup theory, we prove that $\mathcal L$ generates an analytic semigroup, characterize its domain as a weighted Sobolev space and show that it has maximal regularity.
We study elliptic and parabolic problems governed by singular elliptic operators \begin{equation*} \mathcal L =\sum_{i,j=1}^{N+1}q_{ij}D_{ij}+\frac c y D_y \end{equation*} in the half-space $\mathbb{R}^{N+1}_+=\{(x,y): x \in \mathbb{R}^N, y>0\}$ under Neumann boundary conditions at $y=0$. More general operators and oblique derivative boundary conditions will be also considered.
We study elliptic and parabolic problems governed by the singular elliptic operators \begin{equation*} \mathcal L =y^{α_1}Δ_{x} +y^{α_2}\left(D_{yy}+\frac{c}{y}D_y -\frac{b}{y^2}\right), \qquadα_1, α_2 \in\mathbb R \end{equation*} in the half-space $\mathbb R^{N+1}_+=\{(x,y): x \in \mathbb R^N, y>0\}$.
We study elliptic and parabolic problems governed by the singular elliptic operators $$ y^α\left(D_{yy}+\frac{c}{y}D_y\right)-V(y),\qquadα\in\mathbb R $$ in $\mathbb R_+$, where $V$ is a potential having non-negative real part.
We study Sobolev spaces with weights in the half-space $\mathbb{R}^{N+1}_+=\{(x,y): x \in \mathbb{R}^N, y>0\}$, adapted to the singular elliptic operators \begin{equation*} \mathcal L =y^{α_1}Δ_{x} +y^{α_2}\left(D_{yy}+\frac{c}{y}D_y -\frac{b}{y^2}\right). \end{equation*}
We study elliptic and parabolic problems governed by the singular elliptic operators Delta_x+c\yD_y-b\y^2 on the half-space R^{N+1}_+.
We establish two-sided Gaussian bounds for fundamental solutions of general non-divergence form parabolic operators with Hölder continuous coefficients. The result we obtain is essentially based on parametrix method.
Motivated by the recent results in arXiv:1601.05679 about the quark-antiquark potential in $\mathcal N=4$ SYM, we reconsider the problem of computing the asymptotic weak-coupling expansion of the ground state energy of a certain class of 1d Schrödinger operators $-\frac{d^{2}}{dx^{2}}+λ\,V(x)$ with long-range potential $V(x)$. In particular, we consider even potentials obeying $\int_{\mathbb R}dx\, V(x)<0$ with large $x$ asymptotics $V\sim -a/x^{2}-b/x^{3}+\cdots$. The associated Schrödinger operator is known to admit a bound state for $λ\to 0^{+}$, but the binding energy is rigorously non-analytic at $λ=0$. Its asymptotic expansion starts at order $\mathcal O(λ)$, but contains higher corrections $λ^{n}\,\log^{m}λ$ with all $0\le m\le n-1$ and standard Rayleigh-Schrödinger perturbation theory fails order by order in $λ$. We discuss various analytical tools to tame this problem and provide the general expansion of the binding energy at $\mathcal O(λ^{3})$ in terms of quadratures. The method is tested on a soluble potential that is fully under control, and on various non-soluble cases as well. A supersymmetric case, arising in the study of the quark-antiquark potential in $\mathcal N=6$ ABJ(M) theory, is also exploited to provide a further non-trivial consistency check. Our analytical results confirm at third order a remarkable exponentiation of the leading infrared logarithms, first noticed in $\mathcal N=4$ SYM where it may be proved by Renormalization Group arguments. We prove this interesting feature at all orders at the level of the Schrödinger equation for general potentials in the considered class.
We prove heat kernel bounds for the operator (1 + |x|^α)Δin R^N, through Nash inequalities and weighted Hardy inequalities.
In this paper we prove that, under suitable assumptions on α > 0, the operator L = (1 + |x|α)Δadmits realizations generating contraction or analytic semigroups in Lp (RN). For some values of α, we also explicitly characterize the domain of L. Finally, some informations about the location and composition of the spectrum are given.
We study a simple model of a bouncing ball that takes explicitely into account the elastic deformability of the body and the energy dissipation due to internal friction. We show that this model is not subject to the problem of inelastic collapse, that is, it does not allow an infinite number of impacts in a finite time. We compute asymptotic expressions for the time of flight and for the impact velocity. We also prove that contacts with zero velocity of the lower end of the ball are possible, but non-generic. Finally, we compare our findings with other models and laboratory experiments.