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Chiara Spina

Publications and source records attributed to Chiara Spina.

16 recordsLinked to original sources
arXiv2026

Higher order Schr\"odinger operators

In this paper we consider higher order Schr\"odinger operators $$\mathcal L u=Lu+Vu,$$ where $L$ denotes a fourth order operator and $V\geq 0$ a suitable potential. We initiate our analysis by considering the constant coefficients differential operator $L=\Delta^2$. Subsequently, we extend our results to more general operators $L$ featuring suitable variable coefficients. We are interested in domain characterization and generation properties of these operators in $L^p(\mathbb{R}^N)$ for $p \in (1, \infty)$. To address this problems we employ a noncommutative version of the Dore-Venni theorem due to Monniaux and Pr\"uss and we prove that the $L^p$-realization of $\mathcal L$ is quasi sectorial and, consequently, generates an analytic semigroup. Furthermore, this approach allows for a sharp characterization of the operator's domain as the intersection of the domains of the bilaplacian and the multiplication operator. The required assumptions allow to treat potentials that grow at infinity like $|x|^r$ for some $r<4$.

math.AP↗
arXiv2025

Harnack inequality for Bessel operators

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 $y=0$.

math.AP↗
arXiv2024

Sharp kernel bounds for parabolic operators with first order degeneracy

We prove sharp upper and lower estimates for the parabolic kernel of the singular elliptic operator \begin{align*} \mathcal L&=\mbox{Tr }\left(AD^2\right)+\frac{\left(v,\nabla\right)}y, \end{align*} in the half-space $\mathbb{R}^{N+1}_+=\{(x,y): x \in \mathbb{R}^N, y>0\}$ under Neumann or oblique derivative boundary conditions at $y=0$.

math.AP↗
arXiv2024

Gaussian Poincar\'e inequalities on the half-space with singular weights

We prove Rellich-Kondrachov type theorems and weighted Poincar\'e inequalities on the half-space $\mathbb{R}^{N+1}_+=\{z=(x,y): x \in \mathbb{R}^N, y>0\}$ endowed with the weighted Gaussian measure $\mu :=y^ce^{-a|z|^2}dz$ where $c+1>0$ and $a>0$. We prove that for some positive constant $C>0$ one has \begin{align*} \left\|u-\overline u\right\|_{L^2_\mu(\mathbb{R}^{N+1}_+)}\leq C \|\nabla u\|_{L^2_\mu (\mathbb{R}^{N+1}_+)},\qquad \forall u\in H^1_\mu(\mathbb{R}^{N+1}_+) \end{align*} where $\overline u=\frac 1{\mu(\mathbb{R}^{N+1}_+)}\int_{\mathbb{R}^{N+1}_+} u\,d\mu(z)$. Besides this we also consider the local case of bounded domains of $\mathbb{R}^{N+1}_+$ where the measure $\mu$ is $y^cdz$.

math.AP↗
arXiv2024

Fourth-order operators with unbounded coefficients in $L^1$ spaces

We prove that operators of the form $A=-a(x)^2\Delta^{2}$, with suitable growth conditions on the coefficient $a(x)$, generate analytic semigroups in $L^1(\mathbb{R}^N)$. In particular, we deduce generation results for the operator $A :=- (1+|x|^2)^{\alpha} \Delta^{2}$, $0\leq\alpha\leq2$. Moreover, we characterise the maximal domain of $A$ in $L^1(\mathbb{R}^N)$.

math.FA↗
arXiv2024

Regularity theory for parabolic operators in the half-space with boundary degeneracy

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.

math.AP↗
arXiv2024

Fourth-order operators with unbounded coefficients

We prove that operators of the form $A=-a(x)^2Δ^{2}$, with $|D a(x)|\leq c a(x)^\frac{1}{2}$, generate analytic semigroups in $L^p(\mathbb{R}^N)$ for $1<p\leq\infty$ and in $C_b(\mathbb{R}^N)$. In particular, we deduce generation results for the operator $A :=- (1+|x|^2)^α Δ^{2}$, $0\leqα\leq2$. Moreover, we characterize the maximal domain of such operators in $L^p(\mathbb{R}^N)$ for $1<p<\infty$.

math.AP↗
arXiv2023

Singular parabolic problems in the half-space

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.

math.AP↗
arXiv2022

A unified approach to degenerate problems in the half-space

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 $\mathbb R^{N+1}_+=\{(x,y): x \in \mathbb R^N, y>0\}$.

math.AP↗
arXiv2022

Degenerate operators on the half-line

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.

math.AP↗
arXiv2022

Anisotropic Sobolev spaces with weights

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*}

math.AP↗
arXiv2010

Elliptic operators with unbounded diffusion coefficients in Lp spaces

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.

math.AP↗
arXiv2009

Maximal regularity for non-autonomous Schroedinger type equations

In this paper we study the maximal regularity property for non-autonomous evolution equations $\partial_t u(t)+A(t)u(t)=f(t), u(0)=0.$ If the equation is considered on a Hilbert space $H$ and the operators $A(t)$ are defined by sesquilinear forms $ a(t,.,.)$ we prove the maximal regularity under a Holder continuity assumption of $t \to a(t,.,.)$. In the non-Hilbert space situation we focus on Schrodinger type operators $A(t):= -Δ+ m(t, .)$ and prove $L^p-L^q$ estimates for a wide class of time and space dependent potentials $m$.

math.AP↗