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Nicola Garofalo

Publications and source records attributed to Nicola Garofalo.

At least 19 recordsLinked to original sources
arXiv2026

An extension theory for fully fractional Schrödinger equations with memory

We develop a Caffarelli-Silvestre extension theory for the fully fractional Schrödinger operator $\mathcal{L}^s=(\partial_t-iΔ_x)^s$, $0<s<1$, nonlocal in space and time, whose Cauchy problem prescribes a past history rather than data at a single time. Every extension theory so far rests on positivity or sectoriality of the generator; here the semigroup is unitary and the symbol changes sign across the characteristic paraboloid, so neither is at hand. We construct the extension nonetheless. Its Poisson kernel, computed explicitly, is oscillatory rather than positive; its Dirichlet-to-Neumann map is $\mathcal{L}^s$; its normalising constant is the Caffarelli-Silvestre constant times the phase $e^{i\frac{πs}{2}}$; and the theory undergoes a transition at $s=\frac12$. We then identify the intrinsic Hilbert space of histories, on which the Poisson lifting is an isometry. This closes a circle. A lifted history is precisely an initial datum for the singular Schrödinger equation with nonlinear Neumann interaction of our companion paper, whose well-posedness theory therefore transfers to $\mathcal{L}^su=μ|u|^{p-1}u$ with prescribed past. With our earlier work on the Bessel operator on a half-line, the three papers form a single programme, of which the present one is the closing step.

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arXiv2026

Strichartz estimates for Schrödinger equations with nonlinear boundary interactions

We study a Schrödinger equation in the upper half-space with a nonlinear Neumann boundary interaction driven by the Bessel operator $\Ba$, $a>-1$. The problem arises naturally as an extension formulation for a nonlocal NLS with memory and can also be interpreted as a Schrödinger evolution with a nonlinear singular source concentrated on a codimension-one interface. We first develop a complete linear theory for the associated inhomogeneous problem with nonhomogeneous Neumann data. A central ingredient is a new Duhamel representation formula that separates bulk and boundary dynamics and identifies the precise role of the boundary propagator. Using this formula, we establish sharp Strichartz estimates adapted to the geometry of the half-space and the singular structure induced by the Bessel operator. The analysis reveals a basic dichotomy between the regimes $a\ge 0$ and $-1<a<0$: in the former, bulk and boundary exhibit a unified dispersive behavior, whereas in the latter the dispersive structure becomes anomalous, requiring weighted estimates and distinct functional frameworks for the bulk and boundary contributions. As an application of the linear theory, we prove well-posedness results for the nonlinear problem in the $L_a^2$-mass critical and subcritical regimes. For $a\ge 0$, we obtain global well-posedness for sufficiently small critical data, together with local and global results in the subcritical case. In the anomalous range $-1<a<0$, we establish existence and uniqueness on arbitrary finite time intervals for sufficiently small critical data, as well as local well-posedness in the subcritical regime. The results provide a unified dispersive framework for Schrödinger equations with nonlinear boundary interactions and singular extension structures associated with nonlocal-in-time dynamics.

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arXiv2026

A nonlinear Li-Yau inequality and its consequences

We prove a sharp nonlinear version of the celebrated Li-Yau inequality for positive solutions of the normalized parabolic $p$-Laplacian equation $u_t = Δu + (p-2)|\nabla u|^{-2}\nabla^2u(\nabla u,\nabla u)$, $1<p<\infty$, on a closed Riemannian manifold with nonnegative Ricci curvature, the equation being interpreted in the viscosity sense on the critical set of the solution. The constant is best possible, even within the class of closed manifolds: it is attained identically by an explicit self-similar solution in flat $\Rn$, and its sharpness transfers to the compact setting through a large-torus limit along the flat tori $\mathbb R^n/(L\mathbb Z)^n$, $L \to \infty$. The proof rests on an exact Bochner-type identity for a family of uniformly parabolic approximating flows in which the second-order regularization and the first-order eikonal term are decoupled: for this family the maximum principle applies with the sharp constant, uniformly in the regularization parameter, and with no assumption on the critical set of the solution. The identity is moreover form-invariant under the classical $α$-relaxation of the Li-Yau functional; as a consequence we also obtain the corresponding inequality on closed manifolds with $\operatorname{Ric} \ge -κ$ (again with no assumption beyond the Ricci lower bound), and, on complete noncompact manifolds with $\operatorname{Ric} \ge 0$, the sharp inequality for the approximating flows -- again with no assumption on the critical set -- under a cutoff hypothesis on the distance function (automatic in $\Rn$, and, for $p \ge 2$, under nonnegative sectional curvature) and a qualitative polynomial growth condition on the Li-Yau quantity, satisfied, with exponent zero, by the extremal profile. A sharp global Harnack inequality follows.

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arXiv2026

Calculations for a general maximum principle: 1993 notes on the normalized parabolic $p$-Laplacian

This note makes available, in facsimile form, a set of handwritten notes from September 1993 devoted to Bochner-type maximum principle computations for a general class of quasilinear parabolic equations which contains, as a special instance, the normalized parabolic $p$-Laplacian \[ u_t = Δu + (p-2)\,|\nabla u|^{-2}\,\nabla^2u(\nabla u,\nabla u). \] The notes were acknowledged in the introduction of the paper of Jin and Silvestre \cite{JS}, and the scheme they propose has recently been implemented in \cite{BGM} to obtain the sharp Li-Yau inequality for this equation. They are reproduced here verbatim, with no alterations, preceded by a brief introduction explaining their content and the subsequent history of the equation.

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arXiv2026

The Schrödinger Ornstein--Uhlenbeck flow: dispersion, restriction and nonlinear dynamics

We study the Schrödinger evolution generated by the isotropic Ornstein--Uhlenbeck operator \[ \mathscr L=Δ-\langle x,\nabla\rangle \] in $L^2(\mathbb R^d,dγ)$, where $dγ$ is the invariant Gaussian measure. Combining the classical lens correspondence between the free Schrödinger equation and the harmonic oscillator with Gaussian conjugation, we identify the free Schrödinger structure hidden in the Ornstein--Uhlenbeck evolution. The main consequence is a dynamic restriction theorem of spacetime type. After an explicit transformation of spacetime variables, the adjoint Ornstein--Uhlenbeck extension operator becomes an ordinary spacetime Fourier transform restricted to the classical Schrödinger paraboloid. On the Ornstein--Uhlenbeck side this corresponds to an explicitly time-dependent Gaussian spacetime measure. Thus a restriction geometry which is neither suggested by the symbol of $\mathscr L$ nor selected by a scaling symmetry of the Ornstein--Uhlenbeck equation emerges dynamically from the free Schrödinger representation. The same correspondence identifies the weighted Gaussian spaces in the homogeneous and inhomogeneous Ornstein--Uhlenbeck Strichartz estimates and transports the mass-critical nonlinear Schrödinger equation to \[ i\partial_tu+\mathscr Lu = μe^{-|x|^2/d}|u|^{4/d}u. \] We also obtain a global description of the unitary Ornstein--Uhlenbeck group, including its oscillatory representation away from the caustics and its exact values at the caustic times, together with the associated sharp weighted dispersive estimate and Hardy-type uncertainty principles.

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arXiv2026

Quantitative uniqueness for parabolic equations with Hölder potentials

In this note we derive a space-like quantitative uniqueness result for parabolic operators with Hölder zero-order term that interpolates between the Donnelly-Fefferman and the Bourgain-Kenig estimate. This generalizes a recent result of Teng, Wang and Zhu for the time-independent Schrödinger operator with a Hölder potential.

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arXiv2026

A strong unique continuation result for the Baouendi operator

We establish a strong unique continuation property for the subelliptic Baouendi operator under the presence of zero-order perturbations satisfying an almost Hardy-type growth condition. In particular, the admissible class includes both $L^\infty_{\mathrm{loc}}$ and singular potentials. We prove that any solution vanishing to infinite order at a point of the degeneracy manifold of the operator must be identically zero. The result holds extends to variable-coefficient operators with intrinsic Lipschitz regularity. A notable feature of the proof is that it relies exclusively on $L^2$ Carleman estimates combined with the classical Hardy inequality.

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arXiv2025

Fractal Mehler kernels and nonlinear geometric flows

In this paper we introduce a two-parameter family of Mehler kernels and connect them to a class of Baouendi-Grushin flows in fractal dimension. We also highlight a link with a geometric fully nonlinear equation and formulate two questions.

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arXiv2025

Dispersive equations with invariant measures

In mathematical physics it is of interest to study Schrödinger equations with friction and possessing an invariant measure. The focus of this paper is the Cauchy problem for the Schrödinger equation $\p_t f - i \mathscr L f = 0$, where $\mathscr L = Δ- \sa x,\nabla\da$ is the Ornstein-Uhlenbeck operator. We use this as a model to stimulate interest in a new class of possibly degenerate dispersive equations which cannot be treated by the existing theory.

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arXiv2025

Strichartz estimates for a Schrödinger equation on the half-line with a Neumann boundary condition

In this paper we prove some new Strichartz estimates related to the Cauchy problem for the Bessel operator on the half-line and we establish a fractal version of the Tomas-Stein restriction theorem for the Hankel transform. Then we use the proved Strichartz estimates to show global in time well-posedness for a class of nonlinear $L^2_a$-critical problems, and local in time well-posedness in the sub-critical case.

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arXiv2025

Absence of $L^p$ spectrum for asymptotically flat diffusions in region with cavities

We study solutions to variable-coefficient elliptic equations of the form $-\D(A(x) \nabla u) = κu$, $κ>0$, in an exterior domain $\Om\subset \Rn$, where $A(x)$ is uniformly elliptic and asymptotically flat. Extending Rellich's classical $L^2$ result for the Laplacian, we show that if $u\in L^p(\Om)$ for some $0<p<\frac{2n}{n-1}$, then $u\equiv 0$. The proof uses new monotonicity formulas based on weighted energies and vector fields adapted to the geometry of $A(x)$. Our results highlight a sharper integrability threshold in the variable-coefficient setting.

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arXiv2024

On unique continuation in measure for fractional heat equations

We prove a theorem of unique continuation in measure for nonlocal equations of the type $(\partial_t - Δ)^s u= V(x,t) u$, for $0<s <1$. Our main result, Theorem 1.1, establishes a delicate nonlocal counterpart of the unique continuation in measure for the local case $s=1$.

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arXiv2024

Schrödinger semigroups and the Hörmander hypoellipticity condition

We introduce a class of (possibly) degenerate dispersive equations with a drift. We prove that, under the Hörmander hypoellipticity condition, the relevant Cauchy problem can be uniquely solved in the Schwartz class, and the solution operator can be uniquely extended to a strongly continuous semigroup $\{\mathcal T(t)\}_{t\ge 0}$ in $L^2(\Rm)$. Finally, we prove that for $t>0$ the operator $\mathcal T(t)$ satisfies a sharp form of dispersive estimate in $L^p$, for any $1\le p\le 2$, and an uncertainty principle.

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