arXiv subjects

Stefano Biagi

Publications and source records attributed to Stefano Biagi.

47 records · Page 3Linked to original sources

arXiv2019

Existence results for boundary value problems associated with singular strongly nonlinear equations

We consider a strongly nonlinear differential equation of the following general type $$(Φ(a(t,x(t)) \, x'(t)))'= f(t,x(t),x'(t)), \quad \text{a.e. on $[0,T]$}$$ where $f$ is a Carathédory function, $Φ$ is a strictly increasing homeomorphism (the $Φ$-Laplacian operator) and the function $a$ is continuous and non-negative. We assume that $a(t,x)$ is bounded from below by a non-negative function $h(t)$, independent of $x$ and such that $1/h \in L^p(0,T)$ for some $p> 1$, and we require a weak growth condition of Wintner-Nagumo type. Under these assumptions, we prove existence results for the Dirichlet problem associated to the above equation, as well as for different boundary conditions. Our approach combines fixed point techniques and the upper/lower solutions method.

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arXiv2019

Global Heat Kernels for Parabolic Homogeneous Hörmander Operators

The aim of this paper is to prove the existence and several selected properties of a global fundamental Heat kernel $Γ$ for the parabolic operators $\mathcal{H}=\sum_{j=1}^m X_j^2-\partial_t$, where $X_1,\ldots,X_m$ are smooth vector fields on $\mathbb{R}^n$ satisfying Hörmander'snrank condition, and enjoying a suitable homogeneity assumption with respect to a family of non-isotropic dilations. The proof of the existence of $Γ$ is based on a (algebraic) global lifting technique, together with a representation of $Γ$ in terms of the integral (performed over the lifting variables) of the Heat kernel for the Heat operator associated with a suitable sub-Laplacian on a homogeneous Carnot group. Among the features of $Γ$ we prove: homogeneity and symmetry properties; summability properties; its vanishing at infinity; the uniqueness of the bounded solutions of the related Cauchy problem; reproduction and density properties; an integral representation for the higher-order derivatives.

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arXiv2019

Large sets at infinity and Maximum Principle on unbounded domains for a class of sub-elliptic operators

Maximum Principles on unbounded domains play a crucial rôle in several problems related to linear second-order PDEs of elliptic and parabolic type. In this paper we consider a class of sub-elliptic operators $\mathcal{L}$ in $\mathbb{R}^N$ and we establish some criteria for an unbounded open set to be a Maximum Principle set for $\mathcal{L}$. We extend some classical results related to the Laplacian (by Deny, Hayman and Kennedy) and to the sub-Laplacians on stratified Lie groups (by Bonfiglioli and the second-named author).

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arXiv2019

Global estimates in Sobolev spaces for homogeneous Hörmander sums of squares

Let $\mathcal{L}=\sum_{j=1}^m X_j^2$ be a Hörmander sum of squares of vector fields in space $\mathbb{R}^n$, where any $X_j$ is homogeneous of degree $1$ with respect to a family of non-isotropic dilations in space. In this paper we prove global estimates and regularity properties for $\mathcal{L}$ in the $X$-Sobolev spaces $W^{k,p}_X(\mathbb{R}^n)$, where $X = \{X_1,\ldots,X_m\}$. In our approach, we combine local results for general Hörmander sums of squares, the homogeneity property of the $X_j$'s, plus a global lifting technique for homogeneous vector fields.

math.AP↗

arXiv2019

A symmetry result for cooperative elliptic systems with singularities

We obtain symmetry results for solutions of an elliptic system of equation possessing a cooperative structure. The domain in which the problem is set may possess "holes" or "small vacancies" (measured in terms of capacity) along which the solution may diverge. The method of proof relies on the moving plane technique, which needs to be suitably adapted here to take care of the complications arising from the vacancies in the domain and the analytic structure of the elliptic system.

math.AP↗

arXiv2018

On the Baker-Campbell-Hausdorff Theorem: non-convergence and prolongation issues

We investigate some topics related to the celebrated Baker-Campbell-Hausdorff Theorem: a non-convergence result and prolongation issues. Given a Banach algebra $\mathcal{A}$ with identity $I$, and given $X,Y\in \mathcal{A}$, we study the relationship of different issues: the convergence of the BCH series $\sum_n Z_n(X,Y)$, the existence of a logarithm of $e^Xe^Y$, and the convergence of the Mercator-type series $\sum_n {(-1)^{n+1}}(e^Xe^Y-I)^n/n$ which provides a selected logarithm of $e^Xe^Y$. We fix general results and, by suitable matrix counterexamples, we show that various pathologies can occur, among which we provide a non-convergence result for the BCH series. This problem is related to some recent results, of interest in physics, on closed formulas for the BCH series: while the sum of the BCH series presents several non-convergence issues, these closed formulas can provide a prolongation for the BCH series when it is not convergent. On the other hand, we show by suitable counterexamples that an analytic prolongation of the BCH series can be singular even if the BCH series itself is convergent.

math-ph↗

arXiv2018

A symmetry result for polyharmonic problems with Navier conditions

We consider an elliptic polyharmonic problem of any order which takes place in a punctured bounded domain with Navier conditions. We prove that if the domain is convex in one direction and symmetric with respect to the reflections induced by the normal hyperpane to such a direction, then the solution is necessarily symmetric under this reflection and monotone in the corresponding direction.

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arXiv2016

The existence of a global fundamental solution for homogeneous Hörmander operators via a global lifting method

We prove the existence of a global fundamental solution $Γ(x;y)$ (with pole $x$) for any Hörmander operator $\mathcal{L}=\sum_{i=1}^m X_i^2$ on $\mathbb{R}^n$ which is $δ$-homogeneous of degree $2$. By means of a global Lifting method for homogeneous operators proved by Folland in [On the Rothschild-Stein lifting theorem, Comm. PDEs, 1977], there exists a Carnot group $\mathbb{G}$ and a polynomial surjective map $π:\mathbb{G}\to \mathbb{R}^n$ such that $\mathcal{L}$ is $π$-related to a sub-Laplacian $\mathcal{L}_{\mathbb{G}}$ on $\mathbb{G}$. We show that it is always possible to perform a (global) change of variable on $\mathbb{G}$ such that the lifting map $π$ becomes the projection of $\mathbb{G}\equiv \mathbb{R}^n\times\mathbb{R}^p$ onto $\mathbb{R}^n$. If $Γ_{\mathbb{G}}(x,{x}';y,{y}')$ ($x,{x}'\in\mathbb{R}^n$; $y,{y}'\in\mathbb{R}^p$) is the fundamental solution of $\mathcal{L}_{\mathbb{G}}$, we show that $Γ_{\mathbb{G}}(x,0;y,{y}')$ is always integrable w.r.t. ${y}'\in \mathbb{R}^p$, and its integral is a fundamental solution for $\mathcal{L}$.

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arXiv2016

A Hardy Inequality for subelliptic operators with global fundamental solution, and an application to Unique Continuation

This is a chapter from PhD Thesis by Stefano Biagi (advisor: prof. A. Bonfiglioli). We overview existing results showing that it is possible to generalize the classical Hardy's Inequality to more general linear partial differential operators (PDOs, in the sequel), possibly degenerate-elliptic, of the following quasi-divergence form $$ \mathcal{L} = \frac{1}{w(x)}\sum_{i = 1}^N\frac{\partial}{\partial x_i} \left(\sum_{j = 1}^Nw(x)a_{ij}(x)\frac{\partial}{\partial x_j}\right), \quad x \in \mathbb{R}^N, $$ where $w \in C^{\infty}(\mathbb{R}^N,\mathbb{R})$ is a (smooth and) strictly positive function on the whole of $\mathbb{R}^N$ and $A(x) := \begin{pmatrix}a_{ij}(x) \end{pmatrix}$ is a symmetric and positive semi-definite $N\times N$ matrix with real $C^{\infty}$ entries. From such a inequality, it has been derived a result of unique continuation for the solutions of the equation $$ -\mathcal{L} u + Vu = 0, $$ where $\mathcal{L}$ is a left-invariant homogeneous PDO on a homogeneous Lie group $\mathbb{G}$ and $V$ is real-valued function defined on $\mathbb{G}$ and continuous on $\mathbb{G}\setminus\{0\}$.

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arXiv2014

The Strong Maximum Principle and the Harnack inequality for a class of hypoelliptic divergence-form operators

In this paper we consider a class of hypoelliptic second-order partial differential operators $\mathcal{L}$ in divergence form on $\mathbb{R}^N$, arising from CR geometry and Lie group theory, and we prove the Strong and Weak Maximum Principles and the Harnack Inequality for $\mathcal{L}$. The involved operators are not assumed to belong to the Hörmander hypoellipticity class, nor to satisfy subelliptic estimates, nor Muckenhoupt-type estimates on the degeneracy of the second order part; indeed our results hold true in the infinitely-degenerate case and for operators which are not necessarily sums of squares. We use a Control Theory result on hypoellipticity in order to recover a meaningful geometric information on connectivity and maxima propagation, yet in the absence of any Hörmander condition. For operators $\mathcal{L}$ with $C^ω$ coefficients, this control-theoretic result will also imply a Unique Continuation property for the $\mathcal{L}$-harmonic functions. The (Strong) Harnack Inequality is obtained via the Weak Harnack Inequality by means of a Potential Theory argument, and by a crucial use of the Strong Maximum Principle and the solvability of the Dirichlet problem for $\mathcal{L}$ on a basis of the Euclidean topology.

math.AP↗

arXiv2013

On M. Mérigot's theorem on the convergence domain of the Campbell-Baker-Hausdorff-Dynkin series

The aim of this manuscript is to present the proof given by Michel Mérigot in 1974 for an enlarged convergence domain of the Campbell-Baker-Hausdorff-Dynkin series in the Lie algebra of a Banach-Lie group. This proof is based on a theorem, of independent interest, on the lifetime of the solution of a Cauchy problem. We furnish all the details for this ODE result in Appendix A.

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