arXiv subjects

Alberto Cialdea

Publications and source records attributed to Alberto Cialdea.

9 recordsLinked to original sources

arXiv2026

Completeness theorems on the boundary for a parabolic equation

Let $\{v_{\alpha}\}$ be a system of polynomial solutions of the parabolic equation $a_{hk}\partial_{x_{h}x_{k}}u - \partial_t u =0$ in a bounded $C^1$-cylinder $\Omega_{T}$ contained in $\mathbb{R}^{n+1}$. Here $a_{hk}\partial_{x_{h}x_{k}}$ is an elliptic operator with real constant coefficients. We prove that $\{v_{\alpha}\}$ is complete in $L^{p}(\Sigma')$, where $\Sigma'$ is the parabolic boundary of $\Omega_{T}$. Similar results are proved for the adjoint equation $a_{hk}\partial_{x_{h}x_{k}} u+ \partial_t u =0$.

math.AP↗

arXiv2026

On the Robin problem for the Laplace equation in multiply connected domains

This paper complements the existing theory developed in [5] for the Dirichlet and Neumann problems for the Laplace equation, in multiply connected domains. Within the framework of layer potential methods, we study the Laplace equation under Robin boundary conditions, representing the solutions by means of a double layer potential. We observe that the classical approach searches the solutions in terms of a single layer potential.

math.AP↗

arXiv2024

New completeness theorems on the boundary in Elasticity

The completeness on the boundary (in the sense of Picone) of certain systems related to the III and IV BVPs for the elasticity system is proved. The completeness is obtained in both $L^p$ ($1\leq 1<\infty$) and uniform norms.

math.AP↗

arXiv2022

The functional dissipativity of certain systems of partial differential equations

In the present paper we consider the functional dissipativity of the Dirichlet problem for systems of partial differential operators of the form $\partial_{h} ({\mathop{\mathscr A}\nolimits}^{hk}(x)\partial_{k})$ (${\mathop{\mathscr A}\nolimits}^{hk}$ being $m\times m$ matrices with complex valued $L^{1}_{\text{loc}}$ entries). In the particular case of the operator $\partial_{h} ({\mathop{\mathscr A}\nolimits}^{h}(x)\partial_{h})$ (where ${\mathop{\mathscr A}\nolimits}^{h}$ are $m\times m$ matrices) we obtain algebraic necessary and sufficient conditions. We give also three different notions of functional ellipticity and investigate the relations between them and the functional dissipativity for the operators in question.

math.AP↗

arXiv2020

Criterion for the functional dissipativity of second order differential operators with complex coefficients

In the present paper we consider the Dirichlet problem for the second order differential operator $E=\nabla(A \nabla)$,where $A$ is a matrix with complex valued $L^\infty$ entries. We introduce the concept of dissipativity of $E$ with respect to a given function $φ:R^+ \to R^+$. Under the assumption that the $Im\, A$ is symmetric, we prove that the condition $|s\, φ'(s)| \, | \langle Im\, A (x)\, ξ,ξ\rangle |\leq 2\, \sqrt{φ(s)\, [s\, φ(s)]'}\, \langle Re\, A(x) \, ξ,ξ\rangle $ (for almost every $x\inΩ\subset R^N$ and for any $s>0$, $ξ\in R^N$) is necessary and sufficient for the functional dissipativity of $E$.

math.AP↗

arXiv2017

The $L^p$-dissipativity of certain differential and integral operators

The first part of the paper is a survey of some of the results previously obtained by the authors concerning the $L^p$-dissipativity of scalar and matrix partial differential operators. In the second part we give new necessary and, separately, sufficient conditionsfor the $L^p$-dissipativity of the "complex oblique derivative" operator. In the case of real coefficients we provide a necessary and sufficient condition. We prove also the $L^p$-positivity for a certain class of integral operators.

math.AP↗

arXiv2009

A quasi-commutativity property of the Poisson and composition operators

Let $Φ$ be a real valued function of one real variable, let $L$ denote an elliptic second order formally self-adjoint differential operator with bounded measurable coefficients, and let $P$ stand for the Poisson operator for $L$. A necessary and sufficient condition on $Φensuring the equivalence of the Dirichlet integrals of $Φ\circ Ph$ and $P(Φ\circ h)$ is obtained. We illustrate this result by some sharp inequalities for harmonic functions.

math.AP↗

arXiv2006

Criteria for the $L^{p}$-dissipativity of systems of second order differential equations

We give complete algebraic characterizations of the $L^{p}$-dissipativity of the Dirichlet problem for some systems of partial differential operators of the form $\partial_{h}({\mathscr A}^{hk}(x)\partial_{k})$, were ${\mathscr A}^{hk}(x)$ are $m\times m$ matrices. First, we determine the sharp angle of dissipativity for a general scalar operator with complex coefficients. Next we prove that the two-dimensional elasticity operator is $L^{p}$-dissipative if and only if $$ ({1\over 2}-{1\over p})^{2} \leq {2(ν-1)(2ν-1)\over (3-4ν)^{2}}, $$ $ν$ being the Poisson ratio. Finally we find a necessary and sufficient algebraic condition for the $L^{p}$-dissipativity of the operator $\partial_{h} ({\mathscr A}^{h}(x)\partial_{h})$, where ${\mathscr A}^{h}(x)$ are $m\times m$ matrices with complex $L^{1}_{\rm loc}$ entries, and we describe the maximum angle of $L^{p}$-dissipativity for this operator.

math.AP↗

arXiv2004

Criterion for the $L^{p}$-dissipativity of second order differential operators with complex coefficients

We prove that the algebraic condition $|p-2| |< {\mathscr Im}{\mathscr A}ξ,ξ>| \leq 2 \sqrt{p-1} < {\mathscr Re}{\mathscr A}ξ,ξ>$ (for any $ξ\in\mathbb{R}^{n}$) is necessary and sufficient for the $L^{p}$-dissipativity of the Dirichlet problem for the differential operator $\nabla^{t}({\mathscr A}\nabla)$, where ${\mathscr A}$ is a matrix whose entries are complex measures and whose imaginary part is symmetric. This result is new even for smooth coefficients, when it implies a criterion for the $L^{p}$-contractivity of the corresponding semigroup. We consider also the operator $\nabla^{t}({\mathscr A}\nabla)+{\bf b}\nabla +a$, where the coefficients are smooth and ${\mathscr Im}{\mathscr A}$ may be not symmetric. We show that the previous algebraic condition is necessary and sufficient for the $L^{p}$-quasi-dissipativity of this operator. The same condition is necessary and sufficient for the $L^{p}$-quasi-contractivity of the corresponding semigroup. We give a necessary and sufficient condition for the $L^{p}$-dissipativity in $\mathbb{R}^{n}$ of the operator $\nabla^{t}({\mathscr A}\nabla)+{\bf b}\nabla +a$ with constant coefficients.

math.AP↗