arXiv subjects

Umberto Biccari

Publications and source records attributed to Umberto Biccari.

41 records · Page 3Linked to original sources

arXiv2017

A Finite Element approximation of the one-dimensional fractional Poisson equation with applications to numerical control

We present a finite element (FE) scheme for the numerical approximation of the solution to a non-local Poisson equation involving the one-dimensional fractional Laplacian $(-d_x^2)^s$ on the interval $(-L,L)$. In particular, we include the complete computations for obtaining the stiffness matrix, starting from the variational formulation of the problem. The problem being one-dimensional, the values of the matrix can be explicitly calculated, without need of any numerical integration, thus obtaining an algorithm which is very efficient in terms of the computational cost. As an application, we analyze the corresponding parabolic equation from the point of view of controllability properties, employing the penalized Hilbert Uniqueness Method (HUM) for computing the numerical approximation of the null-control, acting from an open subset $ω\subset(-L,L)$. In accordance to the theory, our numerical simulations show: (1) that the method solves the elliptic equation with an acceptable approximation in the natural functional setting, (2) that the parabolic problem is null-controllable for $s>1/2$ and (3) that for $s\leq 1/2$ we only have approximate controllability.

math.AP↗

arXiv2017

Local elliptic regularity for the Dirichlet fractional Laplacian

We analyze the local elliptic regularity of weak solutions to the Dirichlet problem associated with the fractional Laplacian $(-Δ)^s$ on an arbitrary bounded open set $Ω\subset\mathbb{R}^N$. For $1<p<2$, we obtain regularity in the Besov space $B^{2s}_{p,2,\textrm{loc}}(Ω)$, while for $2\leq p<\infty$ we show that the solutions belong to $W^{2s,p}_{\textrm{loc}}(Ω)$. The key tool consists in analyzing carefully the elliptic equation satisfied by the solution locally, after cut-off, to later employ sharp regularity results in the whole space. We do it by two different methods. First working directly in the variational formulation of the elliptic problem and then employing the heat kernel representation of solutions.

math.AP↗

arXiv2017

Local regularity for fractional heat equations

We prove the maximal local regularity of weak solutions to the parabolic problem associated with the fractional Laplacian with homogeneous Dirichlet boundary conditions on an arbitrary bounded open set $Ω\subset\mathbb{R}^N$. Proofs combine classical abstract regularity results for parabolic equations with some new local regularity results for the associated elliptic problems.

math.AP↗

arXiv2016

Boundary null controllability for a heat equation with general dynamical boundary condition

Let $Ω\subset\mathbb R^N$ be a bounded open set with Lipschitz continuous boundary $Γ$. Let $γ>0$, $δ\ge 0$ be real numbers and $β$ a nonnegative measurable function in $L^\infty(Γ)$. Using some suitable Carleman estimates, we show that the linear heat equation $\partial_tu - γΔu = 0$ in $Ω\times(0,T)$ with the non-homogeneous general dynamic boundary conditions $\partial_tu_Γ -δΔ_Γu_Γ+ γ\partial_νu + βu_Γ = g$ on $Γ\times(0,T)$ is always null controllable from the boundary for every $T>0$ and initial data $(u_0,u_{Γ,0})\in L^2(Ω)\times L^2(Γ)$.

math.AP↗

arXiv2016

Null controllability for a heat equation with a singular inverse-square potential involving the distance to the boundary function

This article is devoted to the analysis of control properties for a heat equation with singular potential $μ/δ^2$, defined on a bounded $C^2$ domain $Ω\subset\mathbb{R}^N$, where $δ$ is the distance to the boundary function. More precisely, we show that for any $μ\leq 1/4$ the system is exactly null controllable using a distributed control located in any open subset of $Ω$, while for $μ>1/4$ there is no way of preventing the solutions of the equation from blowing-up. The result is obtained applying a new Carleman estimate.

math.AP↗