In this article, we construct simple and convergent finite element methods for linear and nonlinear elliptic differential equations in non-divergence form with discontinuous coefficients. The methods ...are motivated by discrete Miranda–Talenti estimates, which relate the H2 semi-norm of piecewise polynomials with the L2 norm of its Laplacian on convex domains. We develop a stability and convergence theory of the proposed methods, and back up the theory with numerical experiments.
In this paper we consider PDE’s problems involving the anisotropic Laplacian operator, with Robin boundary conditions. By means of Talenti techniques, widely used in the last decades, we prove a ...comparison result between the solutions of the above-mentioned problems and the solutions of the symmetrized ones. As a consequence of these results, a Bossel–Daners type inequality can be shown in dimension 2.
Nationality swapping in sports is commonly assumed to be a rapidly expanding practice that is indicative of the marketization of citizenship. Sports are said to have become wholesale markets in which ...talent is being traded for citizenship. In this article, we seek to empirically explore such claims by analysing 167 athletes who have competed for two different countries in the Summer Olympic Games. It seems that most switches occurred after the 1990s. Then, following a citizenship as a claims-making approach, we introduce the work of Bourdieu so as to connect citizenship as both legal status and practice with normative claims. The analysis reveals that the practice of nationality switching is shaped by structural conditions of the Olympic field. First, a complex realm of citizenship laws and regulations produces conditions under which athletes make legitimate claims to citizenship. Second, through a mechanism of reverberative causation, prior migrations are often echoed in contemporary nationality swapping . Only a limited number of athletes acquired citizenship via the explicit market principle we call jus talenti. Claiming that instrumental nationality swapping is indicative of the marketization of citizenship obscures the complex interplay between structures of and practices within the Olympic field.
In this paper, we present two type of contributions to the study of two-phases problems. In such problems, the main focus is on optimising a diffusion function $ a $ under $ L^\infty $ and $ L^1 $ ...constraints, this function $ a $ appearing in a diffusive term of the form $ -{{\nabla}} \cdot(a{{\nabla}}) $ in the model, in order to maximise a certain criterion. We provide a parabolic Talenti inequality and a partial bang-bang property in radial geometries for a general class of elliptic optimisation problems: namely, if a radial solution exists, then it must saturate, at almost every point, the $ L^\infty $ constraints defining the admissible class. This is done using an oscillatory method.
Let
Ω
⊂
R
2
be an open, bounded and Lipschitz set. We consider the torsion problem for the Laplace operator associated to
Ω
with Robin boundary conditions. In this setting, we study the equality case ...in the Talenti-type comparison, proved in Alvino et al. (Commun Pure Appl Math 76:585–603, 2023).. We prove that the equality is achieved only if
Ω
is a disk and the torsion function
u
is radial.
We investigate the Henon type equation involving the critical Sobolev exponent with Dirichret boundary condition $$ - \Delta u = \lambda \Psi u + | x |^\alpha u^{2^*-1} $$ in $\Omega$ included in a ...unit ball, under several conditions. Here, $\Psi$ is a non-trivial given function with $0 \leq \Psi \leq 1$ which may vanish on $\partial \Omega$. Let $\lambda_1$ be the first eigenvalue of the Dirichret eigenvalue problem $-\Delta \phi = \lambda \Psi \phi$ in $\Omega$. We show that if the dimension $N \geq 4$ and $0 < \lambda < \lambda_1$, there exists a positive solution for small $\alpha > 0$. Our methods include the mountain pass theorem and the Talenti function.
Talenti inequalities are a central feature in the qualitative analysis of PDE constrained optimal control as well as in calculus of variations. The classical parabolic Talenti inequality states that ...if we consider the parabolic equation
∂
u
∂
t
-
Δ
u
=
f
=
f
(
t
,
x
)
then, replacing, for any time
t
,
f
(
t
,
·
)
with its Schwarz rearrangement
f
#
(
t
,
·
)
increases the concentration of the solution in the following sense: letting
v
be the solution of
∂
v
∂
t
-
Δ
v
=
f
#
in the ball, then the solution
u
is less concentrated than
v
. This property can be rephrased in terms of the existence of a maximal element for a certain order relationship. It is natural to try and rearrange the source term not only in space but also in time, and thus to investigate the existence of such a maximal element when we rearrange the function with respect to the two variables. In the present paper we prove that this is not possible.
This paper presents a nonconforming finite element scheme for the planar biharmonic equation, which applies piecewise cubic polynomials (
P
3
) and possesses
O
(
h
2
)
convergence rate for smooth ...solutions in the energy norm on general shape-regular triangulations. Both Dirichlet and Navier type boundary value problems are studied. The basis for the scheme is a piecewise cubic polynomial space, which can approximate the
H
4
functions with
O
(
h
2
)
accuracy in the broken
H
2
norm. Besides, a discrete strengthened Miranda-Talenti estimate (∇
h
2
·, ∇
h
2
·) = (∆
h
·, ∆
h
·), which is usually not true for nonconforming finite element spaces, is proved. The finite element space does not correspond to a finite element defined with Ciarlet’s triple; however, it admits a set of locally supported basis functions and can thus be implemented by the usual routine. The notion of the finite element Stokes complex plays an important role in the analysis as well as the construction of the basis functions.
23 edizioni alle spalle, circa 3200 ragazze e ragazzi delle scuole superiori coinvolti; 487 lavori invitati all’esposizione annuale; la maggior parte di questi inviati a rappresentare l’Italia nei ...grandi appuntamenti internazionali per gli studenti eccellenti: ecco i dati più significativi de «I giovani e le scienze», l’appuntamento della Fast che seleziona i talenti per il Concorso dell’Unione europea dei giovani scienziati e per gli altri prestigiosi eventi mondiali. Le finalità di questa azione, voluta dalle istituzioni comunitarie quali Parlamento, Commissione e Consiglio, sono l’individuazione dei migliori studenti, il loro avvicinamento alla scienza e alla tecnologia, la valorizzazione delle eccellenze. L’evento ha regole generali uguali in tutti i paesi comunitari.
PDF
