AN ISOPERIMETRIC INEQUALITY FOR THE FIRST STEKLOV-DIRICHLET LAPLACIAN EIGENVALUE OF CONVEX SETS WITH A SPHERICAL HOLE - Gavitone, Nunzia; Paoli, Gloria; Piscitelli, Gianpaolo; Sannipoli, Rossano (Pacific Journal of Mathematics, 2022)
We prove the existence of a maximum for the first Steklov-Dirichlet eigenvalue in the class of convex sets with a fixed spherical hole, under volume constraint. More precisely, if Omega = Omega(0) (B) over bar (R1), where B-R1 is the ball centered at the origin with radius R-1 > 0 and Omega(0) subset of R-n, n >= 2, is an open, bounded and convex set such that BR1 (sic) Omega(0), then the first Steklov-Dirichlet eigenvalue sigma(1)(Omega) has a maximum when R-1 and the measure of Omega are fixed. Moreover, if Omega(0) is contained in a suitable ball, we prove that the spherical shell is the maximum.
CRIS-Team WoS Importer:2023-03-03

Finite Element Approximation of the Hardy Constant - Della Pietra, Francesco; Fantuzzi, Giovanni; Ignat, Liviu I.; Masiello, Alba Lia; Paoli, Gloria; Zuazua Iriondo, Enrique (Journal of Convex Analysis, 2024)
We consider finite element approximations to the optimal constant for the Hardy inequality with exponent p = 2 in bounded domains of dimension n = 1 or n ≥ 3. For finite element spaces of piecewise linear and continuous functions on a mesh of size h, we prove that the approximate Hardy constant converges to the optimal Hardy constant at a rate proportional to 1/|log h|2. This result holds in dimension n = 1, in any dimension n ≥ 3 if the domain is the unit ball and the finite element discretization exploits the rotational symmetry of the problem, and in dimension n = 3 for general finite element discretizations of the unit ball. In the first two cases, our estimates show excellent quantitative agreement with values of the discrete Hardy constant obtained computationally.
CRIS-Team Scopus Importer:2024-04-12
Keywords: finite element method; Hardy constant; Hardy inequality

Finite element approximation of the Hardy constant - Della Pietra, Francesco; Fantuzzi, Giovanni; Ignat, Liviu I.; Masiello, Alba Lia; Paoli, Gloria; Zuazua Iriondo, Enrique (2024)
We consider finite element approximations to the optimal constant for the Hardy inequality with exponent p=2 in bounded domains of dimension n=1 or n≥ 3. For finite element spaces of piecewise linear and continuous functions on a mesh of size h, we prove that the approximate Hardy constant, S_h^n, converges to the optimal Hardy constant S^n no slower than O(1/h). We also show that the convergence is no faster than O(1/h^2) if n=1 or if n≥ 3, the domain is the unit ball, and the finite element discretization exploits the rotational symmetry of the problem. Our estimates are compared to exact values for S_h^n obtained computationally.<br
https://cris.fau.de/converis/publicweb/Publication/308553439

Sharp inequalities involving the Cheeger constant of planar convex sets - Ftouhi, Ilias; Masiello, Alba Lia; Paoli, Gloria (Esaim-Control Optimisation and Calculus of Variations, 2024)
We are interested in finding sharp bounds for the Cheeger constant h via different geometrical quantities, namely the area |€ ¢|, the perimeter P, the inradius r, the circumradius R, the minimal width w and the diameter d. We provide new sharp inequalities between these quantities for planar convex bodies and enounce new conjectures based on numerical simulations. In particular, we completely solve the Blaschke-Santaló diagrams describing all the possible inequalities involving the triplets (P, h, r), (d, h, r) and (R, h, r) and describe some parts of the boundaries of the diagrams of the triplets (w, h, d), (w, h, R), (w, h, P), (w, h, |€ ¢|), (R, h, d) and (w, h, r).
CRIS-Team Scopus Importer:2024-05-31
Keywords: Blaschke-Santaló diagrams; Cheeger constant; Convex sets; Sharp inequalities