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European Mathematical Society Publishing House
2024-03-28 12:33:49
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https://www.ems-ph.org/meta/jmeta-stream.php?jrn=RLM&vol=27&iss=3&update_since=2024-03-28
Rendiconti Lincei - Matematica e Applicazioni
Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur.
RLM
1120-6330
1720-0768
General
10.4171/RLM
http://www.ems-ph.org/doi/10.4171/RLM
subscribers
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society (from 2006)
27
2016
3
Peano on definition of surface area
Gabriele
Greco
Università di Trento, POVO (TRENTO), ITALY
Sonia
Mazzucchi
Università di Trento, POVO (TRENTO), ITALY
Enrico
Pagani
Università di Trento, POVO (TRENTO), ITALY
Grassmann–Peano geometric-vector calculus, area of polygons, bi-vectorial definition of area, Peano’s surface area, Lebesgue’s surface area
In this paper we investigate the evolution of the concept of surface area in Peano’s mathematical research, taking into account the main role played by Grassmann’s geometric-vector calculus and Peano’s theory on derivative of measures. Geometric (in Applicazioni geometriche, 1887) and bi-vector (in Calcolo geometrico, 1888) Peano’s approaches to surface area, culminating into the celebrated Peano’s paper Sulla definizione dell’area d’una superficie, presented by Casorati for publication on Rendiconti dell’Accademia dei Lincei in 1890 and re-proposed in Peano’s textbook Lezioni di analisi infinitesimale (1893), mark the development of this topic during the first half of the last century. Moreover we will present some remarkable contributions on surface area that are inspired and/or closely related to Peano’s definition.
History and biography
Linear and multilinear algebra; matrix theory
Measure and integration
251
286
10.4171/RLM/734
http://www.ems-ph.org/doi/10.4171/RLM/734
Duality for $A_\infty$ weights on the real line
Luigi
D'Onofrio
Università degli Studi di Napoli Parthenope, NAPOLI, ITALY
Arturo
Popoli
Università degli Studi di Napoli Federico II, NAPOLI, ITALY
Roberta
Schiattarella
Università degli Studi di Napoli Federico II, NAPOLI, ITALY
Muckenhoupt weights, Gehring classes
Under the same bounds on $G_q$-constants and $A_p$-constants, the optimal exponents for sharp inclusions between Gehring $G_q$-class of weights and Muckenhoupt $A_p$-class ($1 < p, q < \infty$) are Hölder conjugate, if $p$ and $q$ are conjugate. This is a consequence of a representation theorem of $A_\infty$ weights in terms of $W^{1,r}$-biSobolev maps and a duality result between $G_q$ and $A_p$ classes in dimension one. We prove also that sharp a priori bounds on constants correspond under the Hölder conjugate mapping $\phi(t)= \frac{t}{t-1}$.
Functional analysis
Real functions
287
308
10.4171/RLM/735
http://www.ems-ph.org/doi/10.4171/RLM/735
On maximal and potential operators with rough kernels in variable exponent spaces
Humberto
Rafeiro
Pontificia Universidad Javeriana, BOGOTA, COLOMBIA
Stefan
Samko
University of Algarve, FARO, PORTUGAL
Rough operators, variable exponent Lebesgue spaces, variable exponent Morrey spaces
In the framework of variable exponent Lebesgue and Morrey spaces we prove some boundedness results for operators with rough kernels, such as the maximal operator, fractional maximal operator, sharp maximal operators and fractional operators. The approach is based on some pointwise estimates.
Fourier analysis
Functional analysis
309
325
10.4171/RLM/736
http://www.ems-ph.org/doi/10.4171/RLM/736
A sharp quantitative estimate for the surface areas of convex sets in $\mathbb R^3$
Menita
Carozza
Università del Sannio, BENVENTO, ITALY
Flavia
Giannetti
Università degli Studi di Napoli Federico II, NAPOLI, ITALY
Francesco
Leonetti
Università degli Studi dell'Aquila, L'AQUILA, ITALY
Antonia
Passarelli di Napoli
Università degli Studi di Napoli Federico II, NAPOLI, ITALY
Convex sets, surface areas, Hausdorff distance
Let $E \subset B \subset \mathbb R^3$ be closed, bounded, convex sets. It is known that the monotonicity of the surface areas holds, i.e. $\mathcal{H}^{2}(\partial E) \leqslant \mathcal{H}^{2}(\partial B)$. Here we give a quantitative estimate of the difference of the surface areas from below depending on the Hausdorff distance between $E$ and $B$. Moreover, we construct an example which shows the sharpness of our result.
Convex and discrete geometry
327
333
10.4171/RLM/737
http://www.ems-ph.org/doi/10.4171/RLM/737
Continuity estimates for $p$-Laplace type operators in Orlicz–Zygmund spaces
Fernando
Farroni
Università Telematica Pegaso, NAPOLI, ITALY
Nonlinear elliptic equations, continuity estimates
We study the Dirichlet problem $$\mathrm {div} \: \mathcal A (x, \nabla u) = \mathrm {div} \: f \;\; \mathrm {in} \; \Omega,$$ $$u = 0 \;\; \mathrm {on}\; \partial \Omega,$$ in a bounded Lipschitz domain $\Omega\subset \mathbb R^N$, with $N \ geq 2$. The vector field $\mathcal A \colon \Omega \times \mathbb R^N \rightarrow \mathbb R^N$ satisfies the typical growth and coercivity conditions of the $p$-Laplacian type operator with $p>1$. We prove existence and uniqueness results in the case the vector field $f$ belongs to the Orlicz–Zygmund space $\mathcal L^q \: \mathrm {log}^{–\alpha} \mathcal L (\mathrm {log} \: \mathrm {log} \: \mathcal L)^{– \beta} (\Omega, \mathbb R^N), q = \frac {p} {p-1}, \alpha > 0$ and $\beta \in \mathbb R$ or $\alpha = 0$ and $\beta > 0$. In particular, the gradient of the solution belongs to $\mathcal L^q \: \mathrm {log}^{–\alpha} \: \mathcal L (\mathrm {log} \: \mathrm {log} \: \mathcal L)^{– \beta} (\Omega, \mathbb R^N)$. Further, we provide estimates implying the continuity of the operator which carries any given $f$ into the gradient field $\nabla u$ of the solution.
Partial differential equations
335
354
10.4171/RLM/738
http://www.ems-ph.org/doi/10.4171/RLM/738
Hölder continuity up to the boundary for a class of fractional obstacle problems
Janne
Korvenpää
Aalto University, AALTO, FINLAND
Tuomo
Kuusi
Aalto University, AALTO, FINLAND
Giampiero
Palatucci
Università degli Studi di Parma, PARMA, ITALY
Quasilinear nonlocal operators, fractional Sobolev spaces, nonlocal tail, Caccioppoli estimates, obstacle problem
We deal with the obstacle problem for a class of nonlinear integro-differential operators, whose model is the fractional $p$-Laplacian with measurable coeffcients. In accordance with well-known results for the analog for the pure fractional Laplacian operator, the corresponding solutions inherit regularity properties from the obstacle, both in the case of boundedness, continuity, and Hölder continuity, up to the boundary.
Partial differential equations
Operator theory
355
367
10.4171/RLM/739
http://www.ems-ph.org/doi/10.4171/RLM/739