IRIS Univ. Trentohttps://iris.unitn.itIl sistema di repository digitale IRIS acquisisce, archivia, indicizza, conserva e rende accessibili prodotti digitali della ricerca.Tue, 18 Jan 2022 01:59:38 GMT2022-01-18T01:59:38Z101641On the evolution of ferromagnetic mediahttp://hdl.handle.net/11572/53594Titolo: On the evolution of ferromagnetic media
Sun, 01 Jan 1984 00:00:00 GMThttp://hdl.handle.net/11572/535941984-01-01T00:00:00ZDiscontinuous hysteresis and P.D.E.shttp://hdl.handle.net/11572/78029Titolo: Discontinuous hysteresis and P.D.E.s
Abstract: (G. Dal Maso ed.) Birkhaeuser, Basel
Sun, 01 Jan 2006 00:00:00 GMThttp://hdl.handle.net/11572/780292006-01-01T00:00:00ZErrata-corrige: 'Pattern evolution'http://hdl.handle.net/11572/53231Titolo: Errata-corrige: 'Pattern evolution'
Tue, 01 Jan 1991 00:00:00 GMThttp://hdl.handle.net/11572/532311991-01-01T00:00:00ZThe Preisach model and partial differential equationshttp://hdl.handle.net/11572/53473Titolo: The Preisach model and partial differential equations
Sat, 01 Jan 2005 00:00:00 GMThttp://hdl.handle.net/11572/534732005-01-01T00:00:00ZMotion by mean curvature and nucleation.http://hdl.handle.net/11572/53317Titolo: Motion by mean curvature and nucleation.
Wed, 01 Jan 1997 00:00:00 GMThttp://hdl.handle.net/11572/533171997-01-01T00:00:00ZCurvature flows and related topicshttp://hdl.handle.net/11572/54849Titolo: Curvature flows and related topics
Sun, 01 Jan 1995 00:00:00 GMThttp://hdl.handle.net/11572/548491995-01-01T00:00:00ZQuasilinear parabolic P.D.E.s with discontinuous hysteresishttp://hdl.handle.net/11572/8607Titolo: Quasilinear parabolic P.D.E.s with discontinuous hysteresis
Sun, 01 Jan 2006 00:00:00 GMThttp://hdl.handle.net/11572/86072006-01-01T00:00:00ZA free boundary problem of biological interesthttp://hdl.handle.net/11572/47923Titolo: A free boundary problem of biological interest
Sun, 01 Jan 1989 00:00:00 GMThttp://hdl.handle.net/11572/479231989-01-01T00:00:00ZP.D.E.s with hysteresis operatorshttp://hdl.handle.net/11572/47833Titolo: P.D.E.s with hysteresis operators
Sat, 01 Jan 2000 00:00:00 GMThttp://hdl.handle.net/11572/478332000-01-01T00:00:00ZTowards a two-scale calculushttp://hdl.handle.net/11572/71993Titolo: Towards a two-scale calculus
Abstract: \magnification=1080
\centerline{\bf Towards a Two-Scale Calculus}
\medskip
\noindent\centerline{ESAIM Control Optim. Calc. Var. 12 (2006), 371--397}
\medskip
\noindent\centerline{A. Visintin}
\medskip
\noindent{\bf A.M.S.\ Subject Classification (2000):}
35B27, 35J20, 74Q, 78M40.
\medskip
\noindent{\bf Keywords:}
Two-scale convergence,
Two-scale decomposition,
Sobolev spaces,
Homogenization.
\medskip
\medskip
\noindent\centerline{\bf Abstract}
\medskip
\noindent
More than twenty years ago first Nguetseng and then Allaire introduced the concept of {\sl two-scale
convergence,\/} as a tool for homogenization.
Two-scale convergence also offers a new point of view for {\sl multiscaling\/}
--- a classical topic of applied mathematics that plays a fundamental role in physics and
engineering and nowadays is attracting a renewed interest among mathematicians.
Two-scale convergence may be regarded as a mathematical object
in itself, and may also be equipped with a calculus of its own.
The purpose of this work is to illustrate some possibilities in this direction.
In Sect.\ 1 we reformulate this notion via the procedure of {\sl periodic unfolding.\/}
We represent weak and strong two-scale convergence in
$L^p({\bf R}^N\!\times\! Y)$ for any $p\in[1,+\infty]$,
and in the Fr\'echet space $C^0({\bf R}^N \!\times\! Y)$ (here $Y= [0,1[^N$).
In Sect.\ 2 we derive some properties of two-scale convergence.
Some of these results are already known.
Here we organize their derivation by using the tool of two-scale decomposition,
and also deal with two-scale convergence in $C^0$ and in ${\cal D}'$,
with the Fourier transform, and with two-scale convolution.
In Sect.\ 3 we study weak and strong two-scale compactness.
We prove a two-scale version of a result of Chacon, known as {\sl the biting lemma.\/}
We characterize strong two-scale compactness in $L^p$ and in $C^0$,
generalizing classic criteria of Riesz and Ascoli-Arzel\`a.
Along the same lines, we also extend Vitali's convergence theorem.
Differential properties of two-scale convergence are the main concern of this paper.
The two-scale limit of sequences bounded in $H^1(\Omega)$ had already been studied
by Nguetseng and then Allaire; the present analysis moves towards a different direction.
In Sect.\ 4 we show that it is possible to express the gradient of the two-scale limit without
the need of evaluating the limit itself, via what we name {\sl approximate two-scale
derivatives.\/} More specifically, we define an approximate gradient $\Lambda_\varepsilon$ such
that, denoting the weak two-scale limit by $\lim_{\varepsilon\to 0}{}^{(2)}$,
$$
\lim_{\varepsilon\to 0}{}^{(2)} \Lambda_\varepsilon u_\varepsilon =
(\nabla_x, \nabla_y) \lim_{\varepsilon\to 0}{}^{(2)} u_\varepsilon
\qquad\hbox{ in } L^p({\bf R}^N \!\times\! Y)^{2N}.
$$
By means of these two-scale approximate derivatives, in Sect.\ 5 we define two-scale
convergence in spaces of differentiable functions: $W^{m,p}$, $C^m$, $C^{m,\lambda}$, ${\cal D}$.
We then derive two-scale versions of the Rellich compactness theorem and
of the Sobolev and Morrey imbedding theorems.
Indeed several classic results have a two-scale counterpart, which does not concern single
functions but sequences of functions.
\end
Sun, 01 Jan 2006 00:00:00 GMThttp://hdl.handle.net/11572/719932006-01-01T00:00:00Z