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+<HTML>
+
+<!------------------------------------------------------------------------>
+
+<HEAD>
+<TITLE>Package sobolev</TITLE>
+</HEAD>
+<BODY >
+
+<!------------------------------------------------------------------------>
+
+<P>
+
+<H1 ALIGN="CENTER">
+Package <TT>sobolev</TT><BR>
+<SMALL>
+This is version 2.2, last revised 1998/02/04; documentation date 1998/02/06.
+</SMALL>
+</H1>
+
+<P ALIGN="CENTER">
+<STRONG>
+Author: <I>F. Bosisio</I><BR>
+<SMALL>
+E-mail: <A HREF="MAILTO:bosisio@mate.polimi.it">
+ <TT>bosisio@mate.polimi.it</TT>
+ </A><BR>
+Web page: <A HREF="http://ftp.mate.polimi.it/~bosisio/LaTeX">
+ <TT>http://ftp.mate.polimi.it/~bosisio/LaTeX</TT>
+ </A><BR>
+CTAN location: <A HREF="http://www.tex.ac.uk/tex-archive/macros/latex/contrib/supported/bosisio/">
+ <TT>CTAN:macros/latex/contrib/supported/bosisio/</TT>
+ </A>
+</SMALL>
+</STRONG>
+</P>
+
+<P ALIGN="LEFT"></P>
+
+<!------------------------------------------------------------------------>
+
+<H3>Abstract:</H3>
+<DIV>
+Documentation for the package <TT>sobolev</TT>.
+</DIV>
+
+<HR><BR>
+<!------------------------------------------------------------------------>
+
+<H2>Table of contents</H2>
+
+<P>
+<UL> <!-- Beginning of list -->
+ <LI><A HREF="#SECTION01">Introduction</A>
+ <LI><A HREF="#SECTION02">The options</A>
+ <LI><A HREF="#SECTION03">The commands</A>
+ <UL> <!-- Beginning of sublist -->
+ <LI><A HREF="#SECTION03a">The <TT>\H</TT> command</A>
+ <LI><A HREF="#SECTION03b">The <TT>\Hdiv</TT> command</A>
+ <LI><A HREF="#SECTION03c">The <TT>\L</TT> command</A>
+ <LI><A HREF="#SECTION03d">The <TT>\W</TT> command</A>
+ <LI><A HREF="#SECTION03e">The <TT>\D</TT> command</A>
+ <LI><A HREF="#SECTION03f">The <TT>\Norm</TT> command</A>
+ <LI><A HREF="#SECTION03g">The <TT>\SemiNorm</TT> command</A>
+ <LI><A HREF="#SECTION03h">The <TT>\Scalar</TT> command</A>
+ <LI><A HREF="#SECTION03i">The <TT>\Crochet</TT> command</A>
+ </UL> <!-- End of sublist -->
+</UL> <!-- End of list -->
+</P>
+
+<HR><BR>
+<!------------------------------------------------------------------------>
+
+<H2>
+<A NAME="SECTION01">Introduction</A>
+</H1>
+
+<P>
+This package provides some commands which are useful when dealing with
+Sobolev spaces and their relatives.
+
+<P>
+In particular some commands are redefined, so care should be taken,
+expecially when including this package in an already existent LaTeX file.
+
+<P>
+The redefined commands are <TT>\H</TT> and <TT>\L</TT>.
+The effect of ``<TT>\H</TT>'' (which is a type of accent) can now be achieved
+by the command ``<TT>\HAccent</TT>'', whilst the job of ``<TT>\L</TT>''
+(i.e. print an ``L'' with a superimposed bar) is now done by the command
+``<TT>\Lbar</TT>''.
+
+<H2>
+<A NAME="SECTION02">The options</A>
+</H2>
+
+<P>
+Two options are available at the moment: <TT>DivInBrackets</TT> and
+<TT>DivAsExponent</TT>.
+They only affect the output of the ``<TT>\Hdiv</TT>'' command.
+
+<P>
+The firt options (<TT>DivInBrackets</TT>, which is the default) makes
+<TT>\Hdiv</TT> behave like ``<TT>H(div;</TT>...<TT>)</TT>'', while the
+second one (<TT>DivAsExponent</TT>) makes <TT>\Hdiv</TT> expand to
+``<TT>H^{div}(</TT>...<TT>)</TT>''.
+
+<H1>
+<A NAME="SECTION03">The commands</A>
+</H1>
+
+<P>
+Most of the subsequent space-generating commands have mandatory arguments to
+indicate the type of the space.
+Often this argument consists of a single digit: in this case it is <EM>not</EM>
+necessary to enclose it in brackets, since in LaTeX the names of commands
+consists of letters only, and so a digit following it is certainly an argument.
+This saves a lot of typing and is the only reason that makes these commans
+useful (if you always had to type the brackets, then it would have been simpler
+to type the expansion of the command than the command itself !).
+In other words, you can think as if several commands exist (like <TT>\H</TT>,
+<TT>\H1</TT>, <TT>\H10</TT>, etc.), the ones with the digit beeing a sort of
+abbreviation for the general one.
+
+<H3>
+<A NAME="SECTION03a">The <TT>\H</TT> command</A>
+</H3>
+
+<P>
+The <TT>\H</TT> command is used to generate the symbol of sobolev spaces.
+It takes a mandatory argument, which is used as a superscript, and an optional
+argument, which is used as a subscript.
+
+<P>
+As explained above, if the mandatory argument is a digit, it need not be
+enclosed in brackets.
+Moreover, if the optional argument is the digit ``0'', it can be typed without
+the square brackets.
+
+<P>
+Here are some examples (whith the <TT>\DefaultSet</TT> set to its default
+value <TT>\Omega</TT>):
+<PRE>
+ \H2 ==> H^2(\Omega)
+ \H10 ==> H^1_0(\Omega)
+ \H1[\Gamma_D] ==> H^1_{\Gamma_D}(\Omega)
+ \H^{-1/2} ==> H^{-1/2}(\Omega)
+</PRE>
+
+<H3>
+<A NAME="SECTION03b">The <TT>\Hdiv</TT> command</A>
+</H3>
+
+<P>
+The <TT>\Hdiv</TT> command is used to generate the sobolev space
+called ``H div''.
+It takes only an optional argument, which is used as a subscript and which
+need not to be surrounded by the square brackets if it is the digit ``0''.
+
+<P>
+If the (default) option <TT>DivInBrackets</TT> is in effect, it differs from
+the command <TT>\H</TT> in that the word ``div'' is printed (in roman type)
+inside brackets, before the set.
+If, instead, the option <TT>DivAsExponent</TT> is active, then it is simply
+an abbreviation for <TT>\Hdiv</TT>.
+
+<P>
+Here are some examples:
+<PRE>
+ DivInBrackets DivAsExponent
+ \Hdiv ==> H(\mathrm{div};\Omega) H^{\mathrm{div}}(\Omega)
+ \Hdiv0 ==> H_0(\mathrm{div};\Omega) H^{\mathrm{div}}_0(\Omega)
+ \Hdiv[\Gamma_D] ==> H_{\Gamma_D}(\mathrm{div};\Omega) H^{\mathrm{div}}_{\Gamma_D}(\Omega)
+</PRE>
+
+<H3>
+<A NAME="SECTION03c">The <TT>\L</TT> command</A>
+</H3>
+
+<P>
+The <TT>\L</TT> command is used to generate the symbol of Lebesgue-measurable
+functions.
+It has one argument which is the exponent of the L-space.
+Again, if this argument is a digit (or a single symbol,
+like ``<TT>\infty</TT>'') the surrounding braces are optional.
+Like for the <TT>\H</TT> command, the output of <TT>\DefaultSet</TT>
+is appended.
+
+<P>
+Here are some examples:
+<PRE>
+ \L2 ==> L^2(\Omega)
+ \L10 ==> L^{10}(\Omega)
+ \L ==> L^\infty(\Omega)
+</PRE>
+
+<H3>
+<A NAME="SECTION03d">The <TT>\W</TT> command</A>
+</H3>
+
+<P>
+The <TT>\W</TT> command is completly analogous, except that it prints a ``W''
+insted af an ``L'' and that it has two argument, both printed as a supercript,
+separated by a comma.
+It is used for the generalized Sobolev spaces.
+
+<P>
+Here is an example of how it is used:
+<PRE>
+ \W{k}{p} ==> W^{k,p}(\Omega)
+ \W1 ==> W^{1,\infty}(\Omega)
+</PRE>
+
+<H3>
+<A NAME="SECTION03e">The <TT>\D</TT> command</A>
+</H3>
+
+<P>
+The <TT>\D</TT> command is used in the theory of distributions:
+it prints the space of distributions over the <TT>\DefaultSet</TT> if
+followed by a prime symbol, or its dual space, otherwise.
+<PRE>
+ \D ==> \mathcal{D}(\Omega)
+ \D' ==> \mathcal{D}'(\Omega)
+</PRE>
+
+<H3>
+<A NAME="SECTION03f">The <TT>\Norm</TT> command</A>
+</H3>
+
+<P>
+The <TT>\Norm</TT> command has a mandatory and an optional argument;
+it generates the norm of the mandatory argument, with the optional argument,
+if present, as a whole subscript, to denote the space within which the norm
+is taken.
+
+<P>
+Some examples:
+<PRE>
+ \Normf(x) ==> \left\Vert f(x)\right\Vert
+ \Normg[L^2] ==> \left\Vert g\right\Vert _{L^2}
+</PRE>
+
+<H3>
+<A NAME="SECTION03g">The <TT>\SemiNorm</TT> command</A>
+</H3>
+
+<P>
+The <TT>\SemiNorm</TT> command is completly analoguos, but generates the
+semi-norm instead of the norm.
+
+<P>
+Some examples:
+<PRE>
+ \SemiNorm{f(x)} ==> \left\Vert f(x)\right\Vert
+ \SemiNormg[H^1] ==> \left\vert g\right\vert _{H^1}
+</PRE>
+
+<H3>
+<A NAME="SECTION03h">The <TT>\Scalar</TT> command</A>
+</H3>
+
+<P>
+The <TT>\Scalar</TT> command has two arguments; a third optional argument
+(which is used as a whole subscript) may follow inside square brackets.
+The output consists of the two arguments separated by a comma and enclosed
+in a pair of adjustable-size brackets, with the optional argument placed as
+a subscript (to denote the space inside which the scalr product is taken).
+
+<P>
+Some examples:
+<PRE>
+ \Scalar{f}{g} ==> \left(f,g\right)
+ \Scalar{u}{v}[L^2] ==> left(u,v\right)_{L^2}
+</PRE>
+
+<H3>
+<A NAME="SECTION03i">The <TT>\Crochet</TT> command</A>
+</H3>
+
+<P>
+The <TT>\Crochet</TT> command has two arguments; a third optional argument
+(which is used as a whole subscript) may follow inside square brackets.
+The output consists of the two arguments separated by a comma and enclosed
+in a pair of adjustable-size angular-parenthesys, with the optional argument
+placed as a subscript (to denote the space inside which the duality is taken).
+
+<P>
+Some examples:
+<PRE>
+ \Crochet{f}{g} ==> \left\langle f,g\right\rangle
+ \Crochet{u}{v}[D] ==> \left\langle u,v\right\rangle_D
+</PRE>
+
+<HR><BR>
+<!------------------------------------------------------------------------>
+
+<ADDRESS>
+<I>Francesco Bosisio</I><BR>
+E-mail: <A HREF="MAILTO:bosisio@mate.polimi.it">
+ <TT>bosisio@mate.polimi.it</TT>
+ </A><BR>
+Web page: <A HREF="http://ftp.mate.polimi.it/~bosisio/LaTeX">
+ <TT>http://ftp.mate.polimi.it/~bosisio/LaTeX</TT>
+ </A>
+</ADDRESS>
+
+<!------------------------------------------------------------------------>
+
+</BODY>
+</HTML>