From: Art Werschulz Subject: Re: Sobolev and Hilbert Spaces? Date: 13 Jan 2000 14:10:21 -0500 Newsgroups: sci.math.research Summary: What are they? Hi. Steve McGahey writes: > What exactly is the point/meaning/use of: > > Sobolev space > as well as > Hilbert Space? This will be a really brief introduction. (I hope that there are no typos in what follows.) We'll start with Hilbert space first. A Hilbert space is a complete inner-product space. (An inner product is a generalization of the usual dot product of vectors. A complete space is one in which all Cauchy sequences have a limit.) One particular example of a Hilbert space is the space of all functions whose square has a finite integral (in the sense of Lebesgue). This is called $L_2$ (in the notation of \TeX). The inner product of functions $u$ and $v$ in this case is given by $\int_0^1 uv$. Another example is the space of those functions, all of whose derivatives of order up through $r$, belong to $L_2$. This space is called $H^r$; it is a Sobolev space of order $r$. Here, the inner product is $$\sum_{i=0}^r \int_0^1 u^{(i)} v^{(i)}.$$ Now consider the following problem: given a function $f$ on the interval $(0,1)$, find a function $u$ on $[0,1]$ satisfying the differential equation $$-u'' + u = f$$ in $(0,1)$, with boundary conditions $u'(0) = u'(1) = 0$. This is called the classical formulation of this boundary value problem. Using integration by parts, it is easy to see that we can characterize the solution $u$ as the unique element of $H^1$ such that $$\int_0^1 (u'v' + uv) = \int_0^1 fv$$ for all functions $v\in H^1$. You'll now recognize the left-hand side of this equation as the $H^1$ inner product of $u$ and $v$. Alternatively, $u$ minimizes the functional $$J(v) = \int_0^1 [(v')^2 + v^2 - 2fv]$$ over the space $H^1$. These latter characterizations are often referred to as the weak form and the variational form of the original problem. They are the mathematical starting point for finite element methods, which are a powerful technique for solving elliptic problems. Now, why the Hilbert space stuff? Basically, we want to know when solutions to the weak or variational form exist. The short answer is that we need the completeness property of $H^1$ to be able to infer that this existence holds, e.g., for any $f\in L_2$. Why the need for Sobolev spaces? First of all, they provide us with Hilbert spaces that are useful for presenting the weak/variational forms of elliptic problems. Another reason: it is possible to show that "shift theorems" hold. That is, if $f$ belongs to $H^r$, then $u$ belongs to $H^{r+2}$. This says that the smoother $f$ is, the smoother $u$ will be, at least in the $L_2$ sense. There's another result, called the Sobolev embedding theorem, which says if a function belongs to Sobolev spaces of sufficient smoothness, then it will have a certain number of continuous derivatives. This allows us to say that the variational/weak solution of a problem is also a solution of the classical form. Again, I have gone through this *very* briefly, and skipped many details. There are many books on the topic. My favorite is the book by Ciarlet: @Book{Ciarlet:Elliptic, author = {P. G. Ciarlet}, title = {The Finite Element Method For Elliptic Problems}, publisher = {North-Holand}, year = 1978, address = {New York} } I hope this helps. -- Art Werschulz (8-{)} "Metaphors be with you." -- bumper sticker GCS/M (GAT): d? -p+ c++ l u+(-) e--- m* s n+ h f g+ w+ t++ r- y? Internet: agw@cs.columbia.eduWWW ATTnet: Columbia U. (212) 939-7061, Fordham U. (212) 636-6325 ============================================================================== From: Stephen Montgomery-Smith Subject: Re: Sobolev and Hilbert Spaces? Date: Fri, 14 Jan 2000 04:29:19 GMT Newsgroups: sci.math.research Steve McGahey wrote: > > Hi all, > > I apologise if this is felt to be off topic, but I am finding maths > texts just plain unfriendly of late, and I think that my research (into > Computer Aided Modelling wrt distributed parameter systems (PDEs)) would > benefit from the following definitions... > > What exactly is the point/meaning/use of: > > Sobolev space > as well as > Hilbert Space? > > As well as that nice fuzzy feeling you get deep down inside, I'd also > like you to know that any answers you might give are going 100% towards > helping me and other poor, currently under-mathematical, frustrated, > first-year PhD chemical engineering students around the globe. > If I may be allowed to speak on subjects for which I am not an expert: Sobolev spaces play a big role in proving existence or uniqueness results for solving differential equations. Hilbert spaces come up all over the place. For example, they are very important in control theory. (I know that one expert in this field, N Young, has a book on Hilbert Spaces.) Another place is Quantum Mechanics, which studies operators on Hilbert Spaces. I think that these are particularly important in chemical engineering. You want to solve equations for your chemical reaction (which might be coupled with equations for fluid dynamics). Knowing that the solution exists in a certain Sobolev space will tell you how wild it will get, and give stability results. You also want to control your reaction, so control theory is very useful. -- Stephen Montgomery-Smith stephen@math.missouri.edu 307 Math Science Building stephen@showme.missouri.edu Department of Mathematics stephen@missouri.edu University of Missouri-Columbia Columbia, MO 65211 USA Phone (573) 882 4540 Fax (573) 882 1869 http://www.math.missouri.edu/~stephen