Dirichlet form and Talk:To Do List: Difference between pages

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The porous medium equation is technically a quasi-linear equation. There is now a page about [[semilinear equations]] to clarify the issue.
\newcommand{\dd}{\mathrm{d}}
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A Dirichlet form refers to a quadratic functional defined by an integral of the form
I see, by the way, how come Navier-Stokes is semilinear then? I was sure it was quasi until now.
\[ \iint_{\R^n \times \R^n} (u(y)-u(x))^2 k(x,y)\, \dd x \dd y, \]
for some nonnegative kernel $K$.


If the kernel $K$ satisfies the bound $K(x,y) \leq \Lambda |x-y|^{-n-s}$, then the quadratic form is bounded in $\dot H^{s/2}$. If moreover, $\lambda |x-y|^{-n-s} \leq K(x,y)$, then the form is comparable to the norm in $\dot H^{s/2}$ squared.
---> It is the heat equation plus a first order nonlinear term.


Dirichlet forms are natural generalizations to fractional order of the Dirichlet integrals
== Main page vs no Main page? ==
\[ \int a_{ij}(x) \partial_i u \partial_j u \dd x, \]
where $a_{ij}$ is elliptic.


The Euler-Lagrange equation of a Dirichlet form is a fractional order version of elliptic equations in divergence form. They are studied using variational methods and they are expected to satisfy similar properties <ref name="BBCK"/><ref name="K"/><ref name="CCV"/>.
What is currently the "Main page" should become the  "Community portal", and we can use the Main page as the starting page. What do you guys think? -Nestor.
:Right now I don't have an opinion either for or against. ([[User:Luis|Luis]] 11:03, 5 June 2011 (CDT))


== References ==
I also started playing around with the use of categories (see the "Quasilinear equations" category)
(There should be a lot more references here)
{{reflist|refs=
<ref name="CCV">{{Citation | last1=Caffarelli | first1=Luis | last2=Chan | first2=Chi Hin | last3=Vasseur | first3=Alexis | title= | doi=10.1090/S0894-0347-2011-00698-X | year=2011 | journal=[[Journal of the American Mathematical Society]] | issn=0894-0347 | issue=24 | pages=849–869}}</ref>
<ref name="BBCK">{{Citation | last1=Barlow | first1=Martin T. | last2=Bass | first2=Richard F. | last3=Chen | first3=Zhen-Qing | last4=Kassmann | first4=Moritz | title=Non-local Dirichlet forms and symmetric jump processes | url=http://dx.doi.org/10.1090/S0002-9947-08-04544-3 | doi=10.1090/S0002-9947-08-04544-3 | year=2009 | journal=[[Transactions of the American Mathematical Society]] | issn=0002-9947 | volume=361 | issue=4 | pages=1963–1999}}</ref>
<ref name="K">{{Citation | last1=Kassmann | first1=Moritz | title=A priori estimates for integro-differential operators with measurable kernels | url=http://dx.doi.org/10.1007/s00526-008-0173-6 | doi=10.1007/s00526-008-0173-6 | year=2009 | journal=Calculus of Variations and Partial Differential Equations | issn=0944-2669 | volume=34 | issue=1 | pages=1–21}}</ref>
}}


== Let us try to avoid a big bias ==


{{stub}}
I was talking to Russell today that we have to make an effort not to have a strong bias towards Caffarelli related stuff. Otherwise, the purpose of the wiki will fail. ([[User:Luis|Luis]] 00:58, 8 June 2011 (CDT))
 
Indeed! I think so far we don't have to worry about it since we are just getting started with the pages (naturally we will write first stuff we know best). I hope it won't become a problem. Also, having Moritz will help a lot too. ([[User:Nestor|Nestor]]) (5:42 pm US Eastern Time, 8 June 2011)

Revision as of 16:42, 8 June 2011

The porous medium equation is technically a quasi-linear equation. There is now a page about semilinear equations to clarify the issue.

I see, by the way, how come Navier-Stokes is semilinear then? I was sure it was quasi until now.

---> It is the heat equation plus a first order nonlinear term.

Main page vs no Main page?

What is currently the "Main page" should become the "Community portal", and we can use the Main page as the starting page. What do you guys think? -Nestor.

Right now I don't have an opinion either for or against. (Luis 11:03, 5 June 2011 (CDT))

I also started playing around with the use of categories (see the "Quasilinear equations" category)

Let us try to avoid a big bias

I was talking to Russell today that we have to make an effort not to have a strong bias towards Caffarelli related stuff. Otherwise, the purpose of the wiki will fail. (Luis 00:58, 8 June 2011 (CDT))

Indeed! I think so far we don't have to worry about it since we are just getting started with the pages (naturally we will write first stuff we know best). I hope it won't become a problem. Also, having Moritz will help a lot too. (Nestor) (5:42 pm US Eastern Time, 8 June 2011)

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