|
|
| Line 1: |
Line 1: |
| In [[mathematics]], the '''Schauder estimates''' are a collection of results due to {{harvs|txt|first=Juliusz|last= Schauder|year=1934|year2=1937|author-link=Juliusz Schauder}} concerning the regularity of solutions to linear, uniformly [[Elliptic operator|elliptic]] [[partial differential equation]]s. The estimates say that when the equation has appropriately [[differentiable|smooth]] terms and appropriately smooth solutions, then the [[Hölder condition|Hölder norm]] of the solution can be controlled in terms of the Hölder norms for the coefficient and source terms. Since these estimates do not assume the existence of the solution, they are called [[a priori estimate]]s.
| | It involves expertise and knowledge of various tools and technologies used for creating websites. Offshore expert Word - Press developers high level of interactivity, accessibility, functionality and usability of our website can add custom online to using. Your parishioners and certainly interested audience can come in to you for further information from the group and sometimes even approaching happenings and systems with the church. Keep reading for some great Word - Press ideas you can start using today. You can customize the appearance with PSD to Word - Press conversion ''. <br><br>Choosing what kind of links you'll be using is a ctitical aspect of any linkwheel strategy, especially since there are several different types of links that are assessed by search engines. But as expected the level of support you get with them can be hit or miss based on the developer's free time and desire. This plugin allows a blogger get more Facebook fans on the related fan page. From my very own experiences, I will let you know why you should choose WPZOOM Live journal templates. Moreover, many Word - Press themes need to be purchased and designing your own WP site can be boring. <br><br>You can down load it here at this link: and utilize your FTP software program to upload it to your Word - Press Plugin folder. Word - Press has ensured the users of this open source blogging platform do not have to troubleshoot on their own, or seek outside help. Those who cannot conceive with donor eggs due to some problems can also opt for surrogacy option using the services of surrogate mother. To turn the Word - Press Plugin on, click Activate on the far right side of the list. So, if you are looking online to hire dedicated Wordpress developers, India PHP Expert can give a hand you in each and every best possible way. <br><br>A built-in widget which allows you to embed quickly video from popular websites. Quttera - Quttera describes itself as a 'Saa - S [Software as a Service] web-malware monitoring and alerting solution for websites of any size and complexity. A higher percentage of women are marrying at older ages,many are delaying childbearing until their careers are established, the divorce rate is high and many couples remarry and desire their own children. IVF ,fertility,infertility expert,surrogacy specialist in India at Rotundaivf. If your blog employs the permalink function, This gives your SEO efforts a boost, and your visitors will know firsthand what's in the post when seeing the URL. <br><br>He loves sharing information regarding wordpress, Majento, Drupal and Joomla development tips & tricks. I'm a large fan of using Word - Press to create pretty much any sort of web page. For those who have any kind of issues concerning exactly where and also how you can utilize [http://nkle.in/wordpress_backup_49583 backup plugin], you are able to e-mail us from the web page. In simple words, this step can be interpreted as the planning phase of entire PSD to wordpress conversion process. Extra investment in Drupal must be bypassed as value for money is what Drupal provides. The 2010 voting took place from July 7 through August 31, 2010. |
| | |
| There is both an ''interior'' result, giving a Hölder condition for the solution in interior domains away from the boundary, and a ''boundary'' result, giving the Hölder condition for the solution in the entire domain. The former bound depends only on the spatial dimension, the equation, and the distance to the boundary; the latter depends on the smoothness of the boundary as well.
| |
| | |
| The Schauder estimates are a necessary precondition to using the [[method of continuity]] to proving the existence and regularity of solutions to the [[Dirichlet problem]] for elliptic PDEs. This result says that when the coefficients of the equation and the nature of the boundary conditions are sufficiently smooth, there is a smooth classical solution to the PDE.
| |
| | |
| == Notation ==
| |
| The Schauder estimates are given in terms of weighted Hölder norms; the notation will follow that given in the text of {{harvs|txt|first=D.|last=Gilbarg|last2=Trudinger|first2=Neil|authorlink2=Neil Trudinger|year=1983}}.
| |
| | |
| The supremum norm of a continuous function <math>f \in C(\Omega)</math> is given by
| |
| :<math>|f|_{0;\Omega} = \sup_{x\in\Omega} |f(x)|</math> | |
| For a function which is Hölder continuous with exponent <math>\alpha</math>, that is to say, <math>f \in C^\alpha (\Omega)</math> the usual [[Hölder condition|Hölder seminorm]] is given by
| |
| | |
| :<math>[f]_{0,\alpha;\Omega} = \sup_{x,y\in \Omega} \frac{|f(x)-f(y)|}{|x-y|^\alpha}.</math>
| |
| | |
| The sum of the two is the full Hölder norm of ''f''
| |
| | |
| :<math>|f|_{0,\alpha;\Omega} = |f|_{0;\Omega} + [f]_{0,\alpha;\Omega} = \sup_{x\in\Omega} |f(x)| + \sup_{x,y\in \Omega} \frac{|f(x)-f(y)|}{|x-y|^\alpha}.</math>
| |
| | |
| For differentiable functions ''u'', it is necessary to consider the higher order norms, involving derivatives. The norm in the space of functions with ''k'' continuous derivatives, <math>C^k(\Omega)</math>, is given by
| |
| | |
| :<math>|u|_{k;\Omega} = \sum_{|\beta| \leq k} \sup_{x\in \Omega} |D^\beta u(x)|</math>
| |
| | |
| where <math>\beta</math> ranges over all [[multi-index notation|multi-indices]] of appropriate orders. For functions with ''k''th order derivatives which are Holder continuous with exponent <math>\alpha</math>, the appropriate semi-norm is given by
| |
| | |
| :<math>[u]_{k,\alpha;\Omega} = \sup_{\stackrel{x,y\in \Omega}{|\beta| = k}} \frac{|D^\beta u(x) - D^\beta u(y)|}{|x-y|^\alpha}</math>
| |
| | |
| which gives a full norm of
| |
| | |
| :<math>|u|_{k,\alpha;\Omega} = |u|_{k;\Omega} + [u]_{k,\alpha;\Omega} = \sum_{|\beta| \leq k} \sup_{x\in \Omega} |D^\beta u(x)| + \sup_{\stackrel{x,y\in \Omega}{|\beta| = k}} \frac{|D^\beta u(x) - D^\beta u(y)|}{|x-y|^\alpha}.</math>
| |
| | |
| For the interior estimates, the norms are weighted by the distance to the boundary
| |
| | |
| :<math>d_x = d(x, \partial \Omega)</math>
| |
| | |
| raised to the same power as the derivative, and the seminorms are weighted by
| |
| | |
| :<math>d_{x,y} = \min (d_x,d_y) </math>
| |
| | |
| raised to the appropriate power. The resulting weighted interior norm for a function is given by
| |
| | |
| :<math>|u|^*_{k,\alpha;\Omega} = |u|^*_{k;\Omega} + [u]^*_{k,\alpha;\Omega} = \sum_{|\beta| \leq k} \sup_{x\in \Omega} |d_x^{|\beta|} D^\beta u(x)| + \sup_{\stackrel{x,y\in \Omega}{|\beta| = k}} d_{x,y}^{k+\alpha} \frac{|D^\beta u(x) - D^\beta u(y)|}{|x-y|^\alpha}</math>
| |
| | |
| It is occasionally necessary to add "extra" powers of the weight, denoted by
| |
| | |
| :<math>|u|^{(m)}_{k,\alpha;\Omega} = |u|^{(m)}_{k;\Omega} + [u]^{(m)}_{k,\alpha;\Omega} = \sum_{|\beta| \leq k} \sup_{x\in \Omega} |d_x^{|\beta|+m} D^\beta u(x)| + \sup_{\stackrel{x,y\in \Omega}{|\beta| = k}} d_{x,y}^{m+k+\alpha} \frac{|D^\beta u(x) - D^\beta u(y)|}{|x-y|^\alpha}.</math>
| |
| | |
| ==Formulation==
| |
| The formulations in this section are taken from the text of {{harvs|txt|first=D.|last=Gilbarg|last2=Trudinger|first2=Neil|authorlink2=Neil Trudinger|year=1983}}.
| |
| | |
| ===Interior estimates===
| |
| Consider a bounded solution <math>u \in C^{2,\alpha}(\Omega)</math> on the domain <math>\Omega</math> to the elliptic, second order partial differential equation
| |
| | |
| :<math>\sum_{i,j} a_{i,j}(x) D_i D_j u(x) + \sum_i b_i(x) D_i u(x) + c(x) u(x) = f(x)</math>
| |
| | |
| where the source term satisfies <math>f\in C^\alpha(\Omega)</math>. If there exists a constant <math>\lambda > 0</math> such that the <math>a_{i,j}</math> are strictly elliptic,
| |
| | |
| :<math>\sum a_{i,j}(x) \xi_i \xi_j \geq \lambda |\xi|^2</math> for all <math>x\in \Omega, \xi \in \mathbb{R}^n</math>
| |
| | |
| and the relevant norms coefficients are all bounded by another constant <math>\Lambda</math>
| |
| | |
| :<math>|a_{i,j}|_{0,\alpha;\Omega}, |b_i|^{(1)}_{0,\alpha;\Omega}, |c|^{(2)}_{0,\alpha;\Omega} \leq \Lambda.</math>
| |
| | |
| Then the weighted <math>C^{2,\alpha}</math> norm of ''u'' is controlled by the supremum of ''u'' and the Holder norm of ''f'':
| |
| | |
| :<math>|u|^*_{2,\alpha;\Omega} \leq C(n,\alpha,\lambda,\Lambda) (|u|_{0,\Omega} + |f|^{(2)}_{0,\alpha;\Omega}).</math>
| |
| | |
| ===Boundary estimates===
| |
| Let <math>\Omega</math> be a <math>C^{2,\alpha}</math> domain (that is to say, about any point on the boundary of the domain the boundary surface can be realized, after an appropriate rotation of coordinates, as a <math>C^{2,\alpha}</math> function), with Dirichlet boundary data that coincides with a function <math>\phi(x)</math> which is also at least <math>C^{2,\alpha}</math>. Then subject to analogous conditions on the coefficients as in the case of the interior estimate, the unweighted Holder norm of ''u'' is controlled by the unweighted norms of the source term, the boundary data, and the supremum norm of ''u'':
| |
| | |
| :<math>|u|_{2,\alpha;\Omega} \leq C(n,\alpha,\lambda,\Lambda,\Omega) (|u|_{0,\Omega} + |f|_{0,\alpha;\Omega} + |\phi|_{2,\alpha;\partial\Omega}).</math>
| |
| | |
| When the solution ''u'' satisfies the [[maximum principle]], the first term on the right hand side can be dropped.
| |
| | |
| ==Sources==
| |
| * {{citation|first1=D.|last=Gilbarg|first2=Neil|last2=Trudinger|authorlink2=Neil Trudinger|title=Elliptic Partial Differential Equations of Second Order|publisher=Springer|publication-place=New York|year=1983|isbn=3-540-41160-7}}
| |
| * {{Citation
| |
| |last = Schauder
| |
| |first = Juliusz
| |
| |author-link = Juliusz Schauder
| |
| |year= 1934
| |
| |url=http://www.springerlink.com/content/l5151uv221520628/
| |
| | title = Über lineare elliptische Differentialgleichungen zweiter Ordnung
| |
| | periodical = Mathematische Zeitschrift
| |
| | series =
| |
| | publication-place = Berlin, Germany
| |
| | publisher = Springer-Verlag
| |
| | language = German
| |
| | volume = 38
| |
| | issue = 1
| |
| | pages = 257–282
| |
| |doi = 10.1007/BF01170635
| |
| }} {{MathSciNet|id=1545448}}
| |
| * {{Citation
| |
| |last = Schauder
| |
| |first = Juliusz
| |
| |author-link = Juliusz Schauder
| |
| |year= 1937
| |
| | title = Numerische Abschätzungen in elliptischen linearen Differentialgleichungen
| |
| | periodical = Studia Mathematica
| |
| | series =
| |
| | publication-place = Lwów, Poland
| |
| | publisher = Polska Akademia Nauk. Instytut Matematyczny
| |
| | language = German
| |
| | volume = 5
| |
| | issue =
| |
| | pages = 34–42
| |
| | url = http://matwbn.icm.edu.pl/ksiazki/sm/sm5/sm516.pdf
| |
| }}
| |
| | |
| ==Further reading==
| |
| * {{Citation
| |
| | last1=Courant
| |
| | first1=Richard
| |
| | author1-link=Richard Courant
| |
| | last2=Hilbert
| |
| | first2=David
| |
| | author2-link=David Hilbert
| |
| | title=Methods of Mathematical Physics
| |
| | volume=2
| |
| | edition=1st English
| |
| | publisher=Wiley-Interscience
| |
| | date=1989
| |
| | place=New York
| |
| | isbn=0-471-50439-4
| |
| }}
| |
| * {{citation|first1=Qing|last1=Han|first2=Fanghua|last2=Lin|authorlink2=Lin Fanghua|title=Elliptic Partial Differential Equations|publisher=[[Courant Institute of Mathematical Sciences]]|publication-place=New York|year=1997|isbn=0-9658703-0-8|oclc=38168365}} {{MathSciNet|id=1669352}}
| |
| | |
| {{DEFAULTSORT:Schauder Estimates}}
| |
| [[Category:Elliptic partial differential equations]]
| |
It involves expertise and knowledge of various tools and technologies used for creating websites. Offshore expert Word - Press developers high level of interactivity, accessibility, functionality and usability of our website can add custom online to using. Your parishioners and certainly interested audience can come in to you for further information from the group and sometimes even approaching happenings and systems with the church. Keep reading for some great Word - Press ideas you can start using today. You can customize the appearance with PSD to Word - Press conversion .
Choosing what kind of links you'll be using is a ctitical aspect of any linkwheel strategy, especially since there are several different types of links that are assessed by search engines. But as expected the level of support you get with them can be hit or miss based on the developer's free time and desire. This plugin allows a blogger get more Facebook fans on the related fan page. From my very own experiences, I will let you know why you should choose WPZOOM Live journal templates. Moreover, many Word - Press themes need to be purchased and designing your own WP site can be boring.
You can down load it here at this link: and utilize your FTP software program to upload it to your Word - Press Plugin folder. Word - Press has ensured the users of this open source blogging platform do not have to troubleshoot on their own, or seek outside help. Those who cannot conceive with donor eggs due to some problems can also opt for surrogacy option using the services of surrogate mother. To turn the Word - Press Plugin on, click Activate on the far right side of the list. So, if you are looking online to hire dedicated Wordpress developers, India PHP Expert can give a hand you in each and every best possible way.
A built-in widget which allows you to embed quickly video from popular websites. Quttera - Quttera describes itself as a 'Saa - S [Software as a Service] web-malware monitoring and alerting solution for websites of any size and complexity. A higher percentage of women are marrying at older ages,many are delaying childbearing until their careers are established, the divorce rate is high and many couples remarry and desire their own children. IVF ,fertility,infertility expert,surrogacy specialist in India at Rotundaivf. If your blog employs the permalink function, This gives your SEO efforts a boost, and your visitors will know firsthand what's in the post when seeing the URL.
He loves sharing information regarding wordpress, Majento, Drupal and Joomla development tips & tricks. I'm a large fan of using Word - Press to create pretty much any sort of web page. For those who have any kind of issues concerning exactly where and also how you can utilize backup plugin, you are able to e-mail us from the web page. In simple words, this step can be interpreted as the planning phase of entire PSD to wordpress conversion process. Extra investment in Drupal must be bypassed as value for money is what Drupal provides. The 2010 voting took place from July 7 through August 31, 2010.