Extension of weak derivatives in Bochner spaces Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Are weak derivatives and distributional derivatives different?Weak Convergence of piecewise constant function (Evans PDE, Section 8.7, Q1b)Evans PDE mappings into better spacesReference about Sobolev spacesWhy L belongs to the dual space $H^-1$Bochner-Sobolev space vs. Sobolev space on product via Fubini-Tonelli?Theorem about difference quotients and weak derivatives in Evans - Why not stronger statement?Inequality between differential quotient and Sobolev normProof of the Sobolev Extension Theorem in Evans' PDE bookConfusions in Evans book regarding weak derivatives in Banach spaces

Stars Make Stars

What is the electric potential inside a point charge?

Strange behaviour of Check

Should you tell Jews they are breaking a commandment?

Geometric mean and geometric standard deviation

What was the last x86 CPU that did not have the x87 floating-point unit built in?

Biased dice probability question

Why is there no army of Iron-Mans in the MCU?

Can a monk deflect thrown melee weapons?

How to colour the US map with Yellow, Green, Red and Blue to minimize the number of states with the colour of Green

I'm having difficulty getting my players to do stuff in a sandbox campaign

Direct Experience of Meditation

Working around an AWS network ACL rule limit

Do working physicists consider Newtonian mechanics to be "falsified"?

Can the prologue be the backstory of your main character?

Unable to start mainnet node docker container

Array/tabular for long multiplication

How to politely respond to generic emails requesting a PhD/job in my lab? Without wasting too much time

How is simplicity better than precision and clarity in prose?

How does modal jazz use chord progressions?

What's the difference between (size_t)-1 and ~0?

Two different pronunciation of "понял"

What can I do if my MacBook isn’t charging but already ran out?

Can smartphones with the same camera sensor have different image quality?



Extension of weak derivatives in Bochner spaces



Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Are weak derivatives and distributional derivatives different?Weak Convergence of piecewise constant function (Evans PDE, Section 8.7, Q1b)Evans PDE mappings into better spacesReference about Sobolev spacesWhy L belongs to the dual space $H^-1$Bochner-Sobolev space vs. Sobolev space on product via Fubini-Tonelli?Theorem about difference quotients and weak derivatives in Evans - Why not stronger statement?Inequality between differential quotient and Sobolev normProof of the Sobolev Extension Theorem in Evans' PDE bookConfusions in Evans book regarding weak derivatives in Banach spaces










1












$begingroup$


I am struggling to understand estimate $(15)$ from the following proof from the PDE book by Evans:
enter image description here



He argues that estimate $(15)$ follows from difference quotients, but I can't understand this. In fact, the theory developed in the earlier section of difference quotients is for real-valued functions, not for functions with values in Banach spaces. What theorem did he use in fact? Any ideas?










share|cite|improve this question









$endgroup$
















    1












    $begingroup$


    I am struggling to understand estimate $(15)$ from the following proof from the PDE book by Evans:
    enter image description here



    He argues that estimate $(15)$ follows from difference quotients, but I can't understand this. In fact, the theory developed in the earlier section of difference quotients is for real-valued functions, not for functions with values in Banach spaces. What theorem did he use in fact? Any ideas?










    share|cite|improve this question









    $endgroup$














      1












      1








      1





      $begingroup$


      I am struggling to understand estimate $(15)$ from the following proof from the PDE book by Evans:
      enter image description here



      He argues that estimate $(15)$ follows from difference quotients, but I can't understand this. In fact, the theory developed in the earlier section of difference quotients is for real-valued functions, not for functions with values in Banach spaces. What theorem did he use in fact? Any ideas?










      share|cite|improve this question









      $endgroup$




      I am struggling to understand estimate $(15)$ from the following proof from the PDE book by Evans:
      enter image description here



      He argues that estimate $(15)$ follows from difference quotients, but I can't understand this. In fact, the theory developed in the earlier section of difference quotients is for real-valued functions, not for functions with values in Banach spaces. What theorem did he use in fact? Any ideas?







      functional-analysis pde sobolev-spaces bochner-spaces






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Mar 25 at 19:05









      RichardRichard

      1,2551724




      1,2551724




















          1 Answer
          1






          active

          oldest

          votes


















          0












          $begingroup$

          The method of difference quotients is also true for functions with values in $L^2$. It can be found for instance in:




          J.L. Lions. Equations différentielles opérationnelles et problèmes aux limites. Die Grundlehren der mathematischen Wissenschaften, Bd. 111 Springer-Verlag, Berlin-Göttingen-Heidelberg, 1961.







          share|cite|improve this answer









          $endgroup$












          • $begingroup$
            Sorry, I can't download the book. May you please state the result being used? I still have no idea.
            $endgroup$
            – Richard
            Mar 25 at 22:12










          • $begingroup$
            Give me your email I will send to you some references on the method.
            $endgroup$
            – S. Maths
            Mar 26 at 20:10











          Your Answer








          StackExchange.ready(function()
          var channelOptions =
          tags: "".split(" "),
          id: "69"
          ;
          initTagRenderer("".split(" "), "".split(" "), channelOptions);

          StackExchange.using("externalEditor", function()
          // Have to fire editor after snippets, if snippets enabled
          if (StackExchange.settings.snippets.snippetsEnabled)
          StackExchange.using("snippets", function()
          createEditor();
          );

          else
          createEditor();

          );

          function createEditor()
          StackExchange.prepareEditor(
          heartbeatType: 'answer',
          autoActivateHeartbeat: false,
          convertImagesToLinks: true,
          noModals: true,
          showLowRepImageUploadWarning: true,
          reputationToPostImages: 10,
          bindNavPrevention: true,
          postfix: "",
          imageUploader:
          brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
          contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
          allowUrls: true
          ,
          noCode: true, onDemand: true,
          discardSelector: ".discard-answer"
          ,immediatelyShowMarkdownHelp:true
          );



          );













          draft saved

          draft discarded


















          StackExchange.ready(
          function ()
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3162199%2fextension-of-weak-derivatives-in-bochner-spaces%23new-answer', 'question_page');

          );

          Post as a guest















          Required, but never shown

























          1 Answer
          1






          active

          oldest

          votes








          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          0












          $begingroup$

          The method of difference quotients is also true for functions with values in $L^2$. It can be found for instance in:




          J.L. Lions. Equations différentielles opérationnelles et problèmes aux limites. Die Grundlehren der mathematischen Wissenschaften, Bd. 111 Springer-Verlag, Berlin-Göttingen-Heidelberg, 1961.







          share|cite|improve this answer









          $endgroup$












          • $begingroup$
            Sorry, I can't download the book. May you please state the result being used? I still have no idea.
            $endgroup$
            – Richard
            Mar 25 at 22:12










          • $begingroup$
            Give me your email I will send to you some references on the method.
            $endgroup$
            – S. Maths
            Mar 26 at 20:10















          0












          $begingroup$

          The method of difference quotients is also true for functions with values in $L^2$. It can be found for instance in:




          J.L. Lions. Equations différentielles opérationnelles et problèmes aux limites. Die Grundlehren der mathematischen Wissenschaften, Bd. 111 Springer-Verlag, Berlin-Göttingen-Heidelberg, 1961.







          share|cite|improve this answer









          $endgroup$












          • $begingroup$
            Sorry, I can't download the book. May you please state the result being used? I still have no idea.
            $endgroup$
            – Richard
            Mar 25 at 22:12










          • $begingroup$
            Give me your email I will send to you some references on the method.
            $endgroup$
            – S. Maths
            Mar 26 at 20:10













          0












          0








          0





          $begingroup$

          The method of difference quotients is also true for functions with values in $L^2$. It can be found for instance in:




          J.L. Lions. Equations différentielles opérationnelles et problèmes aux limites. Die Grundlehren der mathematischen Wissenschaften, Bd. 111 Springer-Verlag, Berlin-Göttingen-Heidelberg, 1961.







          share|cite|improve this answer









          $endgroup$



          The method of difference quotients is also true for functions with values in $L^2$. It can be found for instance in:




          J.L. Lions. Equations différentielles opérationnelles et problèmes aux limites. Die Grundlehren der mathematischen Wissenschaften, Bd. 111 Springer-Verlag, Berlin-Göttingen-Heidelberg, 1961.








          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Mar 25 at 21:55









          S. MathsS. Maths

          667116




          667116











          • $begingroup$
            Sorry, I can't download the book. May you please state the result being used? I still have no idea.
            $endgroup$
            – Richard
            Mar 25 at 22:12










          • $begingroup$
            Give me your email I will send to you some references on the method.
            $endgroup$
            – S. Maths
            Mar 26 at 20:10
















          • $begingroup$
            Sorry, I can't download the book. May you please state the result being used? I still have no idea.
            $endgroup$
            – Richard
            Mar 25 at 22:12










          • $begingroup$
            Give me your email I will send to you some references on the method.
            $endgroup$
            – S. Maths
            Mar 26 at 20:10















          $begingroup$
          Sorry, I can't download the book. May you please state the result being used? I still have no idea.
          $endgroup$
          – Richard
          Mar 25 at 22:12




          $begingroup$
          Sorry, I can't download the book. May you please state the result being used? I still have no idea.
          $endgroup$
          – Richard
          Mar 25 at 22:12












          $begingroup$
          Give me your email I will send to you some references on the method.
          $endgroup$
          – S. Maths
          Mar 26 at 20:10




          $begingroup$
          Give me your email I will send to you some references on the method.
          $endgroup$
          – S. Maths
          Mar 26 at 20:10

















          draft saved

          draft discarded
















































          Thanks for contributing an answer to Mathematics Stack Exchange!


          • Please be sure to answer the question. Provide details and share your research!

          But avoid


          • Asking for help, clarification, or responding to other answers.

          • Making statements based on opinion; back them up with references or personal experience.

          Use MathJax to format equations. MathJax reference.


          To learn more, see our tips on writing great answers.




          draft saved


          draft discarded














          StackExchange.ready(
          function ()
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3162199%2fextension-of-weak-derivatives-in-bochner-spaces%23new-answer', 'question_page');

          );

          Post as a guest















          Required, but never shown





















































          Required, but never shown














          Required, but never shown












          Required, but never shown







          Required, but never shown

































          Required, but never shown














          Required, but never shown












          Required, but never shown







          Required, but never shown







          Popular posts from this blog

          How should I support this large drywall patch? Planned maintenance scheduled April 23, 2019 at 00:00UTC (8:00pm US/Eastern) Announcing the arrival of Valued Associate #679: Cesar Manara Unicorn Meta Zoo #1: Why another podcast?How do I cover large gaps in drywall?How do I keep drywall around a patch from crumbling?Can I glue a second layer of drywall?How to patch long strip on drywall?Large drywall patch: how to avoid bulging seams?Drywall Mesh Patch vs. Bulge? To remove or not to remove?How to fix this drywall job?Prep drywall before backsplashWhat's the best way to fix this horrible drywall patch job?Drywall patching using 3M Patch Plus Primer

          Lowndes Grove History Architecture References Navigation menu32°48′6″N 79°57′58″W / 32.80167°N 79.96611°W / 32.80167; -79.9661132°48′6″N 79°57′58″W / 32.80167°N 79.96611°W / 32.80167; -79.9661178002500"National Register Information System"Historic houses of South Carolina"Lowndes Grove""+32° 48' 6.00", −79° 57' 58.00""Lowndes Grove, Charleston County (260 St. Margaret St., Charleston)""Lowndes Grove"The Charleston ExpositionIt Happened in South Carolina"Lowndes Grove (House), Saint Margaret Street & Sixth Avenue, Charleston, Charleston County, SC(Photographs)"Plantations of the Carolina Low Countrye

          Kathakali Contents Etymology and nomenclature History Repertoire Songs and musical instruments Traditional plays Styles: Sampradayam Training centers and awards Relationship to other dance forms See also Notes References External links Navigation menueThe Illustrated Encyclopedia of Hinduism: A-MSouth Asian Folklore: An EncyclopediaRoutledge International Encyclopedia of Women: Global Women's Issues and KnowledgeKathakali Dance-drama: Where Gods and Demons Come to PlayKathakali Dance-drama: Where Gods and Demons Come to PlayKathakali Dance-drama: Where Gods and Demons Come to Play10.1353/atj.2005.0004The Illustrated Encyclopedia of Hinduism: A-MEncyclopedia of HinduismKathakali Dance-drama: Where Gods and Demons Come to PlaySonic Liturgy: Ritual and Music in Hindu Tradition"The Mirror of Gesture"Kathakali Dance-drama: Where Gods and Demons Come to Play"Kathakali"Indian Theatre: Traditions of PerformanceIndian Theatre: Traditions of PerformanceIndian Theatre: Traditions of PerformanceIndian Theatre: Traditions of PerformanceMedieval Indian Literature: An AnthologyThe Oxford Companion to Indian TheatreSouth Asian Folklore: An Encyclopedia : Afghanistan, Bangladesh, India, Nepal, Pakistan, Sri LankaThe Rise of Performance Studies: Rethinking Richard Schechner's Broad SpectrumIndian Theatre: Traditions of PerformanceModern Asian Theatre and Performance 1900-2000Critical Theory and PerformanceBetween Theater and AnthropologyKathakali603847011Indian Theatre: Traditions of PerformanceIndian Theatre: Traditions of PerformanceIndian Theatre: Traditions of PerformanceBetween Theater and AnthropologyBetween Theater and AnthropologyNambeesan Smaraka AwardsArchivedThe Cambridge Guide to TheatreRoutledge International Encyclopedia of Women: Global Women's Issues and KnowledgeThe Garland Encyclopedia of World Music: South Asia : the Indian subcontinentThe Ethos of Noh: Actors and Their Art10.2307/1145740By Means of Performance: Intercultural Studies of Theatre and Ritual10.1017/s204912550000100xReconceiving the Renaissance: A Critical ReaderPerformance TheoryListening to Theatre: The Aural Dimension of Beijing Opera10.2307/1146013Kathakali: The Art of the Non-WorldlyOn KathakaliKathakali, the dance theatreThe Kathakali Complex: Performance & StructureKathakali Dance-Drama: Where Gods and Demons Come to Play10.1093/obo/9780195399318-0071Drama and Ritual of Early Hinduism"In the Shadow of Hollywood Orientalism: Authentic East Indian Dancing"10.1080/08949460490274013Sanskrit Play Production in Ancient IndiaIndian Music: History and StructureBharata, the Nāṭyaśāstra233639306Table of Contents2238067286469807Dance In Indian Painting10.2307/32047833204783Kathakali Dance-Theatre: A Visual Narrative of Sacred Indian MimeIndian Classical Dance: The Renaissance and BeyondKathakali: an indigenous art-form of Keralaeee