About the optimal constant in the parabolic Harnack inequality











up vote
0
down vote

favorite












For the heat equation $partial_t u = Delta u$ there exists the following version of the harnack inequality. Given $0<t_1<t_2$ and $x_1,x_2 in mathbb{R}^d$ it holds that
$$u(t_1,x_1) le u(t_2,x_2) left( frac{t_2}{t_1}right) ^{frac{d}{2}} exp left( frac{|x_2-x_1|_2^2}{4(t_2-t_1)} right). $$
For many authors it is a well known fact that this is sharp. Some argue that for the heat kernel
$$u(t,x) = left( frac{1}{4pi t}right) ^{frac{d}{2}} exp left( frac{-|x|_2^2}{4t} right)$$
above inequality is an equality. When trying to verify that i get stuck at
$$ exp left( frac{|x_2-x_1|_2^2}{4(t_2-t_1)} - frac{|x_2|_2^2}{4t_2} +frac{|x_1|_2^2}{4t_1} right) overset{!}{=} 1$$ which clearly doesn't hold. So my question is how is it to be understand that above Harnack inequality is sharp?










share|cite|improve this question









New contributor




user33 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
























    up vote
    0
    down vote

    favorite












    For the heat equation $partial_t u = Delta u$ there exists the following version of the harnack inequality. Given $0<t_1<t_2$ and $x_1,x_2 in mathbb{R}^d$ it holds that
    $$u(t_1,x_1) le u(t_2,x_2) left( frac{t_2}{t_1}right) ^{frac{d}{2}} exp left( frac{|x_2-x_1|_2^2}{4(t_2-t_1)} right). $$
    For many authors it is a well known fact that this is sharp. Some argue that for the heat kernel
    $$u(t,x) = left( frac{1}{4pi t}right) ^{frac{d}{2}} exp left( frac{-|x|_2^2}{4t} right)$$
    above inequality is an equality. When trying to verify that i get stuck at
    $$ exp left( frac{|x_2-x_1|_2^2}{4(t_2-t_1)} - frac{|x_2|_2^2}{4t_2} +frac{|x_1|_2^2}{4t_1} right) overset{!}{=} 1$$ which clearly doesn't hold. So my question is how is it to be understand that above Harnack inequality is sharp?










    share|cite|improve this question









    New contributor




    user33 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
    Check out our Code of Conduct.






















      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      For the heat equation $partial_t u = Delta u$ there exists the following version of the harnack inequality. Given $0<t_1<t_2$ and $x_1,x_2 in mathbb{R}^d$ it holds that
      $$u(t_1,x_1) le u(t_2,x_2) left( frac{t_2}{t_1}right) ^{frac{d}{2}} exp left( frac{|x_2-x_1|_2^2}{4(t_2-t_1)} right). $$
      For many authors it is a well known fact that this is sharp. Some argue that for the heat kernel
      $$u(t,x) = left( frac{1}{4pi t}right) ^{frac{d}{2}} exp left( frac{-|x|_2^2}{4t} right)$$
      above inequality is an equality. When trying to verify that i get stuck at
      $$ exp left( frac{|x_2-x_1|_2^2}{4(t_2-t_1)} - frac{|x_2|_2^2}{4t_2} +frac{|x_1|_2^2}{4t_1} right) overset{!}{=} 1$$ which clearly doesn't hold. So my question is how is it to be understand that above Harnack inequality is sharp?










      share|cite|improve this question









      New contributor




      user33 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.











      For the heat equation $partial_t u = Delta u$ there exists the following version of the harnack inequality. Given $0<t_1<t_2$ and $x_1,x_2 in mathbb{R}^d$ it holds that
      $$u(t_1,x_1) le u(t_2,x_2) left( frac{t_2}{t_1}right) ^{frac{d}{2}} exp left( frac{|x_2-x_1|_2^2}{4(t_2-t_1)} right). $$
      For many authors it is a well known fact that this is sharp. Some argue that for the heat kernel
      $$u(t,x) = left( frac{1}{4pi t}right) ^{frac{d}{2}} exp left( frac{-|x|_2^2}{4t} right)$$
      above inequality is an equality. When trying to verify that i get stuck at
      $$ exp left( frac{|x_2-x_1|_2^2}{4(t_2-t_1)} - frac{|x_2|_2^2}{4t_2} +frac{|x_1|_2^2}{4t_1} right) overset{!}{=} 1$$ which clearly doesn't hold. So my question is how is it to be understand that above Harnack inequality is sharp?







      heat-equation






      share|cite|improve this question









      New contributor




      user33 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.











      share|cite|improve this question









      New contributor




      user33 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.









      share|cite|improve this question




      share|cite|improve this question








      edited yesterday





















      New contributor




      user33 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.









      asked Nov 10 at 8:22









      user33

      11




      11




      New contributor




      user33 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.





      New contributor





      user33 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.






      user33 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.



























          active

          oldest

          votes











          Your Answer





          StackExchange.ifUsing("editor", function () {
          return StackExchange.using("mathjaxEditing", function () {
          StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
          StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
          });
          });
          }, "mathjax-editing");

          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',
          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
          });


          }
          });






          user33 is a new contributor. Be nice, and check out our Code of Conduct.










           

          draft saved


          draft discarded


















          StackExchange.ready(
          function () {
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2992364%2fabout-the-optimal-constant-in-the-parabolic-harnack-inequality%23new-answer', 'question_page');
          }
          );

          Post as a guest















          Required, but never shown






























          active

          oldest

          votes













          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes








          user33 is a new contributor. Be nice, and check out our Code of Conduct.










           

          draft saved


          draft discarded


















          user33 is a new contributor. Be nice, and check out our Code of Conduct.













          user33 is a new contributor. Be nice, and check out our Code of Conduct.












          user33 is a new contributor. Be nice, and check out our Code of Conduct.















           


          draft saved


          draft discarded














          StackExchange.ready(
          function () {
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2992364%2fabout-the-optimal-constant-in-the-parabolic-harnack-inequality%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

          AnyDesk - Fatal Program Failure

          QoS: MAC-Priority for clients behind a repeater

          Актюбинская область