Showing that the system of a planar curve is Lipschitz in y












2














This question below, which was given in an exam, goes as follows:




A planar curve $y(x)$ is such that its curvature,
$$k(x)=y''(x)/(1+y'(x)^2)^{3/2}$$ is equal to its height $y(x)$. Write
the equation as a pair of first order equations, and show that the
system is Lipschitz in $y$.




Now my working is as follows:
Let $$u=y, v=y'$$
$$implies u'=v, v'=(1+v^2)^{3/2}u$$
which answers the first part of the question. Now to show that it is Lipschitz in $y$:
$$|u_1'-u_2'|=|v_1-v_2|leq |u_1-u_2|+|v_1-v_2|$$



My problem is for the second part:
begin{align}
|v_1'-v_2'|=&|(1+v_1^2)^{3/2}u_1-(1+v_2^2)^{3/2}u_2|\
=&|(1+v_1^2)^{3/2}u_1-(1+v_1^2)^{3/2}u_2+(1+v_1^2)^{3/2}u_2-(1+v_2^2)^{3/2}u_2|\
leq &|1+v_1^2|^{3/2}|u_1-u_2|+|u_2|||(1+v_1^2)^{3/2}-(1+v_2^2)^{3/2}|
end{align}



Now I can always bound the first term, but for the second term, I have no idea how to simplify in such a way to get $K|v_1-v_2|$ for some constant $K$. Any hints? Am I going about this the wrong way?










share|cite|improve this question






















  • By Lipschitz in $y$ do you mean in $u$ in your notation? Or do you mean a bound of the form $K(|u_1 - u_2| + | v_1 -v_2|)$?
    – Arctic Char
    yesterday










  • Yes, exactly like the latter
    – Dylan Zammit
    yesterday






  • 1




    That it is locally Lipschitz follows from the differentiability of the right sides, thus uniqueness of solutions follows. As you found, this is not globally Lipschitz. Integration after multiplication by $2y'$ gives $C=y^2+(1+y'^2)^{-1/2}$ as first integral.
    – LutzL
    yesterday


















2














This question below, which was given in an exam, goes as follows:




A planar curve $y(x)$ is such that its curvature,
$$k(x)=y''(x)/(1+y'(x)^2)^{3/2}$$ is equal to its height $y(x)$. Write
the equation as a pair of first order equations, and show that the
system is Lipschitz in $y$.




Now my working is as follows:
Let $$u=y, v=y'$$
$$implies u'=v, v'=(1+v^2)^{3/2}u$$
which answers the first part of the question. Now to show that it is Lipschitz in $y$:
$$|u_1'-u_2'|=|v_1-v_2|leq |u_1-u_2|+|v_1-v_2|$$



My problem is for the second part:
begin{align}
|v_1'-v_2'|=&|(1+v_1^2)^{3/2}u_1-(1+v_2^2)^{3/2}u_2|\
=&|(1+v_1^2)^{3/2}u_1-(1+v_1^2)^{3/2}u_2+(1+v_1^2)^{3/2}u_2-(1+v_2^2)^{3/2}u_2|\
leq &|1+v_1^2|^{3/2}|u_1-u_2|+|u_2|||(1+v_1^2)^{3/2}-(1+v_2^2)^{3/2}|
end{align}



Now I can always bound the first term, but for the second term, I have no idea how to simplify in such a way to get $K|v_1-v_2|$ for some constant $K$. Any hints? Am I going about this the wrong way?










share|cite|improve this question






















  • By Lipschitz in $y$ do you mean in $u$ in your notation? Or do you mean a bound of the form $K(|u_1 - u_2| + | v_1 -v_2|)$?
    – Arctic Char
    yesterday










  • Yes, exactly like the latter
    – Dylan Zammit
    yesterday






  • 1




    That it is locally Lipschitz follows from the differentiability of the right sides, thus uniqueness of solutions follows. As you found, this is not globally Lipschitz. Integration after multiplication by $2y'$ gives $C=y^2+(1+y'^2)^{-1/2}$ as first integral.
    – LutzL
    yesterday
















2












2








2







This question below, which was given in an exam, goes as follows:




A planar curve $y(x)$ is such that its curvature,
$$k(x)=y''(x)/(1+y'(x)^2)^{3/2}$$ is equal to its height $y(x)$. Write
the equation as a pair of first order equations, and show that the
system is Lipschitz in $y$.




Now my working is as follows:
Let $$u=y, v=y'$$
$$implies u'=v, v'=(1+v^2)^{3/2}u$$
which answers the first part of the question. Now to show that it is Lipschitz in $y$:
$$|u_1'-u_2'|=|v_1-v_2|leq |u_1-u_2|+|v_1-v_2|$$



My problem is for the second part:
begin{align}
|v_1'-v_2'|=&|(1+v_1^2)^{3/2}u_1-(1+v_2^2)^{3/2}u_2|\
=&|(1+v_1^2)^{3/2}u_1-(1+v_1^2)^{3/2}u_2+(1+v_1^2)^{3/2}u_2-(1+v_2^2)^{3/2}u_2|\
leq &|1+v_1^2|^{3/2}|u_1-u_2|+|u_2|||(1+v_1^2)^{3/2}-(1+v_2^2)^{3/2}|
end{align}



Now I can always bound the first term, but for the second term, I have no idea how to simplify in such a way to get $K|v_1-v_2|$ for some constant $K$. Any hints? Am I going about this the wrong way?










share|cite|improve this question













This question below, which was given in an exam, goes as follows:




A planar curve $y(x)$ is such that its curvature,
$$k(x)=y''(x)/(1+y'(x)^2)^{3/2}$$ is equal to its height $y(x)$. Write
the equation as a pair of first order equations, and show that the
system is Lipschitz in $y$.




Now my working is as follows:
Let $$u=y, v=y'$$
$$implies u'=v, v'=(1+v^2)^{3/2}u$$
which answers the first part of the question. Now to show that it is Lipschitz in $y$:
$$|u_1'-u_2'|=|v_1-v_2|leq |u_1-u_2|+|v_1-v_2|$$



My problem is for the second part:
begin{align}
|v_1'-v_2'|=&|(1+v_1^2)^{3/2}u_1-(1+v_2^2)^{3/2}u_2|\
=&|(1+v_1^2)^{3/2}u_1-(1+v_1^2)^{3/2}u_2+(1+v_1^2)^{3/2}u_2-(1+v_2^2)^{3/2}u_2|\
leq &|1+v_1^2|^{3/2}|u_1-u_2|+|u_2|||(1+v_1^2)^{3/2}-(1+v_2^2)^{3/2}|
end{align}



Now I can always bound the first term, but for the second term, I have no idea how to simplify in such a way to get $K|v_1-v_2|$ for some constant $K$. Any hints? Am I going about this the wrong way?







differential-equations lipschitz-functions






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked yesterday









Dylan Zammit

8801416




8801416












  • By Lipschitz in $y$ do you mean in $u$ in your notation? Or do you mean a bound of the form $K(|u_1 - u_2| + | v_1 -v_2|)$?
    – Arctic Char
    yesterday










  • Yes, exactly like the latter
    – Dylan Zammit
    yesterday






  • 1




    That it is locally Lipschitz follows from the differentiability of the right sides, thus uniqueness of solutions follows. As you found, this is not globally Lipschitz. Integration after multiplication by $2y'$ gives $C=y^2+(1+y'^2)^{-1/2}$ as first integral.
    – LutzL
    yesterday




















  • By Lipschitz in $y$ do you mean in $u$ in your notation? Or do you mean a bound of the form $K(|u_1 - u_2| + | v_1 -v_2|)$?
    – Arctic Char
    yesterday










  • Yes, exactly like the latter
    – Dylan Zammit
    yesterday






  • 1




    That it is locally Lipschitz follows from the differentiability of the right sides, thus uniqueness of solutions follows. As you found, this is not globally Lipschitz. Integration after multiplication by $2y'$ gives $C=y^2+(1+y'^2)^{-1/2}$ as first integral.
    – LutzL
    yesterday


















By Lipschitz in $y$ do you mean in $u$ in your notation? Or do you mean a bound of the form $K(|u_1 - u_2| + | v_1 -v_2|)$?
– Arctic Char
yesterday




By Lipschitz in $y$ do you mean in $u$ in your notation? Or do you mean a bound of the form $K(|u_1 - u_2| + | v_1 -v_2|)$?
– Arctic Char
yesterday












Yes, exactly like the latter
– Dylan Zammit
yesterday




Yes, exactly like the latter
– Dylan Zammit
yesterday




1




1




That it is locally Lipschitz follows from the differentiability of the right sides, thus uniqueness of solutions follows. As you found, this is not globally Lipschitz. Integration after multiplication by $2y'$ gives $C=y^2+(1+y'^2)^{-1/2}$ as first integral.
– LutzL
yesterday






That it is locally Lipschitz follows from the differentiability of the right sides, thus uniqueness of solutions follows. As you found, this is not globally Lipschitz. Integration after multiplication by $2y'$ gives $C=y^2+(1+y'^2)^{-1/2}$ as first integral.
– LutzL
yesterday












1 Answer
1






active

oldest

votes


















1














Apply either a mean value theorem or the binomial theorem as in
$$
(1+v_1^2)^{3/2}-(1+v_2^2)^{3/2}=frac{(1+v_1^2)^3-(1+v_2^2)^3}{(1+v_1^2)^{3/2}+(1+v_2^2)^{3/2}}
\
=frac{(v_1-v_2)(v_1+v_2)Bigl((1+v_1^2)^2+(1+v_1^2)(1+v_2^2)+(1+v_2^2)^2Bigr)}{(1+v_1^2)^{3/2}+(1+v_2^2)^{3/2}}
$$

where you get the simple difference as factors and can bound the other factors.






share|cite|improve this answer





















    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',
    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%2f3060548%2fshowing-that-the-system-of-a-planar-curve-is-lipschitz-in-y%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









    1














    Apply either a mean value theorem or the binomial theorem as in
    $$
    (1+v_1^2)^{3/2}-(1+v_2^2)^{3/2}=frac{(1+v_1^2)^3-(1+v_2^2)^3}{(1+v_1^2)^{3/2}+(1+v_2^2)^{3/2}}
    \
    =frac{(v_1-v_2)(v_1+v_2)Bigl((1+v_1^2)^2+(1+v_1^2)(1+v_2^2)+(1+v_2^2)^2Bigr)}{(1+v_1^2)^{3/2}+(1+v_2^2)^{3/2}}
    $$

    where you get the simple difference as factors and can bound the other factors.






    share|cite|improve this answer


























      1














      Apply either a mean value theorem or the binomial theorem as in
      $$
      (1+v_1^2)^{3/2}-(1+v_2^2)^{3/2}=frac{(1+v_1^2)^3-(1+v_2^2)^3}{(1+v_1^2)^{3/2}+(1+v_2^2)^{3/2}}
      \
      =frac{(v_1-v_2)(v_1+v_2)Bigl((1+v_1^2)^2+(1+v_1^2)(1+v_2^2)+(1+v_2^2)^2Bigr)}{(1+v_1^2)^{3/2}+(1+v_2^2)^{3/2}}
      $$

      where you get the simple difference as factors and can bound the other factors.






      share|cite|improve this answer
























        1












        1








        1






        Apply either a mean value theorem or the binomial theorem as in
        $$
        (1+v_1^2)^{3/2}-(1+v_2^2)^{3/2}=frac{(1+v_1^2)^3-(1+v_2^2)^3}{(1+v_1^2)^{3/2}+(1+v_2^2)^{3/2}}
        \
        =frac{(v_1-v_2)(v_1+v_2)Bigl((1+v_1^2)^2+(1+v_1^2)(1+v_2^2)+(1+v_2^2)^2Bigr)}{(1+v_1^2)^{3/2}+(1+v_2^2)^{3/2}}
        $$

        where you get the simple difference as factors and can bound the other factors.






        share|cite|improve this answer












        Apply either a mean value theorem or the binomial theorem as in
        $$
        (1+v_1^2)^{3/2}-(1+v_2^2)^{3/2}=frac{(1+v_1^2)^3-(1+v_2^2)^3}{(1+v_1^2)^{3/2}+(1+v_2^2)^{3/2}}
        \
        =frac{(v_1-v_2)(v_1+v_2)Bigl((1+v_1^2)^2+(1+v_1^2)(1+v_2^2)+(1+v_2^2)^2Bigr)}{(1+v_1^2)^{3/2}+(1+v_2^2)^{3/2}}
        $$

        where you get the simple difference as factors and can bound the other factors.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered yesterday









        LutzL

        56.3k42054




        56.3k42054






























            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.





            Some of your past answers have not been well-received, and you're in danger of being blocked from answering.


            Please pay close attention to the following guidance:


            • 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.


            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%2f3060548%2fshowing-that-the-system-of-a-planar-curve-is-lipschitz-in-y%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

            1300-talet

            1300-talet

            Display a custom attribute below product name in the front-end Magento 1.9.3.8