Let $X={uin C^1[0,1]|u(0)=0}$ and let $I:Xtomathbb{R}$ be defined as $I(u)=int_0^1 (u'^2-u^2)$. Which of the...











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Let $X={uin C^1[0,1]|u(0)=0}$ and let $I:Xtomathbb{R}$ be defined as $I(u)=int_0^1 (u'^2-u^2)$. Which of the following is correct?



$(a)$ I is bounded below



$(b)$ I is not bounded below



$(c)$ I attains its infimum



$(d)$ I does not attain its infimum



Attempt:



Using Euler-Lagrange equations,



$2u+2u'=0$



Therefore $u(x)=c_1sin(x)+c_2 cos(x)$



It is given that $u(0)=0$ so it implies that $c_2=0$.



$therefore u(x)=c_1sin(x)$



What should be the next step? Please give some hints.






calculus calculus-of-variations







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    down vote

    favorite












    Let $X={uin C^1[0,1]|u(0)=0}$ and let $I:Xtomathbb{R}$ be defined as $I(u)=int_0^1 (u'^2-u^2)$. Which of the following is correct?



    $(a)$ I is bounded below



    $(b)$ I is not bounded below



    $(c)$ I attains its infimum



    $(d)$ I does not attain its infimum



    Attempt:



    Using Euler-Lagrange equations,



    $2u+2u'=0$



    Therefore $u(x)=c_1sin(x)+c_2 cos(x)$



    It is given that $u(0)=0$ so it implies that $c_2=0$.



    $therefore u(x)=c_1sin(x)$



    What should be the next step? Please give some hints.






    calculus calculus-of-variations







    share|cite|improve this question
























      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      Let $X={uin C^1[0,1]|u(0)=0}$ and let $I:Xtomathbb{R}$ be defined as $I(u)=int_0^1 (u'^2-u^2)$. Which of the following is correct?



      $(a)$ I is bounded below



      $(b)$ I is not bounded below



      $(c)$ I attains its infimum



      $(d)$ I does not attain its infimum



      Attempt:



      Using Euler-Lagrange equations,



      $2u+2u'=0$



      Therefore $u(x)=c_1sin(x)+c_2 cos(x)$



      It is given that $u(0)=0$ so it implies that $c_2=0$.



      $therefore u(x)=c_1sin(x)$



      What should be the next step? Please give some hints.






      calculus calculus-of-variations







      share|cite|improve this question













      Let $X={uin C^1[0,1]|u(0)=0}$ and let $I:Xtomathbb{R}$ be defined as $I(u)=int_0^1 (u'^2-u^2)$. Which of the following is correct?



      $(a)$ I is bounded below



      $(b)$ I is not bounded below



      $(c)$ I attains its infimum



      $(d)$ I does not attain its infimum



      Attempt:



      Using Euler-Lagrange equations,



      $2u+2u'=0$



      Therefore $u(x)=c_1sin(x)+c_2 cos(x)$



      It is given that $u(0)=0$ so it implies that $c_2=0$.



      $therefore u(x)=c_1sin(x)$



      What should be the next step? Please give some hints.






      calculus calculus-of-variations




      calculus calculus-of-variations






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      share|cite|improve this question











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      asked 2 hours ago









      StammeringMathematician

      2,1631322




      2,1631322






















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          $u(x) =int_0^{x} u'(t) , dt$ so $u(x)^{2} leq xint_0^{x} u'(t) ^{2} , dt leq int_0^{1} u'(t) ^{2} , dt$ which shows (after integration) that $I geq 0$. Hence a) is true, b) is false. The infimum is attained when $u equiv 0$ so c) is true and d) is false.






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            accepted










            $u(x) =int_0^{x} u'(t) , dt$ so $u(x)^{2} leq xint_0^{x} u'(t) ^{2} , dt leq int_0^{1} u'(t) ^{2} , dt$ which shows (after integration) that $I geq 0$. Hence a) is true, b) is false. The infimum is attained when $u equiv 0$ so c) is true and d) is false.






            share|cite|improve this answer

























              up vote
              2
              down vote



              accepted










              $u(x) =int_0^{x} u'(t) , dt$ so $u(x)^{2} leq xint_0^{x} u'(t) ^{2} , dt leq int_0^{1} u'(t) ^{2} , dt$ which shows (after integration) that $I geq 0$. Hence a) is true, b) is false. The infimum is attained when $u equiv 0$ so c) is true and d) is false.






              share|cite|improve this answer























                up vote
                2
                down vote



                accepted







                up vote
                2
                down vote



                accepted






                $u(x) =int_0^{x} u'(t) , dt$ so $u(x)^{2} leq xint_0^{x} u'(t) ^{2} , dt leq int_0^{1} u'(t) ^{2} , dt$ which shows (after integration) that $I geq 0$. Hence a) is true, b) is false. The infimum is attained when $u equiv 0$ so c) is true and d) is false.






                share|cite|improve this answer












                $u(x) =int_0^{x} u'(t) , dt$ so $u(x)^{2} leq xint_0^{x} u'(t) ^{2} , dt leq int_0^{1} u'(t) ^{2} , dt$ which shows (after integration) that $I geq 0$. Hence a) is true, b) is false. The infimum is attained when $u equiv 0$ so c) is true and d) is false.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered 2 hours ago









                Kavi Rama Murthy

                39k31748




                39k31748






























                     

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