Maximum Principle For Gradient Of Poisson Equation

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Ran across this problem for a poisson equation. Looks like some sort ofaximum principle for the gradient. Can't quite wrap my head around it.



Let $Omega$ be an open subset of $mathbbR^n$. Suppose $u in C^2(overlineOmega)$ is a solution of the equation $Delta u = u^3$ with the property that $|nabla u| leq 1$ on $partial Omega$. Show $|nabla u| leq 1$ on all of $Omega$.



Thank you for any help.




pde maximum-principle poissons-equation




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    Ran across this problem for a poisson equation. Looks like some sort ofaximum principle for the gradient. Can't quite wrap my head around it.



    Let $Omega$ be an open subset of $mathbbR^n$. Suppose $u in C^2(overlineOmega)$ is a solution of the equation $Delta u = u^3$ with the property that $|nabla u| leq 1$ on $partial Omega$. Show $|nabla u| leq 1$ on all of $Omega$.



    Thank you for any help.




    pde maximum-principle poissons-equation




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      up vote
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      up vote
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      Ran across this problem for a poisson equation. Looks like some sort ofaximum principle for the gradient. Can't quite wrap my head around it.



      Let $Omega$ be an open subset of $mathbbR^n$. Suppose $u in C^2(overlineOmega)$ is a solution of the equation $Delta u = u^3$ with the property that $|nabla u| leq 1$ on $partial Omega$. Show $|nabla u| leq 1$ on all of $Omega$.



      Thank you for any help.




      pde maximum-principle poissons-equation




      share|cite|improve this question











      Ran across this problem for a poisson equation. Looks like some sort ofaximum principle for the gradient. Can't quite wrap my head around it.



      Let $Omega$ be an open subset of $mathbbR^n$. Suppose $u in C^2(overlineOmega)$ is a solution of the equation $Delta u = u^3$ with the property that $|nabla u| leq 1$ on $partial Omega$. Show $|nabla u| leq 1$ on all of $Omega$.



      Thank you for any help.





      pde maximum-principle poissons-equation





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      asked Aug 3 at 4:30









      richardryder

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          Edit: I will suppose $uin C^3$.



          It suffices to show that $v=|nabla u|^2$ is subharmonic, or $Delta vge 0$, since such functions satisfy the one-sided maximum principle $sup_Omega vle sup_partialOmegav$. We have
          $$
          Delta v=partial_ipartial_i(u_ju_j)=2partial_i(u_ju_ij)=2u_iju_ij+2u_ju_iij.
          $$
          To deal with the last term, we differentiate the Poisson equation: $Delta u_j=3u^2u_j$. Substituting this in, we get
          $$
          Delta v=2u_iju_ij+6u^2u_ju_j=2|D^2u|^2+6u^2|nabla u|^2,
          $$
          which is clearly nonnegative.






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            Edit: I will suppose $uin C^3$.



            It suffices to show that $v=|nabla u|^2$ is subharmonic, or $Delta vge 0$, since such functions satisfy the one-sided maximum principle $sup_Omega vle sup_partialOmegav$. We have
            $$
            Delta v=partial_ipartial_i(u_ju_j)=2partial_i(u_ju_ij)=2u_iju_ij+2u_ju_iij.
            $$
            To deal with the last term, we differentiate the Poisson equation: $Delta u_j=3u^2u_j$. Substituting this in, we get
            $$
            Delta v=2u_iju_ij+6u^2u_ju_j=2|D^2u|^2+6u^2|nabla u|^2,
            $$
            which is clearly nonnegative.






            share|cite|improve this answer

























              up vote
              1
              down vote













              Edit: I will suppose $uin C^3$.



              It suffices to show that $v=|nabla u|^2$ is subharmonic, or $Delta vge 0$, since such functions satisfy the one-sided maximum principle $sup_Omega vle sup_partialOmegav$. We have
              $$
              Delta v=partial_ipartial_i(u_ju_j)=2partial_i(u_ju_ij)=2u_iju_ij+2u_ju_iij.
              $$
              To deal with the last term, we differentiate the Poisson equation: $Delta u_j=3u^2u_j$. Substituting this in, we get
              $$
              Delta v=2u_iju_ij+6u^2u_ju_j=2|D^2u|^2+6u^2|nabla u|^2,
              $$
              which is clearly nonnegative.






              share|cite|improve this answer























                up vote
                1
                down vote










                up vote
                1
                down vote









                Edit: I will suppose $uin C^3$.



                It suffices to show that $v=|nabla u|^2$ is subharmonic, or $Delta vge 0$, since such functions satisfy the one-sided maximum principle $sup_Omega vle sup_partialOmegav$. We have
                $$
                Delta v=partial_ipartial_i(u_ju_j)=2partial_i(u_ju_ij)=2u_iju_ij+2u_ju_iij.
                $$
                To deal with the last term, we differentiate the Poisson equation: $Delta u_j=3u^2u_j$. Substituting this in, we get
                $$
                Delta v=2u_iju_ij+6u^2u_ju_j=2|D^2u|^2+6u^2|nabla u|^2,
                $$
                which is clearly nonnegative.






                share|cite|improve this answer













                Edit: I will suppose $uin C^3$.



                It suffices to show that $v=|nabla u|^2$ is subharmonic, or $Delta vge 0$, since such functions satisfy the one-sided maximum principle $sup_Omega vle sup_partialOmegav$. We have
                $$
                Delta v=partial_ipartial_i(u_ju_j)=2partial_i(u_ju_ij)=2u_iju_ij+2u_ju_iij.
                $$
                To deal with the last term, we differentiate the Poisson equation: $Delta u_j=3u^2u_j$. Substituting this in, we get
                $$
                Delta v=2u_iju_ij+6u^2u_ju_j=2|D^2u|^2+6u^2|nabla u|^2,
                $$
                which is clearly nonnegative.







                share|cite|improve this answer













                share|cite|improve this answer



                share|cite|improve this answer











                answered Aug 3 at 4:46









                user254433

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