Estimating the $L^p$ norm of a second derivative of a solution of the Laplace equation

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Consider Dirichlet boundary value problem on unit disk : $u_xx+u_yy=0$.



Then, is there a constant $c>0$ satisfying the following?



-For every solution $u$ that $int_0^2pileft|fracd^2u(e^it)dt^2right|^p dtle 1$, $|u_xx|_ple c$



(as long as the all integrals are well-defined)



P.S. You may assume $u$ to be either strong or classical(or else), as you please.




pde harmonic-functions boundary-value-problem




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    Looks reasonable. Looking at the tangential derivative, we see it convolved with the Poisson kernel, so that's bounded. The derivative in normal direction is more problematic, but since the integral is over the disk, it should not be too bad.
    – user357151
    Jul 29 at 6:26














up vote
3
down vote

favorite
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Consider Dirichlet boundary value problem on unit disk : $u_xx+u_yy=0$.



Then, is there a constant $c>0$ satisfying the following?



-For every solution $u$ that $int_0^2pileft|fracd^2u(e^it)dt^2right|^p dtle 1$, $|u_xx|_ple c$



(as long as the all integrals are well-defined)



P.S. You may assume $u$ to be either strong or classical(or else), as you please.




pde harmonic-functions boundary-value-problem




share|cite|improve this question

















  • 1




    Looks reasonable. Looking at the tangential derivative, we see it convolved with the Poisson kernel, so that's bounded. The derivative in normal direction is more problematic, but since the integral is over the disk, it should not be too bad.
    – user357151
    Jul 29 at 6:26












up vote
3
down vote

favorite
2









up vote
3
down vote

favorite
2






2





Consider Dirichlet boundary value problem on unit disk : $u_xx+u_yy=0$.



Then, is there a constant $c>0$ satisfying the following?



-For every solution $u$ that $int_0^2pileft|fracd^2u(e^it)dt^2right|^p dtle 1$, $|u_xx|_ple c$



(as long as the all integrals are well-defined)



P.S. You may assume $u$ to be either strong or classical(or else), as you please.




pde harmonic-functions boundary-value-problem




share|cite|improve this question













Consider Dirichlet boundary value problem on unit disk : $u_xx+u_yy=0$.



Then, is there a constant $c>0$ satisfying the following?



-For every solution $u$ that $int_0^2pileft|fracd^2u(e^it)dt^2right|^p dtle 1$, $|u_xx|_ple c$



(as long as the all integrals are well-defined)



P.S. You may assume $u$ to be either strong or classical(or else), as you please.





pde harmonic-functions boundary-value-problem





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited Jul 29 at 4:50
























asked Jul 28 at 8:16









C.Park

1397




1397







  • 1




    Looks reasonable. Looking at the tangential derivative, we see it convolved with the Poisson kernel, so that's bounded. The derivative in normal direction is more problematic, but since the integral is over the disk, it should not be too bad.
    – user357151
    Jul 29 at 6:26












  • 1




    Looks reasonable. Looking at the tangential derivative, we see it convolved with the Poisson kernel, so that's bounded. The derivative in normal direction is more problematic, but since the integral is over the disk, it should not be too bad.
    – user357151
    Jul 29 at 6:26







1




1




Looks reasonable. Looking at the tangential derivative, we see it convolved with the Poisson kernel, so that's bounded. The derivative in normal direction is more problematic, but since the integral is over the disk, it should not be too bad.
– user357151
Jul 29 at 6:26




Looks reasonable. Looking at the tangential derivative, we see it convolved with the Poisson kernel, so that's bounded. The derivative in normal direction is more problematic, but since the integral is over the disk, it should not be too bad.
– user357151
Jul 29 at 6:26















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