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Given a Laplace equation $u_{xx}+u_{yy}=0$ in $\mathbb{R}^2$ with initial conditions $u(x,0)=u_0(x)$, $u_y(x,0)=u_1(x)$. Is this problem uniquely solvable in general? Does someone have a coutnerexample if not?

Thanks very much!

Giuseppe Negro
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JohnSmith
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    You might get somewhere by noting that your equation is the wave equation in imaginary time. So maybe try to use the D'Alembert formula, replacing $t$ with $ix$ everywhere, and see what you get. (Also, a nitpick: this question doesn't actually make sense if you don't specify what spaces $u_0$ and $u_1$ come from.) – Ian Apr 15 '15 at 12:53
  • The problem looks overdetermined. The function $u_0$ alone already determines $u$ in the upper half space via the Poisson integral: http://en.wikipedia.org/wiki/Poisson_kernel#On_the_upper_half-plane – Giuseppe Negro Apr 15 '15 at 12:58
  • I'm sorry, the first order derivative should be w.r.t. $y$ (just the usual Cauchy data) - I have corrected this. – JohnSmith Apr 15 '15 at 13:00
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    @GiuseppeNegro: I think the Poisson integral only gives you a unique $L^p$ solution or something like that. $u_0$ doesn't uniquely determine a solution in general: for $u(x,0) = 0$ we have the solutions $u(x,y) = 0$ and $u(x,y) = y$. – Nate Eldredge Apr 15 '15 at 15:04
  • @NateEldredge That's an effect of the domain being unbounded, if I remember correctly. – Ian Apr 15 '15 at 15:39
  • Hint: compare the difference of the form of the general solutions between http://eqworld.ipmnet.ru/en/solutions/npde/npde2103.pdf and http://eqworld.ipmnet.ru/en/solutions/npde/npde3103.pdf. – doraemonpaul May 09 '15 at 20:39

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You can just use the D'Alembert's formula:

$u(x,y)=\dfrac{u_0(x+iy)+u_0(x-iy)}{2}-\dfrac{i}{2}\int_{x-iy}^{x+iy}u_1(t)~dt$

doraemonpaul
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