0
$\begingroup$

Suppose we consider the heat equation $$\partial_t u = \Delta u, x \in \text{int}D^2, t > 0$$ where $D^2$ is the closed unit disc in $\mathbb{R}^2$, subject to Neumann type boundary conditions $$\partial_\eta u(x, t) = A(t), x \in \partial M, t > 0$$ and the initial condition $$u(x, 0) = u_0(x), x \in D^2$$ Do we necessarily have to have $$\partial_\eta u(x, 0) = \lim_{t \to 0} A(t) ?$$ Is the answer same for the wave equation $$\partial^2_t u = \Delta u$$ with the same initial and boundary conditions?

$\endgroup$
2
  • $\begingroup$ The choice of $u_0$ has nothing whatsoever to do with the bc (say $A\equiv 0$, and take any $u_0$ that does not satisfy Neumann bc's). Maybe I'm not understanding the question correctly? $\endgroup$ Jul 6, 2014 at 1:14
  • $\begingroup$ If the initial condition does not satisfy the boundary condition, your solution will not satisfy the boundary condition for $t=0$, but will satisfy it at other times, if the conditions are appropriate. $\endgroup$ Jul 6, 2014 at 5:49

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.