Background: In our PDE class we explored the heat equation with Dirichlet boundary condition
$$u_t – \Delta u = 0 \;\text{ in } \Omega \subset \mathbb{R}^n \;\text{bounded}\\
u = u_0(x) \;\,\text{at} \; \,t=0\\
u = 0 \;\, \text{at} \;\, \partial \Omega$$
Multiplying by $u$ on both sides and using the divergence theorem gives us
$$\frac12 \frac{d}{dt}\int_{\Omega}u^2 + \int_{\Omega}|\nabla u|^2 = 0 \tag{1}$$
Now since the function $u$ is zero on the boundary, we can employ Poincare's inequality and say that
$$-\int_{\Omega}|\nabla u|^2 \leq \frac{-1}{C_{\Omega}}\int_{\Omega}|u|^2,$$
which together with (1) gives that
$$\frac12 \frac{d}{dt}\int_{\Omega}u^2 \leq \frac{-1}{C_{\Omega}}\int_{\Omega}|u|^2,$$
so $\int_{\Omega}u^2$ decays exponentially to $0$ as $t \to \infty$.
Question: Having concluded that $\int_{\Omega}u^2$ decays exponentially, we are asked to prove an analogous statement for a homogeneous Neumann boundary condition which says that
$$\frac{\partial u}{\partial \vec{\nu}}=0 \;\, \text{at} \;\, \partial\Omega, $$
for $\vec{\nu}$ the normal vector to the boundary. We are given the hint that there is a constant $M_{\Omega}$ with the property that if $u: \Omega \to \mathbb{R}$ has mean value $0$ then $\int_{\Omega} u^2 \, dx \leq M_{\Omega} \int_{\Omega} |\nabla u|^2 \, dx$.
Now, I understand $u$ having mean value zero to mean that $\int_{\Omega} u \, dx=0$. One way to achieve that might be to subtract (the function of $t$) $\int_{\Omega}u\,dx$ from $u$, so that $v : = u – \int_{\Omega}u\,dx$ has mean value zero for all $t$. But this seems to screw things up when we try to square $v$.
How can we use the Neumann boundary condition to find out something regarding the mean value of $u$? For instance, the divergence theorem would tell us
$$\int_{\partial \Omega} u\cdot \vec{\nu} \, ds = \int_{\Omega} \nabla \cdot u \, dx.$$
The left side here is zero, I believe, but I'm not sure that tells us anything about the mean value.
Could someone help me see the next step? I'm stuck.
Best Answer
The divergence theorem is a good idea, but the way you wrote it down doesn't make sense - you're treating $u$ as a vector field when it is actually a scalar function. The vector field you want to study is $\nabla u$ - the divergence theorem is $$\int_{\partial \Omega} \nabla u \cdot \nu = \int_\Omega \Delta u.$$ The LHS is $0$ by the boundary conditions and the RHS is $$\int_\Omega \frac{\partial u}{\partial t} = \frac d{dt} \int_\Omega u$$ by the PDE (and some regularity assumptions + convergence theorem to exchange the derivative and integral). Thus the mean value of $u$ is constant in time.