# Orthonormal basis of $L^2$ and it's impact on the solution to the heat equation.

If we consider the homogeneous Dirichlet eigenvalue problem on a bounded domain $\Omega\subset\Bbb R^n$ - (one part of my question is if we can assume $\Omega$ to be a Lipschitz domain and still expect to be able to solve the eigenvalue problem) - we obtain an orthonormal basis $\{\varphi_k\}_{k=1}^\infty$ of $L^2(\Omega)$ consisting of the eigenfunctions. Since $H^1(\Omega)\subset L^2(\Omega)$ we can express a $H^1$ function as an infinite sum of eigenfunctions.

Consider the following three heat equations, (Dirichlet, Neumann, Mixed)

\left\{\begin{align}u_t-\Delta u&=f,\quad (x,t)\in\Omega\times(0,T),\\ u&=0,\quad (x,t)\in\partial\Omega\times(0,T),\\ u|_{t=0}&=g,\quad x\in\Omega\end{align}\right.

\left\{\begin{align}u_t-\Delta u&=f,\quad (x,t)\in\Omega\times(0,T),\\ \frac{\partial u}{\partial n}&=0,\quad (x,t)\in\partial\Omega\times(0,T),\\ u|_{t=0}&=g,\quad x\in\Omega\end{align}\right.

\left\{\begin{align}u_t-\Delta u&=f,\quad (x,t)\in\Omega\times(0,T),\\ u&=0,\quad (x,t)\in\partial\Omega_1\times(0,T),\\ \frac{\partial u}{\partial n}&=0,\quad (x,t)\in\partial\Omega_2\times(0,T),\\ u|_{t=0}&=g,\quad x\in\Omega\end{align}\right.

In the mixed case $\partial\Omega=\partial\Omega_1\cup\partial\Omega_2$, and $n$ is the unit outward normal to whichever boundary portion is in consideration.

The weak formulation to each problem is posed on the space $H^1_0(\Omega)$, $H^1(\Omega)/\Bbb R=\{v\in H^1(\Omega):\int_\Omega v=0\}$, and $H^1_E(\Omega)=\{v\in H^1(\Omega):v|_{\partial\Omega_1}=0\}$, respectively. In this way I can see how the weak solutions differ in nature.

We can use Galerkin approximation to show existence to the three problems, by taking the orthonormal basis $\{\varphi_k\}_{k=1}^\infty$ from the eigenvalue problem, and writing the solution $u(x,t)=\sum_{k=1}^\infty c_k(t)\varphi_k$, and approximating by $u^N(x,t)=\sum_{k=1}^N c_k(t)\varphi_k$, plugging this into the weak formulation $$\int_\Omega u_t\varphi+\nabla u\cdot\nabla\varphi=\int_\Omega f\varphi$$ for all $\varphi$ in the respective spaces given above, arriving at a system of ODE's for which we determine the values $c_k(t)$, $k=1,\ldots, N$.

From here we have our approximate solution $u^N$, and by obtaining suitable energy estimates, we are allowed to take the limit $N\to\infty$, to obtain the solution $u$.

My question (asides from the necessary domain boundary regularity) is that since the weak formulation is the same, so is the system of PDE's for $c_k(t)$, so in what sense are the solutions to the Dirichlet and mixed Dirichlet-Neumann problem different (the difference between the pure Neumann problem and the others that is clear to me is the fact that the solution integrates to zero)?

To be more concise, it seems that the solutions should satisfy different BC's but the closed form of the solution (written as the sum $\sum_{k=1}^\infty c_k(t)\varphi_k$) seems to be the same.

• What problems do the eigenfunctions $\varpi_{k}$ satisfy? – DisintegratingByParts Jul 24 '15 at 2:09
• @TrialAndError, I've consider them all to solve the homogeneous dirichlet eigenvalue problem on $\Omega$ – Ellya Jul 24 '15 at 7:53
• The eigenfunctions you are using for a particular problem satisfy the homogeneous boundary conditions of that problem, right? That's how the boundary conditions are different. Without such conditions, you don't have an orthonormal basis, but you end up with a union of many orthonormal bases. For the Dirichlet problem, you use the eigenfunctions satisfying the Dirichlet conditions, etc. – DisintegratingByParts Jul 24 '15 at 10:36
• @TrialAndError, but doesn't the galerkin approximation method work even if I take the same Dirichlet eigenvalue problem basis for all problems? – Ellya Jul 24 '15 at 11:13
• @TrialAndError I get it now, thanks! Sometimes it takes saying the problem to realise how it works! – Ellya Jul 24 '15 at 11:14