PDE Estimate for Poisson Equation

Question

Consider the energy functional associated to the Poisson equation $$I[u] = \int_U \left| Du \right|^2 dx + \int fu dx,$$ where $f \in L^2$ and $u \in H_0^1$. I want to ensure that if I take a sequence $u_n = \sum_{k=1}^n (u_n, w_n)_{L^2(U)}^k w_k$, where the $w_n$ are orthonormalised eigenvectors of the Laplacian with $\int_U \nabla u_n \cdot \nabla w_k dx = \int_U fw_k$ that $$I[u_m] \to \inf_{u \in H_0^1(U)}I[u]$$ I've first tried showing that the infimum is finite. I've currently got the estimate $$I[u] \geq \int_U \left| \nabla u \right|^2 dx - \epsilon \int_U u^2 - \frac{1}{4\epsilon} \int_U f^2 dx$$ $$\geq \int_U \left| \nabla u \right|^2 dx - \text{const}\int_U \left| \nabla u \right|^2 - \frac{1}{4\epsilon} \int_U f^2$$ but I can't control the $f$ term since it will blow up as $\epsilon \to 0$.

Answer 1

You don't need to send $\varepsilon$ to zero. I hope $U$ is bounded. Then you have Poincare's inequality $$\int_U|u|^2dx\le c(\Omega)\int_U|\nabla u|^2dx$$ and so you only need to take $\varepsilon=\frac12 c(\Omega)$.

Answer 2

I suspect you are doing an exercise in Evans proving existence of solutions of Poisson's equation via the Galerkin method. The idea is to choose $u_n=\sum_{k=1}^n d_k w_k$ to satisfy

$$\int_U \nabla u_n \cdot \nabla w_k \, dx = \int_U f w_k\, dx \ \ \ \ \ \ \ (*)$$

for $k=1,\dots,n$. If you plug in the definition of $u_n$ you get that

$$d_k\int_U \|\nabla w_k\|^2 \, dx = \int_U f w_k \, dx$$

due to the orthogonality of the $w_k$. This defines $d_k$ as

$$d_k = \frac{\int_U f w_k}{\int_U \|\nabla w_k\|^2}.$$

Now you want to show that $u_n$ converges weakly in $H^1$ to a solution of Poisson's equation. Just show it is bounded in $H^1$ and then extract a subsequence converging weakly, and show the limit of any such subsequence is a weak solution of Poisson's equation. Use uniqueness to show the entire sequence converges. I can give more help if you have specific questions.