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 $\epsilon \to 0$.

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    $\begingroup$ Your indices in your sum for $u_n$ are all messed up. $\endgroup$ – Ted Shifrin May 23 '17 at 0:20

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.

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    $\begingroup$ For the last part, we want to show that the weak limit of the subsequence of $$u_n = \sum_{k=1}^n \frac{\int_U fw_k}{\int_U \| \nabla w_k \|^2} w_k$$ satisfies the Poisson equation? Therefore, looking at $$\lim_{j \to \infty} \int_U \nabla u_{n_j} \cdot \nabla v dx = \lim_{j \to \infty} \int_U \nabla \left( \sum_{k=1}^n \frac{\int_U fw_k}{\int_U \| \nabla w_k \|^2} w_k \right) \nabla v dx,$$ for $v \in H_0^1(U)$? $\endgroup$ – user412674 May 23 '17 at 1:36
  • $\begingroup$ @Jeff Has user412674 deduced the correct thing? I'm not sure. $\endgroup$ – Phillip Wiggins May 26 '17 at 4:23
  • $\begingroup$ @Jeff I'm still confused by this weak convergence, I've tried integrating by parts, but I still don't get the Poisson condition. Also, what do the $w_k \to ?$ $\endgroup$ – Phillip Wiggins May 28 '17 at 3:40
  • $\begingroup$ To show $u$ is a weak solution of Poisson's equation you need to show that $\int_U \nabla u \cdot \nabla v \, dx = \int_U f v\, dx$ for all $v \in H^1(U)$. No integration by parts is required. Using (*) from my answer and weak convergence of $u_{n_j}$ it is immediate. $\endgroup$ – Jeff May 28 '17 at 3:51
  • $\begingroup$ @Bombyxmori Riesz representation theorem is not useful here. The $w_k$ do not converge to anything; they are indeed the eigenfunctions for the Laplacian and so form a basis for $H^1$. That is, your test functions can be taken from the linear span of the $w_k$. $\endgroup$ – Jeff May 28 '17 at 4:10

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 $\varepsilon=\frac12 c(\Omega)$.

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  • $\begingroup$ Yes $U$ is bounded, so the estimate is fine. Since $I[u]$ is the bounded below, is it sufficient to then show that $u_m$ is a decreasing sequence in order to obtain the weak convergence of the minimum? $\endgroup$ – Phillip Wiggins May 23 '17 at 0:20
  • $\begingroup$ I am not sure I understand what you are trying to do. Are you taking $u_m$ in such a way that $I[u_m]\to\inf_{u\in H^1_0(U)} I$? In that case, then the bound from below tells you that the sequence $\{u_m\}$ is bounded in $H^1_0(U)$ and so a subsequence converges weakly $\endgroup$ – Gio67 May 23 '17 at 0:25
  • $\begingroup$ I want to show that the sequence I have defined above has a subsequence that converges weakly to a solution of the Poisson equation with zero Dirichlet boundary conditions. This is an intermediate step $\endgroup$ – Phillip Wiggins May 23 '17 at 0:27
  • $\begingroup$ What is not clear is how you define the sequence. Writing $u_n=\sum (u_n,w_n)^k w_k$ does not define $u_n$. It is circular. You have $u_n$ on both sides. $\endgroup$ – Gio67 May 23 '17 at 0:32
  • $\begingroup$ I'm not sure, this is how Evan's writes it. See page 357 for reference. In particular, he write "As $\{ w_k \}_{k=1}^{\infty}$ is an orthonormal basis of $L^2(U)$, if $u \in H_0^1(U)$, and $\| u \|_{L^2(U)} =1$, we can write $$u = \sum_{k=1}^{\infty} d_kw_k$$ for $d_k = (u,w_k)_{L^2(U)}$". $\endgroup$ – Phillip Wiggins May 23 '17 at 0:44

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