[proofplan] We prove the variational characterisation by the direct method of calculus of variations (extracting a weakly convergent minimising sequence via Rellich-Kondrachov), show any first eigenfunction can be replaced by its absolute value (which also minimises the Rayleigh quotient since the Dirichlet energy is unchanged), apply the [Strong Maximum Principle](/theorems/102) to conclude strict positivity, and deduce simplicity from the fact that two strictly positive eigenfunctions cannot be linearly independent. [/proofplan] [step:Show the infimum of the Rayleigh quotient is achieved] Define $R(u) = B[u,u]/\|u\|_{L^2}^2$ and $\lambda_1 = \inf R(u)$. Let $\{u_m\}$ be a minimising sequence with $\|u_m\|_{L^2} = 1$. [claim:Boundedness of the Minimizing Sequence] $\{u_m\}$ is bounded in $H^1_0(U)$. [/claim] [proof] From ellipticity: $\theta\|\nabla u_m\|_{L^2}^2 \le B[u_m, u_m] + \|c\|_\infty \le (\lambda_1 + 1) + \|c\|_\infty$ for large $m$. [/proof] A subsequence satisfies $u_m \rightharpoonup w_1$ weakly in $H^1_0(U)$ and $u_m \to w_1$ strongly in $L^2(U)$ (by Rellich-Kondrachov). By weak lower semicontinuity, $B[w_1, w_1] \le \lambda_1$, so $R(w_1) = \lambda_1$. [/step] [step:Derive the Euler-Lagrange equation for the minimiser] [claim:Euler-Lagrange Equation] $B[w_1, v] = \lambda_1 (w_1, v)_{L^2}$ for all $v \in H^1_0(U)$. [/claim] [proof] The function $t \mapsto R(w_1 + tv)$ achieves its minimum at $t = 0$. Differentiating $\Phi(t) = B[w_1 + tv, w_1 + tv] - \lambda_1 \|w_1 + tv\|_{L^2}^2$ (with $\Phi(0) = 0$, $\Phi \ge 0$) and setting $\Phi'(0) = 0$ gives $B[w_1, v] = \lambda_1(w_1, v)_{L^2}$. [/proof] [/step] [step:Show the first eigenfunction is strictly positive via the maximum principle] [claim:Absolute Value is Also an Eigenfunction] $|w_1| \in H^1_0(U)$ and $B[|w_1|, |w_1|] = B[w_1, w_1]$. [/claim] [proof] Since $b_i = 0$ and $|\nabla|w_1|| = |\nabla w_1|$ a.e., \begin{align*} B[|w_1|, |w_1|] = \int_U \sum_{i,j} a_{ij}\,\partial_{x_i}|w_1|\,\partial_{x_j}|w_1| + c\,|w_1|^2 \, d\mathcal{L}^n = B[w_1, w_1]. \end{align*} So $|w_1|$ achieves the same Rayleigh quotient and is also a minimiser. [/proof] Replace $w_1$ by $|w_1| \ge 0$ with $L|w_1| = \lambda_1|w_1|$. Since $|w_1| \not\equiv 0$, the [Strong Maximum Principle](/theorems/102) gives $|w_1| > 0$ in $U$. Choose the sign so that $w_1 > 0$ in $U$. [/step] [step:Prove simplicity of $\lambda_1$] [claim:One-Dimensionality of the First Eigenspace] If $\tilde{w}$ is any eigenfunction for $\lambda_1$, then $\tilde{w} = \alpha w_1$ for some $\alpha \in \mathbb{R}$. [/claim] [proof] By the same argument, $|\tilde{w}| > 0$ in $U$. Consider $\phi = \tilde{w} - \alpha w_1$ where $\alpha = \tilde{w}(x_0)/w_1(x_0)$ for fixed $x_0 \in U$. Then $\phi(x_0) = 0$. If $\phi \not\equiv 0$, then $|\phi| > 0$ in $U$ by the maximum principle, contradicting $\phi(x_0) = 0$. Therefore $\phi \equiv 0$, i.e., $\tilde{w} = \alpha w_1$. [/proof] [/step]