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Eigenvalues and Eigenfunctions of Elliptic Operators

Nov 14, 20241 min read

PDEsFunctionalAnalysis Assume is connected and given the differential operator of an Elliptic PDE:

  • Eigenvalues and eigenfunctions for the divergence form (symmetric): For , the eigenvalues of are
    1. Positive and real
    2. Countable with
    3. Corresponding eigenfunctions form an orthonormal basis of
    4. The principle eigenvalue:
Misplaced & \lambda_1 :&= \min_k \{ \lambda_k \} \\ &= \min_{u \in H_0^1; u \neq 0} \{ \frac{B(u,u)}{||u||_{L^2}} \} \\ &= \min_{u \in H_0^1; ||u||_{L^2} = 1} \{ B(u,u) \} \\ \end{aligned}$$ is simple: $\forall u$ s.t. $Lu = \lambda_1 u \implies \exists \space c$ s.t $u = cw_1$ - **Eigenvalues and eigenfunctions for the non-divergence form** (non-symmetric): For $A^{ij}, b^i, c \in C^\infty(\overline{U})$ and $U$ also bounded and connected with $\partial U \in C^\infty$, the eigenvalues/eigenfunctions are in general complex and the principle eigenvalue: 1. $\lambda_1 \in \mathbb{R}^+$ 2. $\Re({\lambda_k}) \geq \lambda_1$ 3. $w_1 \geq 0$ in $U$ 4. $\lambda_1$ is simple: $\forall u$ s.t. $Lu = \lambda_1 u \implies \exists \space c$ s.t $u = cw_1$

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