Non-uniqueness of a Dirichlet's problem

Question

I found a statement in the book "Random Perturbations of Dynamical Systems" written by M.I.Freidlin and A.D. Wentzell that is not justified. Specifically, in a smooth domain $D \subset \mathbb{R}^n$ let us consider the Dirichlet's problem $$\mathcal{L}u(x) + c(x)u(x) = f(x),~~x\in D;\quad u(x)|_{x\in \partial D} = g(x),$$ in which $\mathcal{L}u(x) = \frac 12\sum_{i,j} a^{ij}(x)\frac{\partial^2 u}{\partial x^i\partial x^j}(x) + \sum_i b^i(x)\frac{\partial u}{\partial x^i}(x) $ with $a(x) = (a^{ij}(x))=\sigma(x)\sigma^T(x)$. Here $c(x)$, $f(x)$, for $x \in \mathbb R^n$, and $g(x)$, for $x\in \partial D$, are bounded continuous functions. If we suppose that $\mathcal{L}$ is uniformly elliptic, then under the additional assumption that $c(x) \leq 0$, this problem admits a solution provided by the Feynman-Kac formula, i.e., $$u(x) = -\mathbb{E}_x \int_0^\tau f(X_t)\exp\left\{\int_0^t c(X_s) ds\right\} + \mathbb{E}_x~ g(X_\tau)\exp\left\{\int_0^t c(X_s) ds\right\},$$ where $\tau = \inf\{t\colon X_t\notin D\}$ is the first exit time of the process defined via $dX_t = b(X_t)dt + \sigma(X_t)dW_t$ from the domain $D$. The authors claimed that "If $c(x) > 0$, then as is well known, Dirichlet's problem can "go out to the spectrum"; the solution of the equation $\mathcal{L}u + c(x)u = 0$ with vanishing boundary values may not be unique in this case." May I know why the last statement holds? Can you provide a baby example showing such non-uniqueness of solutions?

Answer 1

The simplest example is to just take $c$ to be a (constant) eigenvalue of the operator. For example, if we take $\mathcal{L} = \Delta$, then we can find an infinite sequence of eigenfunction/eigenvalue solution to $$ \begin{cases} -\Delta u_k = \lambda_k u_k &\text{in } D \\ u_k = 0 & \text{on } \partial D. \end{cases} $$ These can be found in a number of ways. For example, we can minimize the functional $E(u) = \int_D |\nabla u|^2$ over $H^1_0(D)$ subject to the constraint that $J(u) = \int_D |u|^2 =1$ in order to find the principal eigenvalue, $\lambda_0>0$.

Once we have these, we immediately see that we cannot have unique solutions to $$ \begin{cases} \Delta u + \lambda_k u = f &\text{in } D \\ u = 0 & \text{on } \partial D \end{cases} $$ for any $k$.