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Another argument perhaps worth mentioning uses the maximum principle. To illustrate the idea we assume that $\Omega$ is a bounded smooth domain, that $L = \Delta$, and that we are dealing with Dirichlet eigenfunctions.

Let $u_0$ be a positive eigenfunction corresponding to the smallest eigenvalue of $L$. We may assume after multiplying by a positive constant that $u_0$ either touches $u$ from below in $\Omega$, or lies below $u$ and is tangent to $u$ at a boundary point. Applying the strong maximum principle in the first case or the Hopf lemma in the second one to the nonnegative supersolution $u - u_0$, we conclude that $u = u_0$.

Another argument perhaps worth mentioning uses the maximum principle. To illustrate the idea we assume that $\Omega$ is a smooth bounded domain, that $L = \Delta$, and that we are dealing with Dirichlet eigenfunctions.

Let $u_0$ be a positive eigenfunction corresponding to the smallest eigenvalue of $L$. We may assume after multiplying by a positive constant that $u_0$ either touches $u$ from below in $\Omega$, or lies below $u$ and agrees with $u$ to first order at a boundary point. Applying the strong maximum principle in the first case or the Hopf lemma in the second one to the nonnegative supersolution $u - u_0$, we conclude that $u = u_0$.

Another argument perhaps worth mentioning uses the maximum principle. Let $u_0$ be a positive eigenfunction corresponding to the smallest eigenvalue. We may assume after multiplying by a positive constant that $u_0$ either touches $u$ from below in $\Omega$, or lies below $u$ and is tangent to $u$ at a boundary point. Applying the strong maximum principle in the first case or the Hopf lemma in the second one to the nonnegative supersolution $u - u_0$, we conclude that $u = u_0$. (This argument requires sufficient regularity of $\partial \Omega$, e.g. $C^2$, to apply the Hopf lemma).

Another argument perhaps worth mentioning uses the maximum principle. To illustrate the idea we assume that $\Omega$ is a bounded smooth domain, that $L = \Delta$, and that we are dealing with Dirichlet eigenfunctions.

Let $u_0$ be a positive eigenfunction corresponding to the smallest eigenvalue of $L$. We may assume after multiplying by a positive constant that $u_0$ either touches $u$ from below in $\Omega$, or lies below $u$ and is tangent to $u$ at a boundary point. Applying the strong maximum principle in the first case or the Hopf lemma in the second one to the nonnegative supersolution $u - u_0$, we conclude that $u = u_0$.

Another argument perhaps worth mentioning uses the maximum principle. If $u_0$ is a (positive) eigenfunction corresponding to the smallest eigenvalue, then we may assume after multiplying by a positive constant that $u_0$ either touches $u$ from below in $\Omega$, or lies below $u$ and is tangent to $u$ at a boundary point. Applying the strong maximum principle in the first case or the Hopf lemma in the second one to the nonnegative supersolution $u - u_0$, we conclude that $u = u_0$. (This argument requires sufficient regularity of $\partial \Omega$, e.g. $C^2$, to apply the Hopf lemma).

Another argument perhaps worth mentioning uses the maximum principle. Let $u_0$ be a positive eigenfunction corresponding to the smallest eigenvalue. We may assume after multiplying by a positive constant that $u_0$ either touches $u$ from below in $\Omega$, or lies below $u$ and is tangent to $u$ at a boundary point. Applying the strong maximum principle in the first case or the Hopf lemma in the second one to the nonnegative supersolution $u - u_0$, we conclude that $u = u_0$. (This argument requires sufficient regularity of $\partial \Omega$, e.g. $C^2$, to apply the Hopf lemma).