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hotdog
hotdog
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Let $\Omega \subset \mathbb{R}^N$ be bounded domain with smooth boundary. The first eigenvalue of $-\Delta \colon W_0^{2,2} \rightarrow L^2$$-\Delta \colon W_0^{1,2}(\Omega) \cap W^{2,2}(\Omega) \rightarrow L^2(\Omega)$ is strictly positive. This is well known and proved by using the energy method. But I learned that $-\Delta$ can be extended up to $L^1$$L^1(\Omega)$. In this case, is the first eigenvalue still strictly positive?

The first eigenvalue of $-\Delta \colon W_0^{2,2} \rightarrow L^2$ is strictly positive. This is well known and proved by using the energy method. But I learned that $-\Delta$ can be extended up to $L^1$. In this case, is the first eigenvalue still strictly positive?

Let $\Omega \subset \mathbb{R}^N$ be bounded domain with smooth boundary. The first eigenvalue of $-\Delta \colon W_0^{1,2}(\Omega) \cap W^{2,2}(\Omega) \rightarrow L^2(\Omega)$ is strictly positive. This is well known and proved by using the energy method. But I learned that $-\Delta$ can be extended up to $L^1(\Omega)$. In this case, is the first eigenvalue still strictly positive?

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Ben McKay
Ben McKay
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eigenvalues Eigenvalues of laplacianthe Laplacian in L^1$L^1$ space

theThe first eigenvalue of -¥Delta : W_0^2,2 ¥rightarrow L^2$-\Delta \colon W_0^{2,2} \rightarrow L^2$ is strictly positive. thisThis is well known and proved by using the energy method. but iBut I learned that -¥Delta$-\Delta$ can be extended up to L^1$L^1$. inIn this case, is the first eigenvalue still strictly positive?

eigenvalues of laplacian in L^1 space

the first eigenvalue of -¥Delta : W_0^2,2 ¥rightarrow L^2 is strictly positive. this is well known and proved by using energy method. but i learned -¥Delta can be extended up to L^1. in this case, is the first eigenvalue still strictly positive?

Eigenvalues of the Laplacian in $L^1$ space

The first eigenvalue of $-\Delta \colon W_0^{2,2} \rightarrow L^2$ is strictly positive. This is well known and proved by using the energy method. But I learned that $-\Delta$ can be extended up to $L^1$. In this case, is the first eigenvalue still strictly positive?

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hotdog
hotdog
  • 31
  • 3

eigenvalues of laplacian in L^1 space

the first eigenvalue of -¥Delta : W_0^2,2 ¥rightarrow L^2 is strictly positive. this is well known and proved by using energy method. but i learned -¥Delta can be extended up to L^1. in this case, is the first eigenvalue still strictly positive?