I am reading a paper, Is $L^2$ Physics-Informed Loss Always Suitable for Training Physics-Informed Neural Network?, which uses an "application" of Poincaré's inequality. The form I know and can find everywhere online is $∥u∥_p ≤ C∥Du∥_p$ , where $∥.∥_p$ is the $L^p$ norm on some compact domain, but the version they use is $∥u∥_{2,p} ≤ C∥D^2u∥_p$ where $∥.∥_{2,p}$ is the $W^{2,p}$ Sobolev norm on a compact domain and $D^2u$ is the vector: $D^2u_i=\frac{\partial^2u}{\partial x_i^2}$, and I can't figure out how they arrive to that. The assumption that $u$ dissapears on the boundary is still there. I've searched various literature like Jost's PDEs book, Evans PDE book, and Adam's Sobolev Spaces, and can't find the answer anywhere.

I am reading a paper, Is $L^2$ Physics-Informed Loss Always Suitable for Training Physics-Informed Neural Network?, which uses an "application" of Poincaré's inequality. The form I know and can find everywhere online is $$\lVert u\rVert_p ≤ C\lVert Du\rVert_p$$ where $\lVert.\rVert_p$ is the $L^p$ norm on some compact domain, but the version they use is $$\lVert u\rVert_{2,p} ≤ C\lVert D^2u\rVert_p$$ where $\lVert.\rVert_{2,p}$ is the $W^{2,p}$ Sobolev norm on a compact domain and $D^2u$ is the vector $D^2u_i=\frac{\partial^2u}{\partial x_i^2}$, and I can't figure out how they arrive at that. The assumption that $u$ dissapears on the boundary is still there. I've searched various literature like Jost's PDEs book, Evans' PDE book, and Adams' Sobolev Spaces, and can't find the answer anywhere.

I am reading a paper, https://arxiv.org/pdf/2206.02016.pdf, which uses an "application" of Poincaré's inequality. The form I know and can find everywhere online is $∥u∥_p ≤ C∥Du∥_p$ , where $∥.∥_p$ is the $L^p$ norm on some compact domain, but the version they use is $∥u∥_{2,p} ≤ C∥D^2u∥_p$ where $∥.∥_{2,p}$ is the $W^{2,p}$ Sobolev norm on a compact domain and $D^2u$ is the vector: $D^2u_i=\frac{\partial^2u}{\partial x_i^2}$, and I can't figure out how they arrive to that. The assumption that $u$ dissapears on the boundary is still there. I've searched various literature like Jost's PDEs book, Evans PDE book, and Adam's Sobolev Spaces, and can't find the answer anywhere.

I am reading a paper, Is $L^2$ Physics-Informed Loss Always Suitable for Training Physics-Informed Neural Network?, which uses an "application" of Poincaré's inequality. The form I know and can find everywhere online is $∥u∥_p ≤ C∥Du∥_p$ , where $∥.∥_p$ is the $L^p$ norm on some compact domain, but the version they use is $∥u∥_{2,p} ≤ C∥D^2u∥_p$ where $∥.∥_{2,p}$ is the $W^{2,p}$ Sobolev norm on a compact domain and $D^2u$ is the vector: $D^2u_i=\frac{\partial^2u}{\partial x_i^2}$, and I can't figure out how they arrive to that. The assumption that $u$ dissapears on the boundary is still there. I've searched various literature like Jost's PDEs book, Evans PDE book, and Adam's Sobolev Spaces, and can't find the answer anywhere.