高阶L^p空间技术问询：向量场L^p(Ω)^d的范数如何定义？

Great question! Since you already grasp the scalar $L^p$ norm, extending this to vector fields is a natural, intuitive generalization—let’s break down the standard definition clearly.

First, a quick recap of the scalar case for context: for a scalar function $v \in L^p(\Omega)$, the norm is defined as:
$$
\lVert v\rVert_{L^p(\Omega)} = \left[\int_\Omega |v|^p dx \right]^{1/p}
$$

For a vector field $v = (v_1, v_2, ..., v_d) \in L^p(\Omega)d$ (where $d$ is the dimension of the vector space, e.g., $d=2$ for 2D flow fields, $d=3$ for 3D elasticity), the standard $L^p$ norm follows the same core logic as the scalar case—we just first measure the "size" of the vector at each point in the domain, then integrate and apply the p-th root.

Mathematically, this is written as:
$$
\lVert v \rVert_{L^p(\Omega)d} = \left[ \int_\Omega \lVert v(x) \rVert_{\mathbb{R}^d}p dx \right]^{1/p}
$$

Here, $\lVert v(x) \rVert_{\mathbb{R}^d}$ refers to the p-norm of the vector $v(x) \in \mathbb{R}^d$ at a single point $x$, which expands to:
$$
\lVert v(x) \rVert_{\mathbb{R}^d}p = |v_1(x)|^p + |v_2(x)|^p + \dots + |v_d(x)|^p
$$

Substituting this expansion into the norm definition gives an equivalent form that uses the scalar $L^p$ norms of each vector component:
$$
\lVert v \rVert_{L^p(\Omega)d} = \left( \lVert v_1 \rVert_{L^p(\Omega)}p + \lVert v_2 \rVert_{L^p(\Omega)}p + \dots + \lVert v_d \rVert_{L^p(\Omega)}p \right)^{1/p}
$$

Key Details to Keep in Mind: #

• This is the most widely adopted definition in analysis, PDEs, and numerical methods—it retains all the essential properties of the scalar $L^p$ norm (triangle inequality, homogeneity, etc.) for vector-valued functions.
• In finite-dimensional spaces (which is nearly universal for physical applications), other equivalent norms exist, but this one is the most direct extension of the scalar case.
• For $p=2$, this simplifies to the familiar "square-integrable" vector field norm: sum the squared integrals of each component, then take the square root. This is ubiquitous in fields like fluid dynamics and electromagnetism.

If your preliminary idea aligned with this, you were absolutely on the right track! The core intuition is just applying the scalar norm logic to each point's vector, then aggregating that over the entire domain.

内容的提问来源于stack exchange，提问作者osolmaz