Definition
A Beppo‑Levi space is a functional‑analytic space consisting of locally integrable functions whose weak (distributional) derivatives belong to an $L^{p}$ space. Formally, for an open set $\Omega\subset\mathbb{R}^{n}$ and $1\le p<\infty$,
$$ BL^{p}(\Omega)=\Bigl{u\in L^{1}_{\text{loc}}(\Omega): abla u\in L^{p}(\Omega)\Bigr}/!\sim , $$
where the equivalence relation $u\sim v$ holds if $u-v$ is a (real) constant a.e. on $\Omega$. The space is often identified with the homogeneous Sobolev space $\dot{W}^{1,p}(\Omega)$ and equipped with the seminorm
$$ |u|{BL^{p}(\Omega)}:=| abla u|{L^{p}(\Omega)} . $$
Overview
Beppo‑Levi spaces arise in the study of partial differential equations, variational problems, and potential theory as the natural setting for functions whose energy—expressed by the Dirichlet integral $\int_\Omega |
abla u|^{p}$—is finite, while the functions themselves need not be square‑integrable. They are particularly useful when the underlying domain is unbounded, because the removal of additive constants eliminates the necessity of imposing global $L^{p}$ control on the function values.
The case $p=2$ yields a Hilbert space commonly called the energy space; it plays a central role in the theory of harmonic functions and the Dirichlet problem. For $p eq 2$ the spaces are Banach spaces and are employed in nonlinear elliptic theory.
Etymology / Origin
The name honors the Italian mathematician Giuseppe “Beppo” Levi (1875–1961), noted for his contributions to measure theory (including the monotone convergence theorem, often called the Beppo‑Levi theorem) and for early work on function spaces. The term “Beppo‑Levi space” appeared in the mid‑20th‑century literature on Sobolev‑type spaces, reflecting Levi’s influence on the foundational ideas of integrable derivatives and variational methods.
Characteristics
| Property | Description |
|---|---|
| Seminorm | $|u|_{BL^{p}(\Omega)}=| |
| abla u|_{L^{p}(\Omega)}$. The seminorm vanishes precisely on constants, which are identified as the zero element in the quotient. | |
| Completeness | The quotient space equipped with the above seminorm (or the induced norm after factoring out constants) is complete; consequently, $BL^{p}(\Omega)$ is a Banach space, and for $p=2$ a Hilbert space. |
| Density of smooth functions | If $\Omega$ satisfies a mild regularity condition (e.g., Lipschitz boundary), the set $C_{c}^{\infty}(\Omega)$ of smooth compactly supported functions is dense in $BL^{p}(\Omega)$ modulo constants. |
| Embedding | For bounded $\Omega$, the Poincaré inequality implies that the seminorm controls the $L^{p}$ norm modulo constants, yielding continuous embeddings into standard Sobolev spaces $W^{1,p}(\Omega)$. |
| Invariance | The space is invariant under translations and rotations of $\Omega$; the seminorm depends only on the gradient. |
| Relation to Sobolev spaces | $BL^{p}(\Omega)$ coincides with the homogeneous Sobolev space $\dot{W}^{1,p}(\Omega)$. When $\Omega=\mathbb{R}^{n}$ and $p<n$, functions in $BL^{p}(\mathbb{R}^{n})$ embed into the Lebesgue space $L^{p^{}}(\mathbb{R}^{n})$ with the Sobolev conjugate exponent $p^{}=np/(n-p)$. |
| Variational formulation | Energy functionals of the form $\int_{\Omega}F( |
| abla u),dx$ are naturally defined on $BL^{p}(\Omega)$, making the space suitable for weak formulations of elliptic PDEs. |
Related Topics
- Sobolev spaces $W^{k,p}(\Omega)$ and homogeneous Sobolev spaces $\dot{W}^{k,p}(\Omega)$
- Dirichlet integral and energy methods in PDEs
- Weak (distributional) derivatives
- Beppo‑Levi monotone convergence theorem (measure theory)
- Potential theory and harmonic functions
- Poincaré inequality
- Hilbert and Banach spaces
- Variational calculus and elliptic boundary‑value problems
These connections place Beppo‑Levi spaces within the broader framework of functional analysis and the modern theory of partial differential equations.