11.5 Sobolev inequalities on manifolds

Sobolev inequalities on manifolds provide a powerful framework for analyzing functions and their derivatives in geometric settings. They extend classical results to curved spaces, offering crucial tools for studying partial differential equations and variational problems on manifolds.

These inequalities connect function spaces, embedding theorems, and geometric properties of manifolds. They play a key role in proving existence and regularity of solutions to PDEs, as well as in geometric analysis and the study of minimal surfaces and harmonic maps.

Sobolev spaces on manifolds

• Sobolev spaces extend the notion of differentiability to functions on manifolds, allowing for a more general framework to study PDEs and geometric analysis
• Sobolev spaces provide a natural setting for studying variational problems and energy functionals on manifolds

Definition of Sobolev spaces

• Let $M$ be a smooth, compact Riemannian manifold and $1 \leq p \leq \infty$
• For $k \in \mathbb{N}$, the Sobolev space $W^{k,p}(M)$ consists of functions $u \in L^p(M)$ such that $\nabla^j u \in L^p(M)$ for all $0 \leq j \leq k$, where $\nabla$ is the covariant derivative on $M$
• The norm on $W^{k,p}(M)$ is defined as $\|u\|_{W^{k,p}(M)} = \left(\sum_{j=0}^k \|\nabla^j u\|_{L^p(M)}^p\right)^{1/p}$
• For $p=2$, the Sobolev space $H^k(M) := W^{k,2}(M)$ is a Hilbert space with inner product $\langle u, v \rangle_{H^k(M)} = \sum_{j=0}^k \langle \nabla^j u, \nabla^j v \rangle_{L^2(M)}$

Properties of Sobolev spaces

• Sobolev spaces $W^{k,p}(M)$ are Banach spaces for $1 \leq p \leq \infty$ and $k \in \mathbb{N}$
• For $p=2$, $H^k(M)$ is a Hilbert space
• Sobolev spaces are reflexive for $1 < p < \infty$
• The space $C^\infty(M)$ of smooth functions is dense in $W^{k,p}(M)$ for $1 \leq p < \infty$
• Sobolev spaces satisfy interpolation inequalities, allowing for control of intermediate derivatives

Sobolev embedding theorems

• Sobolev embedding theorems relate the integrability of functions and their derivatives on manifolds
• For $M$ a compact Riemannian manifold of dimension $n$, the Sobolev embedding theorem states that if $kp > n$, then $W^{k,p}(M) \hookrightarrow C^{k-[n/p]}(M)$, where $[n/p]$ denotes the integer part of $n/p$
• In particular, for $p > n/k$, functions in $W^{k,p}(M)$ are continuous (Morrey's inequality)
• The Rellich-Kondrachov compactness theorem asserts that for $kp < n$, the embedding $W^{k,p}(M) \hookrightarrow L^q(M)$ is compact for $1 \leq q < p^* = np/(n-kp)$

Sobolev inequalities on manifolds

• Sobolev inequalities on manifolds provide a quantitative description of the Sobolev embedding theorems, giving explicit control on the norms of functions in different spaces
• These inequalities play a crucial role in the study of PDEs, geometric analysis, and the calculus of variations on manifolds

Gagliardo-Nirenberg-Sobolev inequality

• The Gagliardo-Nirenberg-Sobolev inequality on a compact Riemannian manifold $M$ of dimension $n$ states that for $1 \leq p < n$ and $1 \leq q \leq p^* = np/(n-p)$, there exists a constant $C > 0$ such that

$\|u\|_{L^q(M)} \leq C \|\nabla u\|_{L^p(M)}^{\theta} \|u\|_{L^r(M)}^{1-\theta}$ for all $u \in W^{1,p}(M)$, where $\theta = (n/p - n/q)/(1 - n/p + n/r)$ and $1 \leq r \leq \infty$

• This inequality interpolates between the Sobolev and Hölder inequalities, providing a refined control on the integrability of functions
• The Gagliardo-Nirenberg-Sobolev inequality is a key tool in proving existence and regularity of solutions to PDEs on manifolds

Trudinger-Moser inequality

• The Trudinger-Moser inequality is a limiting case of the Sobolev embedding theorem for $p = n$, providing an exponential integrability result
• For $M$ a compact Riemannian manifold of dimension $n$ and $\alpha > 0$, there exists a constant $C > 0$ such that

$\int_M e^{\alpha |u|^{n/(n-1)}} dV_g \leq C$ for all $u \in W^{1,n}(M)$ with $\|\nabla u\|_{L^n(M)} \leq 1$

• This inequality is sharp in the sense that the exponential integrability fails for exponents larger than $n/(n-1)$
• The Trudinger-Moser inequality has applications in the study of conformal geometry and the prescribed curvature problem

Concentration-compactness principle

• The concentration-compactness principle, introduced by P.L. Lions, is a powerful tool for analyzing the behavior of sequences of functions with bounded Sobolev norms
• It states that for a sequence $(u_k)$ bounded in $W^{1,p}(M)$, there are three possible scenarios:
  1. Compactness: a subsequence of $(u_k)$ converges strongly in $L^p(M)$
  2. Concentration: a subsequence of $(u_k)$ concentrates at a point, i.e., a significant portion of the $L^p$ norm is captured in a small neighborhood
  3. Dichotomy: $(u_k)$ can be split into a sum of two sequences, one exhibiting compactness and the other concentration
• The concentration-compactness principle is crucial in proving existence of solutions to PDEs and geometric variational problems, especially in the presence of critical exponents

Applications of Sobolev inequalities

• Sobolev inequalities on manifolds have numerous applications in various areas of mathematics, including partial differential equations, geometric analysis, and the calculus of variations
• They provide a fundamental tool for controlling the behavior of functions and their derivatives, leading to existence, regularity, and qualitative properties of solutions

Existence of solutions to PDEs

• Sobolev inequalities are essential in proving the existence of weak solutions to PDEs on manifolds using variational methods
• For example, consider the semilinear elliptic equation $-\Delta_g u + u = f(u)$ on a compact Riemannian manifold $(M, g)$, where $f$ satisfies certain growth conditions
• The Sobolev embedding theorems and the Trudinger-Moser inequality can be used to show that the associated energy functional satisfies the Palais-Smale condition, leading to the existence of a critical point, which corresponds to a weak solution of the PDE
• Similar techniques apply to a wide range of PDEs, including the Yamabe problem, the prescribed scalar curvature problem, and the Einstein-scalar field equations

Regularity of solutions

• Sobolev inequalities play a crucial role in establishing the regularity of weak solutions to PDEs on manifolds
• The Gagliardo-Nirenberg-Sobolev inequality and the Sobolev embedding theorems can be used in bootstrap arguments to improve the integrability and differentiability of solutions
• For instance, consider a weak solution $u \in H^1(M)$ to a semilinear elliptic equation
• By applying the Gagliardo-Nirenberg-Sobolev inequality and the Sobolev embedding theorems iteratively, one can show that $u \in W^{k,p}(M)$ for all $k \in \mathbb{N}$ and $p < \infty$, implying that $u$ is smooth by the Sobolev embedding theorem
• These techniques are fundamental in the regularity theory of elliptic and parabolic PDEs on manifolds

Geometric applications

• Sobolev inequalities have significant applications in geometric analysis and the study of the relationships between the geometry and topology of manifolds
• In the Yamabe problem, which asks whether every Riemannian metric on a compact manifold is conformal to a metric of constant scalar curvature, the Sobolev inequalities are used to control the conformal factor and prove the existence of a solution
• Sobolev inequalities also play a key role in the study of minimal surfaces and harmonic maps between Riemannian manifolds, providing estimates on the energy and regularity of these objects
• In the context of Ricci flow, Sobolev inequalities are used to control the evolution of geometric quantities and establish convergence results

Relationship with other inequalities

• Sobolev inequalities on manifolds are closely related to and can be derived from other fundamental inequalities in functional analysis and geometry
• These relationships provide additional insight into the properties of Sobolev spaces and the role of geometry in functional inequalities

Poincaré inequality on manifolds

• The Poincaré inequality on a compact Riemannian manifold $(M, g)$ states that there exists a constant $C > 0$ such that

$\|u - \bar{u}\|_{L^2(M)} \leq C \|\nabla u\|_{L^2(M)}$

for all $u \in H^1(M)$, where $\bar{u} = \frac{1}{\text{Vol}(M)} \int_M u dV_g$ is the average of $u$ over $M$

• This inequality implies that the $H^1(M)$ norm is equivalent to the norm $\left(\|\nabla u\|_{L^2(M)}^2 + |\bar{u}|^2\right)^{1/2}$
• The Poincaré inequality can be used to prove the Rellich-Kondrachov compactness theorem for Sobolev spaces on manifolds
• Generalizations of the Poincaré inequality, such as the Poincaré-Sobolev inequality, can be used to derive Sobolev inequalities on manifolds

Nash inequality on manifolds

• The Nash inequality on a compact Riemannian manifold $(M, g)$ of dimension $n$ states that there exists a constant $C > 0$ such that

$\|u\|_{L^2(M)}^{1+2/n} \leq C \|\nabla u\|_{L^2(M)} \|u\|_{L^1(M)}^{2/n}$ for all $u \in H^1(M)$

• This inequality is a key tool in the study of heat kernel estimates and the behavior of the heat equation on manifolds
• The Nash inequality can be used to derive the Sobolev inequality for $p = 2$ via interpolation with the Poincaré inequality

Log-Sobolev inequality on manifolds

• The log-Sobolev inequality on a compact Riemannian manifold $(M, g)$ states that there exists a constant $C > 0$ such that

$\int_M u^2 \log\left(\frac{u^2}{\|u\|_{L^2(M)}^2}\right) dV_g \leq C \|\nabla u\|_{L^2(M)}^2$ for all $u \in H^1(M)$ with $\|u\|_{L^2(M)} = 1$

• This inequality provides a strengthening of the Sobolev inequality for $p = 2$ and has applications in the study of the heat equation, Ricci curvature, and the Gaussian concentration of measure
• The log-Sobolev inequality is related to the notion of hypercontractivity of the heat semigroup and can be used to derive exponential decay estimates for solutions to the heat equation on manifolds

Extensions and generalizations

• Sobolev inequalities on manifolds can be extended and generalized in various directions, adapting to different geometric settings and incorporating fractional or higher-order derivatives
• These extensions provide a broader framework for studying PDEs and geometric problems in more general contexts

Sobolev inequalities on Riemannian manifolds

• Sobolev inequalities can be studied on complete, non-compact Riemannian manifolds with suitable geometric assumptions, such as bounds on the Ricci curvature or the injectivity radius
• In this setting, the inequalities often involve additional terms that depend on the geometry of the manifold, such as the isoperimetric profile or the heat kernel
• For example, on a complete Riemannian manifold with non-negative Ricci curvature, the Sobolev inequality for $p = 2$ takes the form

$\|u\|_{L^{2n/(n-2)}(M)} \leq C \|\nabla u\|_{L^2(M)}$ for all $u \in C_c^\infty(M)$, where the constant $C$ depends on the dimension $n$ and the volume growth of the manifold

• Sobolev inequalities on non-compact manifolds have applications in the study of the Yamabe problem, the structure of Riemannian manifolds with non-negative Ricci curvature, and the behavior of harmonic functions

Sobolev inequalities on sub-Riemannian manifolds

• Sub-Riemannian geometry is a generalization of Riemannian geometry, where the metric structure is defined only on a subbundle of the tangent bundle, leading to a constrained notion of length and geodesics
• Sobolev spaces on sub-Riemannian manifolds are defined using the horizontal gradient, which takes into account only the admissible directions of differentiation
• Sobolev inequalities in the sub-Riemannian setting involve the sub-Riemannian dimension, also known as the homogeneous dimension, which depends on the growth of the volume of sub-Riemannian balls
• For example, on the Heisenberg group $\mathbb{H}^n$, which is a fundamental example of a sub-Riemannian manifold, the following Sobolev inequality holds:

$\|u\|_{L^{Q/(Q-1)}(\mathbb{H}^n)} \leq C \|\nabla_H u\|_{L^1(\mathbb{H}^n)}$ for all $u \in C_c^\infty(\mathbb{H}^n)$, where $Q = 2n+2$ is the homogeneous dimension and $\nabla_H$ denotes the horizontal gradient

• Sub-Riemannian Sobolev inequalities have applications in the study of hypoelliptic PDEs, control theory, and the geometry of sub-Riemannian manifolds

Fractional Sobolev inequalities on manifolds

• Fractional Sobolev spaces on manifolds extend the notion of differentiability to non-integer orders, allowing for a more refined analysis of the regularity of functions
• For $s \in (0, 1)$ and $1 \leq p < \infty$, the fractional Sobolev space $W^{s,p}(M)$ on a compact Riemannian manifold $(M, g)$ is defined as the space of functions $u \in L^p(M)$ such that

$\iint_{M \times M} \frac{|u(x) - u(y)|^p}{d_g(x,y)^{n+sp}} dV_g(x) dV_g(y) < \infty,$

where $d_g$ denotes the Riemannian distance on $M$

• Fractional Sobolev inequalities on manifolds provide a bridge between the classical Sobolev inequalities and the isoperimetric inequality, which corresponds to the limiting case $s = 1$
• For example, on a compact Riemannian manifold of dimension $n$, the following fractional Sobolev inequality holds for $1 \leq p < n/s$:

$\|u\|_{L^{p^*}(M)} \leq C \left(\iint_{M \times M} \frac{|u(x) - u(y)|^p}{d_g(x,y)^{n+sp}} dV_g(x) dV_g(y)\right)^{1/p},$

where $p^* = np/(n-sp)$