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 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
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Let M be a smooth, compact Riemannian manifold and 1≤p≤∞
For k∈N, the Sobolev space Wk,p(M) consists of functions u∈Lp(M) such that ∇ju∈Lp(M) for all 0≤j≤k, where ∇ is the covariant derivative on M
The norm on Wk,p(M) is defined as ∥u∥Wk,p(M)=(∑j=0k∥∇ju∥Lp(M)p)1/p
For p=2, the Sobolev space Hk(M):=Wk,2(M) is a Hilbert space with inner product ⟨u,v⟩Hk(M)=∑j=0k⟨∇ju,∇jv⟩L2(M)
Properties of Sobolev spaces
Sobolev spaces Wk,p(M) are Banach spaces for 1≤p≤∞ and k∈N
For p=2, Hk(M) is a Hilbert space
Sobolev spaces are reflexive for 1<p<∞
The space C∞(M) of smooth functions is dense in Wk,p(M) for 1≤p<∞
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 states that if kp>n, then Wk,p(M)↪Ck−[n/p](M), where [n/p] denotes the integer part of n/p
In particular, for p>n/k, functions in Wk,p(M) are continuous (Morrey's inequality)
The Rellich-Kondrachov compactness theorem asserts that for kp<n, the embedding Wk,p(M)↪Lq(M) is compact for 1≤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 on a compact Riemannian manifold M of dimension n states that for 1≤p<n and 1≤q≤p∗=np/(n−p), there exists a constant C>0 such that
∥u∥Lq(M)≤C∥∇u∥Lp(M)θ∥u∥Lr(M)1−θ
for all u∈W1,p(M), where θ=(n/p−n/q)/(1−n/p+n/r) and 1≤r≤∞
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 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 α>0, there exists a constant C>0 such that
∫Meα∣u∣n/(n−1)dVg≤C
for all u∈W1,n(M) with ∥∇u∥Ln(M)≤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 problem
Concentration-compactness principle
The , 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 (uk) bounded in W1,p(M), there are three possible scenarios:
Compactness: a subsequence of (uk) converges strongly in Lp(M)
Concentration: a subsequence of (uk) concentrates at a point, i.e., a significant portion of the Lp norm is captured in a small neighborhood
Dichotomy: (uk) 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 −Δgu+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∈H1(M) to a semilinear elliptic equation
By applying the Gagliardo-Nirenberg-Sobolev inequality and the Sobolev embedding theorems iteratively, one can show that u∈Wk,p(M) for all k∈N and p<∞, 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 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 on a compact Riemannian manifold (M,g) states that there exists a constant C>0 such that
∥u−uˉ∥L2(M)≤C∥∇u∥L2(M)
for all u∈H1(M), where uˉ=Vol(M)1∫MudVg is the average of u over M
This inequality implies that the H1(M) norm is equivalent to the norm (∥∇u∥L2(M)2+∣uˉ∣2)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 on a compact Riemannian manifold (M,g) of dimension n states that there exists a constant C>0 such that
∥u∥L2(M)1+2/n≤C∥∇u∥L2(M)∥u∥L1(M)2/n
for all u∈H1(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 on a compact Riemannian manifold (M,g) states that there exists a constant C>0 such that
∫Mu2log(∥u∥L2(M)2u2)dVg≤C∥∇u∥L2(M)2
for all u∈H1(M) with ∥u∥L2(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∥L2n/(n−2)(M)≤C∥∇u∥L2(M)
for all u∈Cc∞(M), where the constant C depends on the dimension n and the volume growth of the manifold
Sobolev inequalities on non- 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 Hn, which is a fundamental example of a sub-Riemannian manifold, the following Sobolev inequality holds:
∥u∥LQ/(Q−1)(Hn)≤C∥∇Hu∥L1(Hn)
for all u∈Cc∞(Hn), where Q=2n+2 is the homogeneous dimension and ∇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∈(0,1) and 1≤p<∞, the fractional Sobolev space Ws,p(M) on a compact Riemannian manifold (M,g) is defined as the space of functions u∈Lp(M) such that
∬M×Mdg(x,y)n+sp∣u(x)−u(y)∣pdVg(x)dVg(y)<∞,
where dg denotes the Riemannian distance on M
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≤p<n/s:
∥u∥Lp∗(M)≤C(∬M×Mdg(x,y)n+sp∣u(x)−u(y)∣pdVg(x)dVg(y))1/p,
where p∗=np/(n−sp)
Key Terms to Review (26)
Boundary conditions: Boundary conditions are specific constraints or requirements placed on the values of a function or its derivatives at the boundaries of a domain. They are crucial in various mathematical and physical contexts, as they help define the behavior of solutions, particularly in problems involving differential equations and variational principles.
Compact manifolds: Compact manifolds are topological spaces that are both compact and smooth, meaning they are closed and bounded, which makes them finite in extent and without edges. These structures are crucial because they allow for a variety of powerful mathematical results and tools to be applied, particularly in geometric analysis and differential geometry. The compactness property often ensures that certain limits exist, leading to significant implications in both volume comparison theorems and the establishment of Sobolev inequalities.
Complete Riemannian Manifolds: Complete Riemannian manifolds are smooth manifolds equipped with a Riemannian metric that allows for every geodesic to be extended indefinitely. This completeness ensures that any Cauchy sequence of points in the manifold converges to a limit within the manifold, making them crucial for understanding geometric properties and implications, such as volume comparison and Sobolev inequalities.
Concentration-Compactness Principle: The concentration-compactness principle is a fundamental concept in the analysis of partial differential equations and variational methods, which addresses the issue of compactness in the context of weak convergence. This principle helps to identify the conditions under which sequences of functions can exhibit concentration phenomena, allowing for the identification of limits and minimizing behaviors of functionals, particularly in the study of Sobolev inequalities on manifolds.
Curvature: Curvature is a measure of how a geometric object deviates from being flat or straight, often quantified in terms of the bending of surfaces or curves in a space. It helps to understand the intrinsic and extrinsic properties of shapes and spaces, revealing how they relate to concepts such as distance, angles, and the overall structure of geometric forms.
Diameter: The diameter of a metric space is defined as the greatest distance between any two points within that space. This concept not only helps in understanding the geometric properties of spaces but also connects to essential results regarding curvature and volume in the context of Riemannian geometry and analysis on manifolds.
Dimensional Constraints: Dimensional constraints refer to limitations or conditions that define the structure and properties of geometric objects within a specific dimensional space. These constraints are essential in understanding how Sobolev inequalities behave on manifolds, as they dictate the relationships between various function spaces and the dimensions of the underlying manifold.
Existence of Minimizers: The existence of minimizers refers to the property that certain functionals, particularly those defined on Sobolev spaces, attain their minimum values under specified conditions. This concept is fundamental in the analysis of variational problems on manifolds, where one seeks to find functions that minimize an integral functional, often related to energy or distance.
Fractional Sobolev inequalities on manifolds: Fractional Sobolev inequalities on manifolds provide a way to estimate norms of functions based on their fractional derivatives, extending classical Sobolev inequalities to accommodate more complex geometries. These inequalities are crucial for understanding the behavior of functions in a manifold setting, particularly in relation to embedding theorems and regularity properties of solutions to partial differential equations.
Friedrichs Inequality: Friedrichs Inequality is a mathematical statement that provides a bound on the norm of a function in terms of the norm of its gradient. This inequality plays a significant role in the theory of Sobolev spaces, particularly in establishing compact embeddings and various functional inequalities on manifolds, which are essential for understanding the behavior of functions under certain constraints.
Gagliardo-Nirenberg-Sobolev Inequality: The Gagliardo-Nirenberg-Sobolev inequality is a fundamental result in the field of functional analysis and partial differential equations, providing bounds for norms of functions in terms of their derivatives. This inequality extends the classical Sobolev inequalities to include more general settings, like those on manifolds, facilitating the study of the regularity and existence of solutions to various equations. It plays a crucial role in understanding embeddings of function spaces, which is essential when working with differential equations on curved spaces.
L^p norm: The l^p norm is a mathematical concept that generalizes the idea of measuring the size or length of a vector in a normed vector space. It is defined as the p-th root of the sum of the absolute values of its components raised to the power of p, represented mathematically as $$||x||_p = \left( \sum_{i=1}^{n} |x_i|^p \right)^{1/p}$$ for a vector x in n-dimensional space. This concept is crucial in functional analysis and plays a significant role in Sobolev inequalities, particularly on manifolds, where it aids in understanding the behavior of functions in various spaces.
L^p spaces: l^p spaces are a class of vector spaces defined by the use of the p-norm, which generalizes the concept of measuring the size of sequences or functions. These spaces are fundamental in analysis and functional analysis, particularly in understanding the properties of Sobolev inequalities on manifolds, where they provide a framework for discussing convergence, integrability, and duality among functions.
Log-Sobolev inequality: The log-Sobolev inequality is a fundamental result in the analysis of functions on Riemannian manifolds, which relates the entropy of a probability measure to its gradient. This inequality provides a powerful tool for establishing functional inequalities and is crucial for understanding the behavior of diffusions on manifolds. In this context, it helps bridge geometric properties and analysis, linking the notion of curvature with functional inequalities and the behavior of heat equations.
Moser-Trudinger Inequality: The Moser-Trudinger inequality is a mathematical result that provides a way to control certain Sobolev norms of functions in terms of their exponential integrability. Specifically, it asserts that for a bounded domain in a Riemannian manifold, if a function has bounded Sobolev norm, it also has a controlled growth condition which implies the function cannot blow up too fast. This inequality connects deeply with Sobolev inequalities on manifolds, highlighting the relationship between the geometry of the manifold and the behavior of functions defined on it.
Nash Inequality: The Nash Inequality is a mathematical inequality that provides a relationship between the norms of functions and their derivatives in the context of Sobolev spaces. It is essential in establishing embeddings of Sobolev spaces into L^p spaces, which allows for the analysis of functions defined on manifolds. This inequality plays a pivotal role in understanding the behavior of Sobolev functions, particularly regarding their integrability and differentiability properties.
Poincaré Inequality: The Poincaré Inequality is a fundamental result in analysis that establishes a relationship between the average value of a function and its gradient on a given domain, particularly within the context of Riemannian manifolds. It asserts that if a function has small average oscillation, then its gradient must also be small in some integral sense. This inequality plays a crucial role in the theory of Sobolev spaces, linking the geometry of the underlying manifold with functional analysis concepts.
Regularity of Solutions: Regularity of solutions refers to the smoothness and well-behaved nature of solutions to differential equations, particularly in the context of Sobolev spaces. This concept is crucial because it helps determine whether solutions exist and if they possess certain desirable properties, such as continuity and differentiability, which are essential for various mathematical analyses and applications.
Riemannian metric: A Riemannian metric is a smoothly varying positive definite inner product on the tangent spaces of a differentiable manifold, allowing one to measure distances and angles on that manifold. This concept forms the backbone of Riemannian geometry, enabling the exploration of curves, surfaces, and their properties in various contexts, from smooth manifolds to curvature concepts and beyond.
Sobolev Embedding Theorem: The Sobolev Embedding Theorem is a fundamental result in functional analysis that establishes a relationship between Sobolev spaces and continuous function spaces. This theorem shows how functions that are sufficiently smooth and have bounded energy can be embedded into spaces of continuous functions, thus connecting the geometric properties of the underlying space with the analysis of functions defined on it.
Sobolev inequalities on Riemannian manifolds: Sobolev inequalities on Riemannian manifolds are mathematical statements that provide bounds on the norms of functions based on their derivatives, allowing the study of function spaces in a geometric setting. These inequalities are essential in understanding how the geometry of the manifold influences the analysis of partial differential equations and functional spaces, connecting geometry and analysis through the study of embeddings and compactness properties.
Sobolev Inequalities on Sub-Riemannian Manifolds: Sobolev inequalities on sub-Riemannian manifolds are mathematical statements that relate the norms of functions defined on these manifolds, providing essential tools for studying the regularity and existence of solutions to partial differential equations. These inequalities generalize classical Sobolev inequalities by taking into account the unique geometric structure and constraints of sub-Riemannian spaces, which can have non-standard notions of distance and integration.
Trudinger-Moser Inequality: The Trudinger-Moser Inequality is a crucial result in the theory of Sobolev spaces, specifically providing a bound for certain functions in terms of their Sobolev norms. This inequality highlights the relationship between the growth of a function and its integrability, especially in the context of non-linear analysis on Riemannian manifolds. It is often used to demonstrate the existence of weak solutions to variational problems involving critical Sobolev embeddings.
Volume Element: A volume element is a mathematical construct used to define the infinitesimal volume in a given space, particularly in the context of integration on manifolds. It allows for the measurement of volume in a curved space by providing a way to express how volume changes with respect to the geometry of the manifold, which is influenced by the metric tensor. Understanding volume elements is crucial when applying Sobolev inequalities, as they help establish relationships between different function spaces on the manifold.
W^{k,p} norm: The w^{k,p} norm is a specific type of function norm that measures the size of a function in terms of its weak derivatives up to order k, taking into account both local behavior and integrability conditions via the p-norm. This norm is particularly relevant when dealing with Sobolev spaces, allowing for a way to quantify the smoothness and integrability of functions defined on manifolds.
W^{k,p} spaces: w^{k,p} spaces, also known as Sobolev spaces, are a type of functional space that includes functions with certain degrees of smoothness and integrability. They are essential in the study of partial differential equations and variational problems, as they provide a framework to work with weak derivatives of functions, allowing for more flexibility compared to classical derivatives. These spaces help in understanding the behavior of functions defined on manifolds, especially when considering Sobolev inequalities.