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Sobolev Inequalities and Riemannian Manifolds University of Connecticut OpenCommons@UConn Honors Scholar Theses Honors Scholar Program Spring 5-1-2021 Sobolev Inequalities and Riemannian Manifolds John Reever [email protected] Follow this and additional works at: https://opencommons.uconn.edu/srhonors_theses Part of the Analysis Commons, and the Geometry and Topology Commons Recommended Citation Reever, John, "Sobolev Inequalities and Riemannian Manifolds" (2021). Honors Scholar Theses. 824. https://opencommons.uconn.edu/srhonors_theses/824 Sobolev Inequalities and Riemannian Manifolds Jack Reever A thesis presented to the Department of Mathematics in partial fulfillment of the requirements of a Bachelors of Science with Honors Storrs, Connecticut Spring 2021 Abstract Sobolev inequalities, named after Sergei Lvovich Sobolev, relate norms in Sobolev spaces and give insight to how Sobolev spaces are embedded within each other. This thesis begins with an overview of Lebesgue and Sobolev spaces, leading into an introduction to Sobolev inequalities. Soon thereafter, we consider the behavior of Sobolev inequalities on Riemannian manifolds. We discuss how Sobolev inequalities are used to construct isoperimetric inequal- ities and bound volume growth, and how Sobolev inequalities imply families of other Sobolev inequalities. We then delve into the usefulness of Sobolev inequalities in determining the geometry of a manifold, such as how they can be used to bound a manifold's number of ends. We conclude with examples of real-world applications Sobolev inequalities. 1 Introduction to Sobolev Spaces We will begin by introducing Sobolev spaces, and to do so we start with Lebesgue spaces. Let p 2 [1; 1). The space of functions which are Lebesgue integrable on a set Ω p-many times is denoted Z Lp(Ω) = ff : jfjp(x)dx < 1g: Ω These Lebesgue spaces have the norm Z 1=p p jjfjjLp(Ω) = jfj (x)dx : Ω If p = 1, the space L1(Ω) is the space of all functions such that jf(x)j < 1 \almost everywhere" on Ω, which is to say that jf(x)j can be infinite on only a set of measure zero. Here, the norm is the essential supremum of jf(x)j for x 2 Ω [3]. The elements of a Lebesgue space are equivalence classes, and functions are considered equivalent if they differ only on a set of Lebesgue measure zero. Lebesgue spaces are complete normed spaces. Let k 2 N [ f0g and p 2 [1; 1]. The Sobolev Space W k;p(Ω) is defined by W k;p(Ω): = fu 2 Lp(Ω) : Dαu 2 Lp(Ω) 8α with jαj ≤ kg: 1 This space has the norm X α jjujjW k;p(Ω) := jjD ujjLp(Ω) jα|≤k as discussed in [3]. For a function u to be included in a Sobolev space, all derivatives Dαu must exist and must be represented by an element of Lp(Ω). Note that if k = 0, the space W 0;p(Ω) = Lp(Ω) identically, as the restriction on the derivatives of each element is gone. Sobolev spaces are vector spaces, as elements can be added together and multiplied by scalars, with the resultant functions being elements of the Sobolev space. In addition, the k;p 1 Sobolev space W0 (Ω) is the completion of C0 (Ω) (the space of infinitely differentiable functions with compact support) with respect to the norm of W k;p(Ω) [3]. Equipped with the definition of Sobolev spaces, we can intuitively embed Sobolev spaces in one another: let Ω ⊂ Rd be a domain with p 2 [1; 1) and k ≤ m. Then, W m;p(Ω) ⊂ W k;p(Ω). This is clear because if a function has m derivatives represented in Lp(Ω), then it must have k derivatives in Lp(Ω). Alternatively, for k ≥ 0 and p; q 2 [1; 1] with q > p it is clear that W k;q(Ω) ⊂ W k;p(Ω): if we can integrate q−many times, we can integrate p−many times. Now, we will consider the Sobolev Embedding Theorem. Consider W k;p(Rn) with k 2 N+, p 2 [1; 1). If we have two real numbers l; q such that k > l, 1 ≤ p < q < 1, and 1 k 1 l − = − ; p n q n then W k;p(Rn) ⊆ W l;q(Rn), and this embedding is continuous. 2 2 Introduction to Sobolev Inequalities When we are working with smooth functions f : R ! R with compact support, we have the inequality 1 Z 1 jf(x)j ≤ jf 0(t)jdt: 2 −∞ This inequality bounds the value of jf(x)j using its derivative. As we move up to multidi- mensional spaces, the notions of absolute value and the derivative become more complicated, and we cannot use this inequality to bound our functions. This brings us to the topic of Sobolev inequalities: how can we construct an inequality that bounds the norm of a function using partial derivatives? While working in Rn, let p 2 [1; n). The Sobolev conjugate of p is np p∗ := 2 (p; 1): n − p Theorem 2.1: When n ≥ 2, the Sobolev inequality in Rn states Z 1=p∗ Z 1=p jf(x)jp∗ dx ≤ C(n; p) jrf(x)jpdx (1) Rn Rn for all smooth functions f with compact support, for any p 2 [1; n). Note that C(n; p) is a constant that depends on n and p [1]. ∗ p n Note that this inequality can be written as jjfjjLp (Rn) ≤ CjjrfjjL (R ); and moving forward we will notate these norms as jjfjjp∗ ≤ Cjjrfjjp. To prove Theorem 2.1, we will illustrate Gagliardo and Nirenberg's proof of the Sobolev inequality in Rn for p = 1, and then expand upon that proof to show this inequality holds n for all real p 2 [1; n). Note that µ = µn denotes the Lebesgue measure in R . Proposition 2.2: When n ≥ 2; Z 1=1∗ Z 1=1 jf(x)j1∗ dx ≤ C(n; 1) jrf(x)j1dx Rn Rn for all smooth functions f with compact support. Note that C(n; 1) is a constant that depends on n. 0 1 1 p p0 Proof. Let 1 ≤ p; p ≤ 1 with 1 = p + p0 . For f 2 L (µ); g 2 L (µ), H¨older'sinequality states that for positive measure µ, Z fgdµ ≤ jjfjjpjjgjjp0 : 3 pi 1 1 1 By induction, for fi 2 L ; 1 ≤ i ≤ k; 1 ≤ pi ≤ 1; + + ::: + = 1, we see p1 p2 pk Z k Y f1f2:::fkdµ ≤ jjfijjpi : (2) i=1 1 n Fix f 2 C0 (R ): a smooth function with compact support that. By the inequality 1 Z 1 jf(x)j ≤ jf 0(t)jdt; 2 −∞ for any x = (x1; :::; xn) and any integer i 2 [1; n] we observe 1 Z 1 jf(x)j ≤ j@xi f(x1; x2; :::; xi−1; t; xi+1; :::; xn)jdt: (3) 2 −∞ Fix Z 1 Fi(x) = j@xi f(x1; x2; :::; xi−1; t; xi+1; :::; xn)jdt; −∞ which depends on n − 1 coordinates of x with the integration variable replacing the ith coordinate. Additionally, fix 8R 1 R 1 < −∞ ::: −∞ j@xi f(x)jdx1:::dxm when i ≤ m Fi;m(x) = R 1 R 1 : −∞ ::: −∞ Fi(x)dx1:::dxm when i > m which depends on n − m or n − m − 1 variables, for all integers m 2 [1; n]. Now, by (3) we have the inequality 1 jfjn ≤ (F :::F ): 2n 1 n Raising both sides of this equality to the power of 1=(n − 1), we see 1 jfjn=(n−1) ≤ (F :::F )1=(n−1): 2n=(n−1) 1 n 1 1 1 1 Setting k = n − 1; p1 = p2 = ::: = pn−1 = n − 1, we see + + ::: + = (n − 1) = 1. p1 p2 pk n−1 By (2) with induction on m ≤ n we see Z Z 1 ::: jf(x)jn=(n−1)dx :::dx ≤ (F (x):::F (x))1=(n−1): 1 m 2n=(n−1) 1;m n;m When n = m, we see n 1=n Y jjfjjn=(n−1) ≤ (1=2) jj@xi fjj1 : i=1 Note that both sides were raised to the power of (n − 1)=n when taking the norm in p n=(n−1) n Qn 1=n 1 Pn Pn L (R ). Because ( 1 ai) ≤ n 1 ai for ai > 0 and n 2 Z, and 1 j@xi fj ≤ njrfj, this becomes n 1 X 1 jjfjj ≤ jj@ f(x)jj dx ≤ p jjrfjj : n=(n−1) 2n xi 1 2 n 1 i=1 This proves Proposition 2.2, the Sobolev Inequality in Rn for p = 1. [1] 4 Next, we are going to build off of the above proof to show that Theorem 2.1 (1) holds for p ≥ 1. Proposition 2.3: If Theorem 2.1, the Sobolev Inequality in Rn, holds for p = 1, then it holds for all p 2 [1; n). Proof. Knowing that (1) holds for p = 1, or that 1 8f 2 C1( n) jjfjj ≤ p jjrfjj ; (4) 0 R n=(n−1) 2 n 1 1 n α 1 fix a p > 1. For any α > 1 and function f 2 C0 (R ), jfj is C0 and satisfies jrjfjαj = αjfjα−1jrfj: α The function jfj can be approximated by a sequence of smooth functions (fi) 2 C0 such α α that rfi ! rjfj . This allows us to substitute jfj into (4): 1 Z jjfjjα ≤ p α jf(x)α−1jrf(x)jdx αn/(n−1) 2 n 1=p0 1=p 1 Z 0 Z ≤ p α jf(x)j(α−1)p dx jrf(x)jpdx 2 n 0 (n−1)p with 1=p + 1=p = 1.
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