FUNCTION SPACES AND GENERAL INEQUALITIES Syafiq Johar syafi[email protected] Contents 1 Definitions 1 1.1 Ck and H¨olderSpaces . .1 1.2 Sobolev Spaces . .2 2 Sobolev Inequalities 4 2.1 Case: 1 ≤ p < d ..........................................4 2.2 Case: d < p < 1 .........................................5 2.3 General Sobolev inequalities . .5 2.4 Consequences: Sobolev Embeddings . .5 2.5 Graphical Representation of Sobolev Embeddings . .6 2.6 Compactness and Rellich-Kondrachov Theorem . .7 3 Other Inequalities 7 3.1 Poincar´e'sInequality . .7 3.2 Difference Quotients . .7 3.3 Other Useful Inequalities . .8 1 Definitions 1.1 Ck and H¨olderSpaces Definition 1.1 (Ck Spaces). The space Ck(Ω) is the set of all k-times continuously differentiable func- tions such that the following norm is finite: X α jjujjCk(Ω) = sup jD (x)j; x2Ω jα|≤k α k for which D is just the usual derivative. The space (C (Ω); jj · jjCk(Ω)) is a Banach space. Some special sets: • C0(Ω) and C1(Ω) are the set of all bounded continuous and smooth functions with bounded derivatives on Ω respectively, • Ck(Ω) is set of elements of C1(Ω) with bounded derivatives up to order k, k k k k • Cloc(Ω) is the set of all C (!) for all ! compact subsets of Ω. C (Ω) $ Cloc(Ω). For example, x k ∗ k ∗ e 2 Cloc(R) for all k 2 N but not in C (R) for any k 2 N . k k • Cc (Ω) is the subset of C (Ω) with compact supports. 1 Definition 1.2 (Ck,γ Spaces). We define the γ-H¨olderseminorm of u :Ω ! R for γ 2 (0; 1] to be ju(x) − u(y)j 0,γ jujC (Ω) = sup γ ; x;y2Ω jx − yj x6=y and we define the k; γ-H¨oldernorm of u as X α jjujjCk,γ (Ω) = jjujjCk (Ω) + jD ujC0,γ (Ω): jαj=k k The k; γ-H¨olderspace is the subset of Cb (Ω) defined by k,γ k C (Ω) = fu : jjujjCk,γ (Ω) < 1g ⊂ C (Ω): This space with the corresponding k; γ-H¨oldernorm is also a Banach space. Remark 1.1. For H¨olderspaces, we only consider the case for which 0 < γ ≤ 1. The cases for γ > 1 are uninteresting because any function in Ck,γ (Ω) for γ > 1 are functions which are constant on each connected component of Ω. C0;1(Ω) is called the set of Lipschitz continuous functions. Lemma 1.1. [TT] For Ω a bounded subset of Rd and k 2 N∗, we have the following inclusions: Ck+1(Ω) ⊂ Ck,β(Ω) ⊂ Ck,α(Ω) ⊂ Ck(Ω) for 0 < α ≤ β ≤ 1 1.2 Sobolev Spaces Definition 1.3 (Lp Spaces). The Lp(Ω) space for 1 ≤ p < 1 is the space of functions for which their p-th power is Lebesgue integrable over Ω i.e. ( 1 ) p p p L (Ω) = u : jjujjLp(Ω) = juj dµ < 1 : ˆΩ The L1 space is defined as the space of functions with finite essential supremum i.e. p L (Ω) = fu : jjujjL1(Ω) = ess supΩjuj < 1g; where ess sup is defined to be ess supΩjuj = inffC ≥ 0 : ju(x)j ≤ C for a.e. x 2 Ωg: Remark 1.2. We usually think of the class of Lp(Ω) functions to be up to equivalence. In other words, two functions u and v of Lp(Ω) are the \same" function in Lp(Ω) if they are equal almost everywhere. Remark 1.3. Equivalently, the Lp(Ω) space can also be defined as the completion of the C1(Ω) space with respect to the jj · jjLp(Ω) norm. Lemma 1.2. For Ω a set with finite measure (i.e. µ(Ω) < 1), we have the following inclusion: Lq(Ω) ⊂ Lp(Ω) for 1 ≤ p ≤ q ≤ 1 There are more functions in Lp spaces than continuous functions and thus they are the natural ob- jects to study in PDEs as candidate solutions. We would like to extend the differentiability of continuous functions to Lp functions. However, Lp functions are not necessarily continuous and thus not differ- entiable! Thus, we need a weaker notion of differentiation which is called the weak derivative. Using weak derivatives, we can define spaces which are analogous to Ck spaces, which we call the Sobolev spaces. This would then allow us to combine Lemmas 1.1 and 1.2 for more general inclusions. 2 Definition 1.4 (Sobolev space). The Sobolev space W k;p(Ω) consists of all locally summable functions u :Ω ! R such that for each multi-index α with jαj ≤ k, Dαu exists in the weak sense and is in Lp(Ω). Definition 1.5. If u 2 W k;p(Ω), we define its norm to be: 8 P α p 1=p < jα≤k Ω jD uj dx if 1 ≤ p < 1; jjujjW k;p(Ω) = P ´ α : jα|≤k ess supΩjD uj if p = 1: The Sobolev spaces W k;p(Ω) for each k 2 N∗ and 1 ≤ p ≤ 1 with this norm are Banach spaces. Clearly, W 0;p(Ω) = Lp(Ω). Remark 1.4. Equivalently, the Sobolev space W k;p(Ω) can also be defined as the completion of the 1 space C (Ω) with respect to the jj · jjW k;p(Ω) norm. k;p 1 k;p Definition 1.6. We denote by W0 (Ω) the closure of Cc (Ω) in W (Ω). Lemma 1.3. Analogous to Lemma 1.2, for Ω a set of finite measure and k 2 N∗, we have the inclusion: W k;q(Ω) ⊂ W k;p(Ω) for 1 ≤ p ≤ q ≤ 1 k;p k;p k 1 Remark 1.5. The spaces W (Ω), Wloc (Ω), C (Ω) and C (Ω) are all separable (i.e. there is a countable dense subset for each space, which are the polynomials with rational coefficients). Theorem 1.1. For each k 2 N and 1 ≤ p ≤ 1, the Sobolev space W k;p(Ω) is a Banach space. For p = 2, the Sobolev space W k;2(Ω) is a Hilbert space and denoted Hk(Ω) with the obvious inner product. Usually, during computations, we might need to take derivatives. Since W k;p(Ω) only have weak derivatives, it is useful to use the density of smooth functions in this space and take limits: Theorem 1.2 (Global approximation by smooth functions). Assume Ω is bounded and @Ω is C0;1. k;p 1 k;p Suppose as well that u 2 W (Ω) for some 1 ≤ p < 1. Then there exist functions um 2 Cc (Ω)\W (Ω) k;p such that um ! u in W (Ω). d 1 d If we replace Ω with and open subset of R , the above still holds. The closure of Cc (R ) in the W k;1(Rd) norm is contained in Ck+1(Rd) which is smaller and not dense in W k;1(Rd) [TT]. Thus, the previous approximation does not hold for p = 1. Theorem 1.3 (Extension Theorem). [ELC] Assume Ω is bounded and @Ω is C0;1. Let Ω0 be a bounded open set such that Ω0 ⊂⊂ Ω. Then, there exists a bounded linear operator E : W 1;p(Ω) ! W 1;p(Rd) for 1 ≤ p ≤ 1 such that for each u 2 W 1;p(Ω): (i) Eu = u a.e. in Ω. (ii) Eu has support within Ω0. 0 1;p n 1;p (iii) jjEujjW (R ) ≤ C(p; Ω; Ω )jjujjW (Ω). Theorem 1.4 (Trace Theorem). [ELC] Assume Ω is bounded and @Ω is C0;1. Then, there exists a bounded linear operator T : W 1;p(Ω) ! Lp(@Ω) for 1 ≤ p ≤ 1 such that: 1;p (i) T u = uj@Ω if u 2 W (Ω) \ C(Ω).¯ 1;p (ii) jjT ujjLp(@Ω) ≤ C(p; Ω)jjujjW 1;p(Ω) for each u 2 W (Ω). Theorem 1.5 (Trace-zero functions in W 1;p). Assume Ω is bounded and @Ω is C0;1. Suppose further- more that u 2 W 1;p(Ω). Then: 1;p u 2 W0 (Ω) () T u = 0 on @Ω 3 2 Sobolev Inequalities 2.1 Case: 1 ≤ p < d Definition 2.1. If 1 ≤ p < d, the Sobolev conjugate of p is defined as: 1 1 1 dp = − ) p∗ = > p p∗ p d d − p 1 n Theorem 2.1 (Gagliardo-Nirenberg-Sobolev inequality). Assume 1 ≤ p < d. Then, for all u 2 Cc (R ): jjujj p∗ d ≤ C(p; d)jjDujj p d L (R ) L (R ) Remark 2.1. Why are the numbers p and p∗ very specific? This is to ensure that the inequality is invariant by scaling the space variable. For example say x = λy for some constant λ > 0, then by changing the integration with respect to the x variable to the y variable, we have: x d p∗ u = λ jju(y)jjLp∗ ( d); p∗ R λ L (R) x d−p Du = λ p jjDu(y)jjLp( d): p R λ L (R) Thus, to ensure that the inequality is invariant under scaling of the space variable, the constants must match up. Otherwise, you can find some large enough λ such that the Gagliardo-Nirenberg-Sobolev d d−p ∗ dp inequality is violated. Thus, we must have: p∗ = p ) p = d−p . The following are two important theorems in Sobolev inequalities: Theorem 2.2 (Estimates for W 1;p for 1 ≤ p < d).