LECTURE NOTES ON SOBOLEV SPACES FOR CCA WILLIE WAI-YEUNG WONG 0.1. References. Before we start, some references: D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second • order, Springer. Ch. 7. L. Evans, Partial differential equations, American Math. Soc. Ch. 5. • M. E. Taylor, Partial differential equations I, Springer. Ch. 4. (Note: this • presentation is based on heavy doses of Fourier analysis and functional analysis.) H. Triebel, Theory of function spaces, Birkhauser. Ch. 2. • R. Adams, Sobolev Spaces, Academic Press. • 0.2. Notations. We will work in Rd. p0: given p 1 a real number, we define p0 to be the positive real number ≥ 1 1 satisfying p− + (p0)− = 1; p0 is called the Hölder conjugate of p Ω: open set in Rd ∂Ω: the boundary of Ω, Ω¯ Ω α \ N d ∂ : partial derivative of multi-index α. α = (α1,...,αd) ( 0) , with norm P α α ∈ α = α . ∂α = ∂ 1 ∂ d | | i 1 · d Ω1 b Ω2: there exists a compact set K such that Ω1 K Ω2 Dα: weak derivative (see §1.3) of multi-index α ⊂ ⊂ suppf : for a function f , this denotes the support set, i.e. the set on which f , 0 C(Ω): continuous functions taking value in the reals defined on Ω (though most of what we say will be valid for functions taking value in a Hilbert space) C(Ω¯ ): the subset of C(Ω) consisting of functions that extend continuously to ∂Ω ¯ C0(Ω): the subset of C(Ω) consisting of functions which vanish on ∂Ω Ck(Ω): functions f such that ∂αf C(Ω) for every α k. k is allowed to be (in which case f is smooth) or ω∈ (in which case f| is| ≤ analytic). Analogously∞ k ¯ k ω we define C (Ω) and C0(Ω) (note that the set C0 (Ω) contains only the zero functions) k k b Cc (Ω): subset of C (Ω) such that suppf Ω p p L (Ω),Lloc(Ω): Lebesgue spaces (see §1.1) k,p k,p s,p W (Ω),Wloc (Ω),W0 (Ω): Sobolev spaces (see §1.4) : Lp norm (see §1.1) k · kp : W k,p norm (see §1.4) k · kp,k Version as of October 27, 2010. 1 2 W. W. WONG 1. Basic Definitions In this first part Ω can be taken to be any open subset of Rd. Throughout dx will be the standard Lebesgue measure. By a measurable function we’ll mean a representative of an equivalence class of measurable functions which differ on Ω in a set of measure 0. Thus sup and inf should be mentally replaced by esssup and essinf when appropriate. 1.1. Lebesgue spaces. For > p 1, Lp(Ω) denotes the set of p-integrable mea- surable functions, with norm∞ ≥ 1/p Z p (1) u p;Ω = u dx . k k | | Ω If u takes values in some normed linear space, then will be the Banach space norm. | · | For p = , L (Ω) denotes the essentially bounded functions ∞ ∞ (2) u ;Ω = sup u . k k∞ Ω | | For 1 p , the spaces Lp(Ω) are Banach spaces. The space L2(Ω) is a Hilbert space, with≤ inner-product≤ ∞ Z (3) u,v = u,v Ω = uv dx . h i h i0; Ω If u,v take values in a Hilbert space with norm , , then the integrand R H h· ·iH should be replaced by u,v d(x). In particular, if they are complex valued, Ω h iH the integrand should be uv¯ . For the sake of completeness, let me recall the definitions of Banach and Hilbert spaces. Let V be a linear space over R. With the obvious substitutions, you can also do over C A norm on V assigns to elements of V nonnegative real numbers, such that for v,w V|: · | ∈ (1) v 0, with equality iff v = 0; (2) |sv| ≥= s v , for any scalar s R; (3) |v +| w| | | v| + w (triangle ineq.)∈ | | ≤ | | | | The pair (V, ) is called a normed linear space. We can define a metric on (V, ) by d(v,w) = v |w · | . | · | | − | Exercise 1. Check that d(v,w) is indeed a metric. Then we can use the usual notion of convergence in metric spaces. If the normed linear space (V, ) is complete as a metric space, that is, all Cauchy sequences converge, then we| · | say (V, ) is a Banach space. That the Lp norms are in fact | · | SOBOLEV SPACES 3 norms follows easily from property of the Euclidean absolute value, and Hölder’s inequality (6) below. Exercise 2. Prove that Lp(Ω) is a Banach space. That is, show that if u Lp(Ω) are i ∈ a sequence of functions satisfying u u Ω 0 as i,j , then there exists k i − j kp; → → ∞ u Lp(Ω) such that u u. ∈ i → Now let V be an R-linear space again. An inner product on V is a map V V R denoted by , , satisfying × → h· ·i (1) v,w = w,v ; (2) hav +ibw,xh =ia v,x + b w,x for all a,b R; h i h i h i ∈ (3) v,v 0, with equality iff v = 0. h i ≥ We call the pair (V, , ) an inner product space or pre-Hilbert space. h· ·i If V were over C, the symmetric property needs to be replaced by Hermitian, and the linearity needs to become sesquilinearity. Exercise 3. Writing v = √ v,v on an inner product space (V, , ), show that k k h i h· ·i v,w v w (Schwarz) • |hv + wi| ≤ k vk k+ kw (Triangle) • kv + wk2 ≤+ k vk wk 2k = 2 v 2 + 2 w 2 (Parallelogram identity) • k k k − k k k k k This implies that every inner product space is a normed linear space. If the induced norm is complete, we say an inner product space is a Hilbert space. In some texts a Hilbert space is also required to be separable: that there exists a countable subset of vectors v1,v2,... whose linear space is dense in V ; this will in particular imply the Hilbert{ space} has a countable orthonormal basis. While the spaces we encounter in this note are all, in fact, separable, we will not make this assumption. 2 Exercise 4. Check that (L (Ω), , 0;Ω) is an inner product space. Check that the inner product defined in (3) indeedh· ·i leads to the norm defined in (1). Therefore by Exercise 2, L2 is a Hilbert space. Recall Young’s inequality: let a,b be positive real numbers, and p 1 real, then ≥ ap bp0 (4) ab + ap + bp0 . ≤ p p0 ≤ The special case p = p = 2 is known as Cauchy’s inequality. By replacing a a 0 → and b 1b, we also get the interpolated version of Young’s inequality → − p p p p (5) ab a + − 0 b 0 . ≤ Using Young’s inequality we can prove Hölder’s inequality Z (6) uv dx u Ω v Ω ≤ k kp; k kp0; Ω when the right hand side is well defined. 4 W. W. WONG Proof. The inequality is homogeneous in scaling of u and v. Therefore it suffices to prove for u = v = 1. Write k kp k kp0 Z Z Z u p v p0 uv dx u v dx | | + | | dx ≤ | | | | ≤ p p0 Ω Ω Ω 1 p 1 p0 1 1 u p;Ω + v p ;Ω = + = 1 ≤ p k k p0 k k 0 p p0 This homogeneous scaling trick is very useful in analysis of partial differen- tial equations. Hölder’s inequality is very useful for deriving facts about the Lp spaces. First we p define the localised Lebesgue spaces: a function u is said to be in Lloc(Ω) if for every open set Ω b Ω, u Lp(Ω ). 0 ∈ 0 α Consider the functions on R given by x − . These functions are not in any Lebesgue spaces, because for pα 1, it decays| | too slowly at infinity, while for pα 1, it blows up too fast at the≤ origin. The localised spaces allows one to distinguish≥ divergences at the boundary of Ω, and singularities in the interior of Ω. Also note that the local Lebesgue spaces are not normed spaces. Proposition 1. (1) Lq(Ω) Lp(Ω) if q p and Ω < ; (2) Lp(Ω) Lq(Ω) Lr (Ω) if p r q. ⊂ ≥ | | ∞ ∩ ⊂ ≤ ≤ q p Note that point (1) in particular implies that Lloc(Ω) Lloc(Ω) for p < q and any Ω. ⊂ Proof. What we will prove are slightly stronger, quantitative versions of the above statements. For the first claim, let χΩ be the characteristic function of Ω, i.e. q χΩ(x) = 1 if x Ω and 0 otherwise. Let u L (Ω). Observe ∈ ∈ Z Z p p p u = u dx = χΩ u dx k kp;Ω | | | | Ω Ω p (by Hölder) χΩ 1/(1 p );Ω u q/p;Ω ≤ k k − q k k 1 p p = Ω − q u | | k kq;Ω and so we have 1 1 p q (7) u Ω Ω − u Ω k kp; ≤ | | k kq; which proves the claim. For the second statement, we write Z r λr (1 λ)r λr (1 λ)r u r;Ω = u u − dx = u p/(λr);Ω u − q/(1 λ)r;Ω k k | | | | k k k k − Ω SOBOLEV SPACES 5 1 λ 1 λ for 0 λ 1 and = + − . Noting that if p r q, one can always find such a λ, ≤ ≤ r p q ≤ ≤ we have λ 1 λ (8) u Ω u u − k kr; ≤ k kp;Ωk kq;Ω and proving the claim.