Notes on weak convergence (MAT Spring 2006)


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1 Notes on weak convergence (MAT Spring 2006) Kenneth H. Karlsen (CMA) February 2, Weak convergence In what follows, let denote an open, bounded, smooth subset of R N with N 2. We assume 1 p < and let p be the conjugate exponent, i.e., 1 p + 1 p = 1 (p := when p = 1). A sequence {u n } n 1 L p () converges weakly to u L p (), in which case we write if u n u u n v dx uv dx, in L p (), v L p (). When p =, we say that a sequence {u n } n 1 L p () converges weakly  to u L (), and we write u n u in L (), if u n v dx uv dx, v L 1 (). We remark that when Ω is bounded the weak  convergence of u n in L (Ω) to some u L (Ω) implies weak convergence of u n to u in any L p (Ω), 1 p <. Theorem 1.1 (boundedness of weakly converging sequences). Suppose 1 p < and u n u in L p (Ω) ( in L (Ω) if p = ). and u n is bounded in L p (Ω) u L p (Ω) lim inf u n L p (Ω) 1
2 We have the following compactness theorem: Theorem 1.2 (Weak convergence in L p ). Suppose 1 < p < and the sequence {u n } n 1 is bounded in L p (). there is a subsequence, still denoted by {u n } n 1, and a function u L p () such that u n u in L p (). If p =, the result still holds with replaced by. Theorem 1.2 is false for p = 1 since L 1 () is not the dual of L (). But a good substitute result exists by regarding L 1 () as a subset 1 of the space of (signed) Radon measures on with finite mass, a space which we denote by M(), and using the weak  topology on M(). Let C c () the denote the space of continuous functions on with compact support. If µ M(), then µ, v = v dµ, v C c (). Recall that µ M() if and only if µ, v C v L (), v C 0(). We define { } µ M() = sup µ, v : v C c (), v L () 1. The space ( M(), M() ) is a Banach space and it is isometrically isomorphic to the dual space of ( C c (), L ()). A sequence {µ n } n 1 M() converges weakly  to µ M(), in which case we write µ n µ in M(), if Theorem 1.3. Suppose v dµ n v dµ, µ n µ in M(Ω). lim sup µ n (K) µ(k), v C c (). 1 We recall that each u L 1 () defines a linear functional on C c() via Z v uv dx, v C c(). 2
3 for each compact set K Ω, and for each open set O Ω. µ(o) lim inf µ n(o), We have the following compactness theorem for measures: Theorem 1.4 (Weak compactness in M()). Suppose the sequence {µ n } n 1 is bounded in M(). there is a subsequence, still denoted by {µ n } n 1, and a measure µ M() such that µ n µ in M(). Theorem 1.5 (Characterizations of weak convergence in L p ). Let u n : Ω R be a sequence in L p (Ω) and u L p (Ω), 1 < p <. Suppose the sequence u n is equibounded in L p (Ω) the following statements are equivalent: 1. u n u in L p (Ω). 2. u n u in M(Ω). 3. u n u in D (Ω). 4. For any Borel set E Ω, E > 0, (u n ) E := 1 E 5. For any cube Q Ω, Q > 0, (u n ) E := 1 E E E u n dx (u) E := 1 u dx. E E u n dx (u) E := 1 u dx. E E Remark 1.1. Similar equivalences hold also if p =, replacing in (1) weak convergence with weak  convergence in L (Ω). Remark 1.2. Condition (5) expresses the intuitive idea of weak convergence as convergence of mean values. The next lemma is simple but quite useful in a number of situations. Lemma 1.1 (products of weakstrong converging sequences). Let 1 < p <, u n : Ω R be a sequence in L p (Ω), and u L p (Ω). Let v n : Ω R be a sequence in L p (Ω), 1 p + 1 p = 1 (or p = p p 1 ). Suppose u n u in L p (Ω), v n u in L p (Ω). u n v n uv in L 1 (Ω). 3
4 2 Some typical behaviors of weakly converging sequences Relevant for our study of nonlinear partial differential equations and calculus of variation, we will illustrate some typical behaviors of sequences that converge weakly but not strongly. 2.1 Oscillations Sequences of rapidly oscillating functions provide examples of weakly but not strongly converging sequences. Letting u n (x) = sin(nx), x (0, 2π), n = 1, 2,..., one can easily check that but u n u := 0 in L p (0, 2π) p 1 ( in L (0, 2π) if p = ), u n L p (0,2π) = C(p) > 0. Hence u n does not converge strongly in L p (0, 2π) for any 1 p. Furthermore, we have u L p (0,2π) < lim inf u n L p (0,2π). Recall that if u n u in L p, then by the weak lower semicontinuity of the L p norms we have always u L p (0,2π) lim inf u n L p (0,2π). (1) If u n u in L p, then we have instead (trivially) equality in (1), but be aware that we can have this equality under mere weak convergence, as the the next examples shows. Let u n (x) = 1 + sin(nx), x (0, 2π), n = 1, 2,..., then one can easily check that u n u := 1 in L p (0, 2π) p 1 ( in L (0, 2π) if p = ) and so that 2π 0 2π 0 u n dx = 2π 0 u n u dx = u dx = 2π, 2π 0 n, sin(nx) dx = 4, n, u n converges weakly but strongly to u in L 1 (0, 2π). 4
5 On the other hand, we have 2π 0 u p dx < lim inf u n p dx, p > 1, since otherwise, passing to a subsequence if necessary, we would have for all p > 1 u n u in L p (0, 2π) and u n L p (0,2π) u L p (0,2π), which implies (via BrezisLieb s refinement of Fatous s lemma) u n u in L p (0, 2π). More generally, the following theorem about weak limits of rapidly oscillating periodic functions is well known: Theorem 2.1. Let 1 p and u be a Y periodic function in L p (Y ). For n = 1, 2,..., set u n (x) := u(nx)., if 1 p <, as n, u n 1 u(y) dy in L p (O) for any bounded open set O R N. Y If p =, as n, 2.2 Concentrations Y 1 u n u(y) dy in L (R N ). Y Y For example, Dirac masses arise by concentration. But concentration phenomena arise also in the context of functions. Define u n : ( 1, 1) R by { n, if x [0, 1/n], u n (x) = 0, otherwise once can easily check that u n δ0 in M( 1, 1). (2) One can also smooth out u n and (2) still holds. Observe here also that while u n 0 in M(0, 1), we do not have that u n converges weakly to 0 in L 1 (0, 1) (in view of DunfordPettis there is a lack of equiintegrability). 5
6 Here is another example. Define u n : ( 1, 1) R by n, if x ( 1/n, 0), u n (x) = n, if x (0, +1/n), 0, otherwise. once can check that u n u := 0 in M( 1, 1), but no subsequence of u n converges weakly in L 1 ( 1, 1) (again there is a lack of equiintegrability). Indeed, for ϕ = χ (0,1) L ( 1, 1), we have 1 1 u n ϕ dx = 1 0 u n dx = 1, while 1 1 uϕ dx = 0. In other words, the sequence is not precompact in weak topology of L 1 ( 1, 1), although 1 1 u n dx = 2 n, i.e., the sequence u n is uniformly bounded (equibounded) in L 1 ( 1, 1). Observe also that there is a loss of energy : 0 = u L 1 ( 1,1) < lim inf u n L 1 ( 1,1) = 2. Here the energy disappears as a measure. Finally, note that u n L p ( 1,1) for any p > 1. Of course, concentration phenomena show up also in L p spaces with p > 1. For p > 1, define u n : ( 1, 1) R by { n 1 p, if x [0, 1/n], u n (x) = 0, otherwise. we have u n L p ( 1,1) = 1 for all n and u n 0 in L p ( 1, 1), but u n does not converge strongly in L p ( 1, 1) since the sequence u n L p ( 1,1) concentrates. But we have u n 0 in L q ( 1, 1) for any 1 q < p. Indeed, we have 1 1 u n q dx = n q p 1 as q p < 1. Hence the sequence the sequence u n L p ( 1,1) does not concentrate. 6
7 2.3 Nonlinearity destroys weak convergence If u n converges weakly but not strongly to u, then for a generic nonlinear function the sequence f(u n ) does NOT converge weakly to f(u). Here is an example. Let u n (x) = sin(nx), x (0, 2π), f(ξ) = ξ 2 for ξ R. but u n u := 0 in L p (0, 2π) p 1 ( in L (0, 2π) if p = ), f(u n ) = sin 2 (nx) = 1 2 (1 cos(2nx)) 1 f(u) = 0. 2 Observe here also that f(u n ) is uniformly bounded (equibounded) in L (0, 2π). Another example is f(ξ) = max(0, ξ), ξ R. If then f(u n ) 1 π f(u) = 0 in L1 (0, 2π). f(ξ) = ξ, ξ R, f(u n ) 2 π f(u) = 0 in L1 (0, 2π). Here is a final example. Let f be any continuous (nonlinear) function and select two number a < b such that ( ) a + b f(a) + f(b) f. 2 2 Define a sequence of functions u n : (0, 1) R by { a, if x [ i/n, (i + 1 u n (x) = 2 )/n], i =, 0,..., n 1, b, otherwise. one can easily check that but u n u := a + b 2 in L (0, 1) and hence weakly in any L p (0, 1), 1 p <, f(u n ) f := f(a) + f(b) 2 ( a + b f(u) = f 2 ). 7
8 3 Weak compactness in L 1 Let us now turn to the more delicate issue of weak compactness in L 1. We start with a definition. Definition 3.1 (equiintegrability). Let Ω R N and L 1 (Ω) be a family of integrable functions. We say that is an equiintegrable family if the following two conditions hold: 1. For any ε > 0 there exists a measurable set A with A < such that u < ε, Ω\A for all u. (This condition is trivially satisfied if Ω <, since then we can take A = Ω.) 2. For any ε > 0 there exists δ > 0 such that for every measurable set E with E < δ there holds u < ε for all u. E We have the following three equivalent formulations of the equiintegrability property. Lemma 3.1. Let Ω R N and L 1 (Ω) be a family of integrable functions. 1. is equiintegrable if and only if for any sequence of measurable sets E n with E n there holds lim sup u dx = 0. u E n 2. If Ω < and is bounded in L 1 (Ω), then is equiintegrable if and only if { } u L 1 (Ω) : Ψ( u ) dx 1, for some increasing function Ψ : [0, ) [0, ] satisfying Ω Ψ(ξ) lim. ξ ξ 3. If Ω < and is bounded in L 1 (Ω), then is equiintegrable if and only if lim sup u dx = 0. ξ u { u >ξ} 8
9 Here is a restatement of (2). If Ω < and is bounded in L 1 (Ω), then the family is equiintegrable if and only if sup Ψ( u ) dx <, u Ω for some increasing function Ψ : [0, ) [0, ] satisfying Ψ(ξ) lim. ξ ξ Let us give an example illustrating how to use this restatement in the context of sequences of functions. Let Ω < and u n : Ω R be a sequence of functions that are equibounded (uniformly bounded) in L 1 (Ω), i.e., u n dx C, n. Ω a sufficient condition for the sequence u n to be equiintegrable is that there exists a constant C, independent of n, such that u n 1+θ dx C, Ω for some θ > 0. The next theorem gives a necessary and sufficient condition (namely equiboundedness and equiintegrability) for compactness with respect to the weak convergence in L 1. Theorem 3.1 (DunfordPettis). Let u n : Ω R be a sequence in L 1 (Ω). Suppose 1. the sequence u n is equibounded in L 1 (Ω), i.e., 2. the sequence u n is equiintegrable. sup u n L n 1 (Ω) <, there exists a subsequence of u n that converges weakly in L 1 (Ω). Conversely, if u n converges weakly in L 1 (Ω), then 1. and 2. hold. The following theorem is the L 1 analog of Theorem 1.5. Theorem 3.2 (Characterizations of weak convergence in L 1 ). Let u n : Ω R be a sequence in L 1 (Ω) and u L 1 (Ω). Suppose the sequence u n is equibounded in L 1 (Ω) and equiintegrable. the following statements are equivalent: 1. u n u in L 1 (Ω). 9
10 2. u n u in M(Ω). 3. u n u in D (Ω). 4. For any Borel set E Ω, E > 0, (u n ) E (u) E. 5. For any cube Q Ω, Q > 0, (u n ) Q (u) Q. The next lemma is simple but still very useful. Lemma 3.2 (products of weakstrong converging sequences). Let u n, u, v n, v : Ω R be measurable functions. 1. If u n u a.e. in Ω, u n L (Ω) C for all n, and v n v in L 1 (Ω), then u n v n uv in L 1 (Ω). 2. If u n u in L 1 (Ω), u n L (Ω) C for all n, and v n v in L 1 (Ω), then u n v n uv in L 1 (Ω). The next lemma is useful in many applications. Lemma 3.3 (Vitali). Let Ω R N be bounded and u n : Ω R be a sequence in L 1 (Ω). Suppose 1. lim u n (x) exists and is finite for a.e. x Ω, 2. the sequence u n is equiintegrable. lim u n (x) dx = Ω Ω lim u n(x) dx. A typical application of Vitali s lemma is provided by the next simple lemma. Lemma 3.4. Let Ω R N be bounded and u n : Ω R be a sequence in L 1 (Ω). Suppose 1. u n u a.e. in Ω, 2. the sequence u n is bounded in L p (Ω) for some p > 1. u n u in L r (Ω) for all 1 r < p. 10
11 Proof. By a previous theorem, Define and u L p (Ω) (and also u n u in L p (Ω)). v n = u u n r, r < p. v n 0 a.e. in Ω v n is bounded in L p r (Ω) and p/r > 1. Hence the sequence v n is equiintegrable, so that by Vitali s lemma v n dx = 0, that is u n u in L r (Ω). lim The next lemma recall the well known fact that convex (concave) functions are lower (upper) semicontinuous with respect to the weak convergence. Lemma 3.5 (weak lower semicontinuity of convex functions). If F : R R is convex and u n u in L 1, then then If F : R R is concave and F (u) dx lim inf u n u in L 1, F (u) dx lim sup F (u n ) dx. F (u n ) dx. 4 Additional reading  collection of some results Definition 4.1 (convergence in measure). Let u n, u : Ω R be measurable functions. We say that u n u in measure if lim {x Ω : u n(x) u(x) > ε} = 0, ε > 0. 11
12 The a.e. convergence and convergence in measure can be easily compared. Indeed, the following statements hold: 1. If Ω < and u n u a.e., then u n u in measure. 2. If u n u in measure, then a subsequence of u n converges a.e. to u. We also remark that if u n u in L p (Ω), then a subsequence of u n converges a.e. to u. This is trivial for p =, and follows for 1 p < by Chebyshev s inequality, stating that {x Ω : g(x) > ε} 1 g dx, ε > 0, 0 g L 1 (Ω), ε which implies immediately that Ω u n u in measure, and hence (2) applies. The following lemma exploits convergence of the L p norm to get strong convergence from a.e. convergence. Lemma 4.1. Let u n : Ω R be a sequence in L p (Ω), 1 p <, and suppose 1. u n u a.e. in Ω, 2. u n L p (Ω) u n L p (Ω). u n u in L p (Ω). Let us now look at a refinement of Fatou s lemma. Suppose { u n u in L p (Ω), u n u a.e. in Ω. (3) As explained before, concentration phenomena will arise when we have the weak convergence in (3) simultaneously with the a.e. convergence in (3). From the a.e. convergence and Fatou s lemma, u L p (Ω) lim inf u n L p (Ω). But we know already this from the weak convergence. Brezis and Lieb have analyzed this situation more carefully and proved the following sharpened assertion: 12
13 Theorem 4.1 (BrezisLieb). Suppose (3) holds with 1 p <. ( ) u n p L p (Ω) u n u p L p (Ω) = u L p (Ω). lim The message is that u n in the limit (as measured in the L p norm) decouples into u n u and u. Note that the case p = 2 is immediate and does need the a.e. convergence in (3). Indeed, if u n u in L 2 (Ω), then we have ( ) lim u n 2 dx u n u 2 dx Ω Ω ( ) = lim u n 2 u n 2 + 2u n u u 2 dx Ω ( ) = lim 2u n u u 2 dx = u 2 dx. Ω Ω If Ω <, then a.e. convergence is equivalent to uniform convergence, up to arbitrarily small sets. This is the content of Egoroff s theorem. Theorem 4.2 (Egoroff). Suppose Ω < and that a sequence of measurable functions u n : Ω R converges to u a.e. in Ω. for any ε > 0 there exists a measurable set Ω ε such that Ω \ Ω ε < ε and u n u uniformly in on Ω ε. We recall next some deeper results showing that under additional assumptions the convergence of a sequence in L p can be improved. The first result says that convergence a.e. or in measure implies weak convergence if the norms are uniformly bounded. Theorem 4.3. Let u n : Ω R be a sequence in L p (Ω), 1 < p <, converging a.e. or in measure to u, with u n L p (Ω) C, n. u L p (Ω) and u n u in L p (Ω). sing the uniform convexity of the L p spaces, 1 < p <, it is not hard to prove the following result: Theorem 4.4 (RadonRiesz). With 1 < p <, let u n : Ω R be a sequence in L p (Ω) converging weakly to u L p (Ω) and u n L p (Ω) u L p (Ω). u n u in L p (Ω). 13
14 This theorem shows that if a weakly converging sequence in L p, 1 < p <, does not converge strongly in L p, then there is must be a loss in the p  norm energy. As a corollary of the previous theorem or by a direct proof, here is a weaker form of the RadonRiesz theorem. Corollary 4.1. Let u n : Ω R be a sequence in L p (Ω) and u L p (Ω), 1 < p <. Suppose u n u a.e. in Ω, u n L p (Ω) u L p (Ω). u n u in L p (Ω). Final remarks A large part of this section is based on Evans lecture notes [1], see also Evans book [2]. References [1] L. C. Evans. Weak convergence methods for nonlinear partial differential equations, volume 74 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC, [2] L. C. Evans. Partial differential equations. American Mathematical Society, Providence, RI,
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