Sobolev spaces. May 18

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1 Sobolev spaces May Weak derivatives The purpose of these notes is to give a very basic introduction to Sobolev spaces. More extensive treatments can e.g. be found in the classical references [1] and [3]. Sobolev spaces are natural generalisations of the Lebesgue spaces L p. Simply stated, if d is an open set, 1 p, and m is a positive integer, the Sobolev space W m,p () consists of functions (or equivalence classes of functions) in L p () whose partial derivatives up to order m are in L p (). We will start by giving a more precise definition of what we mean by partial derivatives, since these are not defined in the classical sense. Throughout these notes, dx refers to Lebesgue measure on d and we assume that all functions are real-valued (analogous results hold for complex-valued functions). We will also identify elements of L p with functions. Before discussing weak derivatives, we introduce some notation. The class of continuous, real-valued functions on with compact support is denoted C c () and the subset of smooth functions is denoted Cc (). 1 Some results concerning approximation of L p -functions by functions in Cc are collected in the appendix. We denote by L p loc () the set of measurable functions f such that f K L p (K) for any compact subset K of.. Note that L p loc () L1 loc () whenever 1 p. If f and ϕ are C 1 functions on and ϕ has compact support, then f k ϕ dx = k fϕ dx, where k = / xk. This can e.g. be proved using Fubini s theorem and the usual integration by parts formula in one dimension. We take this as our definition of weak derivatives. Definition. Let f L 1 loc (). We say that f is weakly partially differentiable with respect to x k if there exists a function g L 1 loc () such that f k ϕ dx = gϕ dx 1 The notation C 0 () and C 0 () is also common. 1

2 for every ϕ Cc (). If f is weakly partially differentiable with respect to x k, we call g the weak partial derivative of f with respect to x k and denote it by k f. The fact that we took ϕ to be C instead of just C 1 in the definition is not essential, but is convenient is some situations. It s not difficult to show that if f is weakly partially differentiable, then the integration by parts formula also holds if ϕ is only C 1. For the above definition to make sense, we should prove that g is unique (up to a.e. equality). Proposition 1 (Uniqueness of weak derivatives). Assume that g and h are both weak partial derivatives with respect to x k of f L 1 loc (). Then g = h a.e. Proof. By linearity we have that (g h)ϕ dx = 0 for all ϕ Cc (). Setting u = g h in and u 0 in c, we find that d uϕ dx = 0 for all ϕ C c ( d ) such that supp ϕ. Hence, if x, we have that d u(y)j ε (x y) dy = 0 for sufficiently small ε > 0, where J ε is the family of mollifiers constructed in the appendix (J ε (x ) is supported in B ε (x) if ε > 0 is small). Letting ε 0 + we obtain by Theorem 12 in the appendix that u = 0 a.e in, which concludes the proof. Example. Let f : be defined by f(x) = x. Then it s clear that f L 1 loc (). Moreover, if ϕ C c () with supp ϕ [a, b], a < 0 < b, then b 0 x ϕ (x) dx = xϕ (x) dx xϕ (x) dx 0 b = = 0 ϕ(x) dx + a 0 a sgn(x)ϕ(x) dx. Hence, the weak derivative of x is 1, x > 0, sgn(x) = 0, x = 0, 1, x < 0. ϕ(x) dx + [xϕ(x)] b 0 [xϕ(x)] 0 a Note that the weak derivative in this example is simply the pointwise derivative wherever it exists. This will be generalised in Section 3. 2

3 2 Sobolev spaces In this section we define Sobolev spaces and discuss some of their basic properties. For simplicity we restrict ourselves to Sobolev spaces of order one, although the definitions and results are easily generalised to higher order spaces. Definition. Let d an open, non-empty set and let 1 p. The Sobolev space W 1,p () is defined as the set of f L p () such that all the weak derivatives k f, k = 1,..., d, exist and belong to L p (). We equip it with the norm when 1 p <, and when p =. f 1,p = ( f p p + f 1, = f + d k f p p k=1 ) 1 p d k f There are several equivalent ways of defining the norm (in the sense that they generate the same topology). When p = 2, ( ) d (f, g) = fg + k f k g dx defines an inner product on W 1,2 () and 1,p is the induced norm with our choice of definition. This space is so important that it has a special notation: H 1 () = W 1,2 (). Theorem 2. W 1,p () is complete for any p [1, + ]. In particular, H 1 () is a Hilbert space. Proof. Let {f n } be a Cauchy sequence in W 1,p (). Then {f n } is also a Cauchy sequence in L p (). Hence, it converges to some f in L p (). Similarly, all the partial derivatives { k f n } converge to some function g k in L p (). It suffices to prove that g k is the weak partial derivative of f with respect to x k. But this follows by taking limits in the identity f n k ϕ dx = k f n ϕ dx, ϕ Cc (), which is possible by Hölder s inequality since ϕ and k ϕ belong to L q (), where 1/p + 1/q = 1. k=1 k=1 We end this section with a natural approximation result. Theorem 3. C c ( d ) is dense in W 1,p ( d ) whenever 1 p <. 3

4 Proof. Let f W 1,p ( d ) and let J ε be as in the appendix. Using the fact that k (J ε f) = J ε k f (differentiate under the integral sign and move the derivative to f using the definition of weak derivatives), we obtain that f ε = J ε f is in W 1,p ( d ) C ( d ) with f ε f as ε 0 +. The only problem is that J ε f doesn t have compact support. Now let ϕ Cc ( d ) be a cut-off function as in Corollary 14. Then ϕ(x/n) is supported in the ball of radius 2n and is equal to one in the ball of radius n. We have that ϕ( /n)f ε f ε p p f ε (x) p dx 0 x n as n. Moreover, xk ϕ(x/n)f ε (x) = 1 n ( kϕ)(x/n)f ε (x) + ϕ(x/n) k f ε (x). The second term converges to f ε in L p as n by the previous argument, whereas the first converges to 0 since ( k ϕ)( /n)f ε (x) p k ϕ f ε (x) p. The result follows by first taking ε sufficiently small and then taking n sufficiently large. 3 elation to absolutely continuous functions If I is an interval, we let AC(I) be the class of real-valued absolutely continuous functions f : I. ecall that f AC([a, b]) if and only if f (x) exists for a.e. x [a, b], f is integrable on [a, b] and f can be recovered from f by the formula (1) f(x) = f(a) + x a f (x) dx (we can of course replace a in this formula by any other point in [a, b]). If f is absolutely continuous it is also continuous. Similarly, we let AC loc (I) be the class of locally absolutely continuous functions, where f : I is locally absolutely continuous on I if it s absolutely continuous on any compact subinterval [a, b] I. Note that AC loc ([a, b]) = AC([a, b]) if [a, b] is a compact interval. It s straightforward to see that f AC loc (I) if and only if f is differentiable a.e. on I, f L 1 loc (I) and f can be recovered from f from the formula (1) where a is an arbitrary point in I. The goal of this section is to relate absolutely continuous functions to Sobolev functions, but we first need a lemma. Lemma 4. Assume that f L 1 loc () has weak derivative 0. Then f is constant (a.e.). Proof. Suppose that ϕ Cc (). Then x ϕ(y) dy is smooth, vanishes when x a, for some a, and is constantly equal to ϕ dx when x b, for some b. Hence, it belongs to Cc () if and only if ϕ dx = 0 Let ϕ 0 be a fixed function in Cc () with ϕ 0 dx = 1. For a general ϕ Cc (), we let I(ϕ) = ϕ dx. Then I(ϕ I(ϕ)ϕ 0 ) = I(ϕ) I(ϕ)I(ϕ 0 ) = 0. 4

5 Hence, ψ(x) = x (ϕ(y) I(ϕ)ϕ 0(y)) dy belongs to Cc (). By assumption we have that 0 = fψ dx = f(ϕ I(ϕ)ϕ 0 ) dx ( ) ( ) = fϕ dx ϕ dx = (f I(fϕ 0 ))ϕ dx. fϕ 0 dx Since this holds for all ϕ Cc (), we find as in the proof of Proposition 1 that f = I(fϕ 0 ) a.e. and this concludes the proof. Theorem 5. f W 1,p () if and only if f AC loc () with f, f L p (), where f is the a.e. pointwise derivative of f. emark: in the above statement we mean that there is a representative of f W 1,p which is locally absolutely continuous. Proof. f AC loc () with f, f L p (), then using the integration by parts formula for absolutely continuous functions (see [2, Cor ], we obtain that f has weak derivative f L p (). Assume instead that f W 1,p (). Let g be the weak derivative of f and set G(x) = x 0 g(y) dy. Then G is locally absolutely continuous with G (x) = g(x) a.e. Hence G and f both have weak derivative g. From Lemma 4 we obtain that G f is constant a.e. It follows that f AC loc () with f = g. One can show a similar result in the case d 2 where absolute continuity is replaced by absolute continuity on a.e. line parallel to the coordinate axes (see [3]). 4 Sobolev embedding theorems In Section 3 we saw that if f W 1,p (), for some p [1, + ], then f is in fact absolutely continuous. In particular, it is continuous. One can actually show quite a bit more. ecall that a function f : d is called (uniformly) Lipschitz continuous if there exists a constant C 0 such that f(x) f(y) C x y for all x, y d. 5

6 Similarly, f is called (uniformly) Hölder continuous with exponent α (0, 1) if there is a constant C 0 such that f(x) f(y) C x y α for all x, y d. For simplicity, we restrict ourselves to one dimension in these notes. Throughout this section we identify an element f W 1,p () with its locally absolutely continuous representative. We begin by discussing the case p =. Proposition 6. If f W 1, (), then f is Lipschitz continuous. Proof. This follows from the calculation x f(x) f(y) = f (t) dt f x y. y One can actually show a converse: if f is bounded and Lipschitz, then f W 1, () (see [3]). We next consider the case 1 < p <. Proposition 7. Let 1 < p <. If f W 1,p (), then f is Hölder continuous with exponent 1 1/p. Moreover, f is bounded with lim x ± f(x) = 0. Proof. Let q be the conjugate exponent of p and α = 1/q. Then α = 1 1/p and x f(x) f(y) = f (t) dt f p x y α by Hölder s inequality. This shows that f is Hölder continuous. Let f C c y (). Note that f p is continuously differentiable and that d dx f(x) p = p f(x) p 1 sgn(f(x))f (x). Integrating this equality from to x and using Young s inequality, we obtain that x f(x) p p f(y) p 1 f (y) dy ( f(y) q(p 1) p + f ) (y) p dy q p where B = max{1, p/q}. Hence, B f p 1,p, (2) f C f 1,p with C = B 1/p. 6

7 Now let f W 1,p () and let {f n } C c () with f n f in W 1,p. Then f n f m C f n f m 1,p, so f n converges uniformly to some continuous and bounded function g. Since f n f in L p we must have g = f (a.e.) and since f n f while f n 1,p f 1,p, we obtain that (2) also holds for f. Let ε > 0 and choose n such that f f n < ε. Choose such that f n (x) = 0 for x. Then f(x) = f(x) f n (x) < ε when x. Hence, lim x ± f(x) = 0. When p = 1, one can still show that following result, whose proof we leave to the reader. Proposition 8. If f W 1,1 (), then f is absolutely continuous, of finite variation and vanishes at ±. The results above are examples of Sobolev embedding theorems, showing that certain Sobolev spaces are embedded in other spaces. Similar results hold for d 2 and for Sobolev spaces of higher order (see [1, 3]). In particular, these are useful in the study of partial differential equations, where they can be used to show that weak solutions are in fact classical solutions (see the next section). 5 An application Weak derivatives and Sobolev spaces are very useful for studying partial differential equations. The idea is that it allows one to separate the questions of existence and regularity. One starts by reformulating the equation in a weak sense and proves the existence of a weak solution (which a priori doesn t have enough regularity to satisfy the equation in the classical sense). One then shows that the solution in fact has enough regularity to satisfy the equation in the classical sense. As an example we consider the (elliptic) partial differential equation (3) f + f = g, where = d 2 is the Laplace operator. Here g is a given function and we want to find a solution f. For simplicity we consider the equation in d, although one can also consider it in some domain d with suitable boundary conditions. In order to find a solution, we begin by relaxing what we mean by a solution. Assume that g L 2 ( d ). Multiplying the equation by a test function ϕ Cc ( d ) and integrating by parts, we obtain that (4) ( f ϕ + fϕ) dx = d gϕ dx. d Note that this equation makes sense even if f is just in H 1 ( d ). 7

8 Definition. We say that f H 1 ( d ) is a weak solution of (3) if (4) holds for each ϕ C c ( d ). We next note that f H 1 ( d ) is a weak solution if and only if (4) holds for each ϕ H 1 ( d ). Indeed, one direction is clear since Cc ( d ) H 1 ( d ) and the other direction follows by approximating ϕ H 1 ( d ) by a sequence of functions in Cc ( d ). With this at hand, we can prove the following theorem. Theorem 9. For each g L 2 ( d ), there exists a unique weak solution f H 1 ( d ) of (3). Proof. Note that ϕ gϕ dx is a bounded linear functional on H 1 ( d ) by d Hölder s inequality. Hence the Fréchet-iesz representation theorem (see Exercise in [2]) shows that there is a unique function f H 1 ( d ) such that (f, ϕ) = gϕ dx, d where (f, ϕ) = (fϕ + f ϕ) dx d is the H 1 inner product of f and ϕ. But this means that f is a weak solution of (3). One can in fact show that the above solution belongs to the space H 2 ( d ) consisting of functions f L 2 ( d ) having weak partial derivatives of all orders less than or equal to 2 which belong to L 2 ( d ) and that (3) holds a.e. This can e.g. be done using Fourier analysis. In general, one can show that if g H n ( d ), for some integer n 1, then the solution f H n+2 ( d ). Combining this with a suitable Sobolev embedding, one can show that f is C 2 and satisfies (3) in the classical sense if n is sufficiently large (depending on d). This is, however, somewhat unsatisfactory. One would guess that if g is continuous then f is a classical solution (i.e. f C 2 and satisfies the equation pointwise). Somewhat surprisingly perhaps, this turns out to be false in general. However, if one imposes a little bit more regularity and requires that g is Hölder continuous, then f is indeed a classical solution. The previous result can also be obtained using Fourier analysis, but the above method has the advantage that it generalises easily to elliptic PDEs with variable coefficients and to other domains. Appendix This appendix contains some results on approximation by smooth functions. It is often convenient to be able to approximate an L p function by a smooth function and in particular by a smooth function with compact support. ecall that a 8

9 function f : d is said to have compact support if it vanishes outside a compact set. The support of f is defined as supp f = {x: f(x) 0}. This definition isn t very useful for L p functions, since the support can change if f is altered on a set of measure zero. The essential support, ess supp f, as the smallest closed set outside of which f = 0 a.e., i.e. ess supp f = d \ {U d : U is open and f(x) = 0 for a.e. x U}. Clearly ess supp f = ess supp g if f = g a.e. Moroever, if f is continuous, then ess supp f = supp f since then f(x) = 0 for a.e. x U if and only if f(x) = 0 for all x U. In the future we will simply talk about the support of f, although what we really mean is the essential support. The class of continuous, real-valued functions on d with compact support is denoted C c ( d ). and the subset of smooth functions is denoted C c ( d ) Before approximating with smooth functions, we recall that any L p function can be approximated by continuous functions with compact support. Lemma 10. C c ( d ) is dense in L p ( d ) whenver 1 p <. This is discussed for d = 1 in Ch. 3.4 of Cohn. The general case is proved in a similar way. We should also show that C c ( d ) is not empty, a fact which is not obvious. ( d )). There exists a non-negative func- Lemma 11 (Existence of functions in Cc tion ϕ Cc ( d ) with supp ϕ = B 1 (0). Proof. Note first that the function f(t) = { e 1/t t > 0 0 t 0 is C on with support [0, ). Indeed, for t > 0 all the derivatives are polynomials in 1/t times e 1/t and t a e 1/t 0 as t 0 + for any a. The function ϕ defined by ϕ(x) = f(1 x 2 ) satisfies the conditions. The basic idea in the approximation argument is to use convolutions with smooth functions. ecall that the convolution f g of two measurable functions f and g is defined by (f g)(x) = f(x y)g(y) dy d 9

10 whenever the integral is defined. We can think of this as a sort of average of f with weight g. Thus, convolutions tend to smear out at a function, making it smoother. Using the change of variables y x y, we find that (f g)(x) = (g f)(x) = f(y)g(x y) dy. d By Hölder s inequality, we obtain that f g L if f L p and g L q with = 1, p, q [1, ]. Moreover, 1 p + 1 q (5) f g f p g q. From Minkowski s inequality for integrals, we obtain that (6) f g p f p g 1, 1 p, so that f g L p if f L p and g L 1. If f and g have compact support, it also follows from the definition that supp(f g) supp f + supp g. Finally, a very important property of convolutions is the following. If g L 1 ( d ) and f C 1 ( d ) is bounded with bounded partial derivatives, then f g C 1 with k (f g) = k f g. This follows from the definition of the convolution and differentiation under the integral sign. We can now construct a smooth approximation of the identity as follows. After dividing ϕ in Lemma 11 by d ϕ dx, we can assume that d ϕ dx = 1. We set J ε (x) = ε d ϕ(ε 1 x), ε > 0. Then J ε Cc ( d ) with supp J ε = B ε (0) and J d ε (x) dx = 1 for all ε > 0 (change of variables). One can think of J ε as converging to a unit mass at the origin. This can be made sense of using distribution theory. The family {J ε } ε>0 is an example of mollifiers. Theorem 12. (i) If f L 1 loc (d ), then J ε f C ( d ). (ii) If f L 1 ( d ) has compact support, then J ε f C c ( d ). (iii) If f L 1 loc (d ), then (J ε f)(x) f(x) whenever x is a Lebesgue point of f. In particular, J ε f f a.e. (iv) If f C( d ), then J ε f f uniformly on any compact subset of d. 10

11 (v) If f L p ( d ), where 1 p <, then J ε f p f p and lim ε 0 + J ε f f p. Proof. The first statement follows by repeated differentiation under the integral sign. The second statement follows since supp J ε supp f + supp J ε and both f and J ε have compact supports. We have that (J ε f)(x) f(x) = J ε (x y)(f(y) f(x)) dy d = ε d J(ε 1 (x y))(f(y) f(x)) dy and therefore B ε(x) (J ε f)(x) f(x) λ(b 1 (0)) J 1 λ(b ε (x)) if x is a Lebesgue point of f. This proves (iii). B ε(x) f(y) f(x) dy 0 We have that (J ε f)(x) f(x) = J ε (y)(f(x y) f(x)) dy d = J(y)(f(x εy) f(x)) dy B 1 (0) Since f is uniformly continuous on any compact subset K d, it follows that sup (x,y) K B1 (0) f(x εy) f(x) 0 as ε 0+, proving (iv). The first part of (v) is a consequence of (6). Let δ > 0. Using Lemma 10 we can find g C c ( d ) such that f g p < δ and it follows from (ii) that J ε g g has support in a compact set which is independent of ε (sufficiently small) and hence from (iv) that J ε g g uniformly. It follows that J ε g g p < δ if ε is sufficiently small. For ε sufficiently small, we therefore have This proves (v). J ε f f p J ε f J ε g p + J ε g g p + g f p 2 g f p + J ε g g p < 3δ. Corollary 13. C c ( d ) is dense in L p ( d ) for 1 p <. Proof. This follows by approximating f L p ( d ) with a function g C c ( d ) using Lemma 10 and then approximating g with a function in Cc ( d ) using Theorem

12 The following construction of a smooth cut-off function is often useful. Corollary 14. There exists a function ϕ C c ( d ) with 0 ϕ 1, ϕ = 1 on B 1 (0) and supp ϕ B 2 (0). Proof. Take f = χ B3/2 (0)(x) and ϕ(x) = (J 1/2 f)(x) = x y 3/2 J 1/2 (y) dy. Since d J 1/2 (y) dy = 1 and J 1/2 0, it follows that 0 ϕ 1. Moreover, if x 1, then supp J 1/2 = B 1/2 (0) B 3/2 (x). Hence, ϕ(x) = 1 on B 1/2 (0). Finally, supp ϕ supp J 1/2 + supp f B 2 (0). eferences [1]. A. Adams and J. J. F. Fournier, Sobolev spaces, vol. 140 of Pure and Applied Mathematics (Amsterdam), Elsevier/Academic Press, Amsterdam, second ed., [2] D. L. Cohn, Measure theory, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser/Springer, New York, second ed., [3] L. C. Evans and. F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CC Press, Boca aton, FL,