THE L 2 HODGE THEORY AND REPRESENTATION ON R n


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1 THE L 2 HODGE THEORY AND REPRESENTATION ON R n BAISHENG YAN Abstract. We present an elementary L 2 Hodge theory on whole R n based on the minimization principle of the calculus of variations and some basic results of real analysis and Sobolev spaces. We also use the elementary and selfcontained proofs to provide some representation results on curlfree and divergencefree fields in terms of local functions.. Introduction The classical Hodge theory deals with decomposition of a differential form into closed and coclosed forms [4, 5]. For vector fields on R n, the closed forms become the curlfree fields and coclosed forms become divergencefree fields. These fields also present a useful tool in studying many important physical and applied problems, such as the Maxwell equations, in particular, the electromagnetics, and the sourceless or incompressible fluids. It is wellknown that the classical L p Hodge theory on the whole Euclidean space can be established by the Riesz transforms and potentials [3, 5, 7]. In this short note, we present an elementary L 2 Hodge theory on whole R n based only on the minimization principles of the calculus of variations and some basic results of real analysis and Sobolev spaces. Let X = L 2 (R n ; R n ) denote the Hilbert space of real functions u = (u,, u n ), u i L 2 (R n ), with the inner product and norm defined by (u,v) = (u v + + u n v n )dx = u v dx; u = (u,u) /2. R n R n For u L 2 (R n ; R n ), we define divu, Curlu = (Curlu) ij as distributions as follows: divu, ϕ = (u ϕ x + + u n ϕ xn )dx = R n u ϕ dx; R n (Curlu) ij, ϕ = (u i ϕ xj u j ϕ xi )dx, R n ϕ C (R n ). 2 Mathematics Subject Classification. 49J45, 49J2, 35G3. Key words and phrases. Hodge decomposition, divergencefree, curlfree.
2 2 BAISHENG YAN If we denote u = (u i x j ) to be the n n matrix of distributional derivatives of u, where u i x j = u i / x j denotes the distributional partial derivative, then divu = tr u = u = n i= ui x i ; Curlu = u ( u) T = (u i x j u j x i ) n i,j=. Note that in the sense of distribution, the Laplacian operator of any L 2 field m (defined for each component of the field) can be written as (.) m = (divm) + Div(Curlm), where, for any matrixvalued distribution A = (a ij ), Div A denotes the vectorvalued distribution defined by (Div A) i = n j= (a ij) xj. In the case n = 2 or n = 3, the operator Curlu can be identified as follows: Curlu u = u = div(u ) = u x 2 u 2 x (n = 2); Curlu curlu = u = (u 3 x 2 u 2 x 3, u x 3 u 3 x, u 2 x u x 2 ) (n = 3). Define the subspaces of divergencefree and curlfree fields as follows: X div = {u L 2 (R n ; R n ) divu = in the sense of distribution}; X Curl = {u L 2 (R n ; R n ) Curlu = in the sense of distribution}. Then we have that X div X Curl = {} (see Lemma 2.) and the wellknown Hodge decomposition theorem: L 2 (R n ; R n ) = X div X Curl (see Theorem 2.5). One of the main purposes of this paper is to characterize the space X Curl, which, in the case n = 2, also characterize the space X div. Another main result is to provide a similar result for X div when n = 3. To do so, we introduce the linear function spaces: and, for n = 3, Y = {f L 2 loc (Rn ) f X} M = {m L 2 loc (R3 ; R 3 ) curlm = m L 2 (R 3 ; R 3 )}. It is easy to see that f X Curl for all f Y and, for n = 3, curlm X div for all m M. The converse is also true; we have the following results. Theorem.. There exists a uniform constant C n > such that, for every v X Curl, there exists a f Y satisfying that (.2) (a) v = f; (b) sup R R n+2 f 2 dx C n v R 2 dx, n B R () where B R (y) = {x R n x y < R} denotes the ball centered y of radius R.
3 THE L 2 HODGE THEORY AND REPRESENTATION ON R n 3 Theorem.2. Let n = 3. There exists a uniform constant C > such that, for every w X div, there exists a m M satisfying that (.3) (a) w = m; (b) sup R R 3 m 2 dx C w 2 dx, R n Q R () where Q R (y) = {x R 3 x i y i < R, i =, 2, 3} denotes the cubes centered y of sidelength 2R. The estimate (.2b) in Theorem. above is not sharp as we will obtain some better estimates later. However, the estimate (b) in both (.2) and (.3) does provide a way to represent a curlfree or divergencefree field v or w by some local function f or m, with f being viewed as the potential function of v and m the velocity of w; initially, these local functions have only been defined as the Schwartz distributions [4, 5]. The estimate (.2b) also suggests that we equip the space Y with the norm f defined by (.4) f 2 = R n f(x) 2 dx + sup R Let Y be the subspace of Y defined by R n+2 f(x) 2 dx. B R Y = {f H loc (Rn ) f < }. It can be easily shown that Y is indeed a Banach space with the norm defined. In what follows, we shall try to find the minimal subspace Z of Y for which the gradient operator : Z X Curl is bijective. In fact such a Z can be completely determined when n 3. Theorem.3. Let Y 2 be the closure of C (Rn ) in Y. Then, for n 3, Y 2 = {f L 2n n 2 (R n ) f L 2 (R n ; R n )}. Furthermore, Y 2 has the equivalent norms f f L 2 (R n ) f Y 2, and the gradient operator : Y 2 X Curl is bijective. The proof of this theorem relies on the SobolevGagliardoNirenberg inequality for H (R n ) functions when n 3 (see [2]); note that in this case the finite number 2 = 2n n 2 is the Sobolev conjugate of n. In the case n = 2, there is no such a SobolevGagliardoNirenberg inequality; instead, there is a JohnNirenbergTrudinger type of BMOestimates for functions with gradient in L 2 (R 2 ) (see [2, 7]). However, we shall try to avoid the BMOestimates. One of the minimal subspaces of Y on which the gradient operator is bijective can be characterized as follows.
4 4 BAISHENG YAN Theorem.4. Let n = 2 and Z be the closure in Y of the subspace S = {ϕ ϕ ρ ϕ H (R 2 )}, where ϕ ρ = R ϕ(x)ρ(x)dx and ρ(x) is the weight function defined by 2 (.5) ρ(x) = ( χ 2π { x } + ) x 4 χ { x >}. Then Z has the equivalent norms f f L 2 (R 2 ) f Z, and the gradient operator : Z X Curl is bijective. 2. Variational principles and the Hodge decomposition We first prove the following useful result. Lemma 2.. Let X div, X Curl be defined as above. Then X div X Curl = {}. Proof. Let m X div X Curl. Then Curlm = divm = in the sense of distributions. Hence, by (.) above, m = also in the sense of distributions. Hence m C (R n ; R n ) is harmonic and each of its components m i is a harmonic function in R n which also belongs to L 2 (R n ). Then the mean value property and Hölder s inequality imply that m i (x) B R (x) B R (x) m i dy c R n/2 m 2 for any x R n and R >, where B R (x) = {y R n y x < R} denotes the ball of radius R and center x. Letting R shows m i = and hence m =. Let Ω be any bounded domain in R n and denote by H (Ω) the usual Sobolev space that is the closure of C (Ω) under the usual H (Ω)norm. We always consider functions in H (Ω) as extended on the whole Rn by zero outside Ω. Given any u X = L 2 (R n ; R n ), for each R >, let B R = B R () and consider the following minimization problem: (2.) inf ϕ u 2 dx. ϕ H (B R) B R Standard direct method of the calculus of variations shows that this problem has a unique solution, which we denote by ϕ R, also extended to all R n. This sequence {ϕ R } is of course uniquely determined by u X. It also satisfies the following properties: (2.2) (2.3) ϕ R L 2 (R n ) u, R n ( ϕ R u) ζ dx = ζ H (Ω), Ω B R.
5 THE L 2 HODGE THEORY AND REPRESENTATION ON R n 5 Theorem 2.2. Given u X, it follows that ϕ R v in X as R and that v X Curl is uniquely determined by u. Moreover, this v satisfies v u = therefore, v = u if u X Curl. min v u ; v X Curl Proof. First of all, we claim ϕ R v weakly in X as R. Let v,v be the weak limits of any two subsequences { ϕ R } and { ϕ R }, where R, R are two sequences going to. We would like to show v = v, which shows that ϕ R v as R. Note that v,v X Curl and, by (2.3) above, for all bounded domains Ω R n, (2.4) (v u) ζ dx = (v u) ζ dx = R n R n for all ζ H (Ω). This implies div(v v ) =. Hence v v X div X Curl = {} by Lemma 2. above. We denote this weak limit by v X Curl. Note that, by (2.4), div(v u) =. Hence if Curlu = then v u X div X Curl = {}; hence v = u. We now prove ϕ R v in X as R. Taking ζ = ϕ R H (B R) in (2.4) and letting R we have (2.5) (v u) v dx =. R n Using ζ = ϕ R in (2.3), taking R and by weak limit, we have lim R R n ϕ R 2 dx = R n v u = R n v 2 dx. This implies ϕ R v strongly in L 2 (R n ; R n ). Finally, let us show (2.6) v u = min v X Curl v u. Given any v X Curl, choose the sequence ϕ R corresponding to v v. Since div(v u) =, it easily follows that (v v,v u) = lim R ( ϕ R,v u) =. Hence v u 2 = v v 2 + 2(v v,v u) + v u 2 v u 2 ; this proves (2.6). The proof is completed. Corollary 2.3. For every u X, there exist unique elements v X Curl, w X div such that u = v + w. Proof. Given u X, let v X Curl be defined as above, and let w = u v. Then u = v +w and, by (2.4) above, w X div. We now show that v,w are unique. Suppose u = v + w for another pair v X Curl and w X div. Then m = v v = w w X Curl X div. Hence v = v and w = w. Corollary 2.4. X Curl = X div, X div = X Curl.
6 6 BAISHENG YAN Proof. It suffices to prove X Curl = Xdiv. Given any v X Curl and w X div, let ϕ R H (B R) be the sequence determined by u = v as above. Since ϕ R v and (w, ϕ R ) = w(x) ϕ R (x) =, R n it follows easily that (w,v) = ; hence v Xdiv. This shows X Curl Xdiv. Assume u Xdiv. We will show u X Curl. Let u = v + w, v X Curl, w X div, be the Hodge decomposition in the previous corollary. Then = (u,w) = (v,w) + w 2 = w 2 ; hence w = and u = v X Curl. This proves X Curl = Xdiv. Finally, the following Hodge decomposition theorem is the combination of Corollaries 2.3 and 2.4 above. Theorem 2.5. X = L 2 (R n ; R n ) = X div X Curl. 3. Proofs of Theorem. and Theorem.2 In this section we prove Theorem. and Theorem.2. We state Theorem. slightly differently as follows. Theorem 3.. Let Y be the space with the norm defined above. Then Y is a Banach space. Moreover, the gradient operator : Y X Curl is surjective; more precisely, for any v X Curl, there exists a f Y such that v = f, f C n v. Proof. The proof that Y is a Banach space follows directly by the definition and will not be given here. We prove the rest of the theorem. Given v X Curl, let v ǫ = v ρ ǫ be the smooth approximation of v. Then v ǫ X Curl C (R n ; R n ). Define f ǫ (x) = v ǫ (tx) x dt. Then one can easily verify that f ǫ (x) = v ǫ (x) for all x R n. Therefore, for all x, y R n, Hence f ǫ (x + y) f ǫ (y) = f ǫ (x + y) f ǫ (y) 2 = v ǫ (y + tx) x dt. v ǫ (y + tx) x dt x 2 v ǫ (y + tx) 2 dt. 2
7 THE L 2 HODGE THEORY AND REPRESENTATION ON R n 7 Integrating this inequality over x B R () = B R, we obtain ( ) f ǫ (z) f ǫ (y) 2 dz R 2 v ǫ (y + tx) 2 dx dt B R (y) B R ( ) = R 2 v ǫ (z) 2 dz t n dt B tr (y) ( ) = R n+2 v ǫ (z) 2 dz dt B tr (y) R n+2 M( v ǫ 2 )(y), B tr (y) where M(h) is the maximal function of h (see Stein [7]). Since v ǫ 2 L (R n ), it follows that m{y R n M( v ǫ 2 )(y) > α} 5n v ǫ 2 dx 5n v 2 dx. α R n α R n Let where we choose E ǫ = {y B M( v ǫ 2 )(y) α }, α = 2 5n B R n v 2 dx. Then it follows that E ǫ 2 B for all ǫ. Therefore, it is a simple exercise to show that there exists a sequence ǫ k and a point y B such that y k= E ǫ k ; that is, M( v ǫk 2 )(y ) α = 2 5n B v 2, k =, 2,. Using this y we define a new sequence Then, for all R, we have g k (z) 2 dz B R g k (z) = f ǫk (z) f ǫk (y ), z R n. B 2R (y ) g k (z) 2 dz (2R) n+22 5n B v 2. By using diagonal subsequences, there exists a subsequence g kj and a function f L 2 loc (Rn ) such that g kj f weakly as k j on all balls B R (), R >. This function f must satisfy f = v L 2 (R n ; R n ) and sup R R n+2 B R f(x) 2 dx C n hence f C v. This completes the proof. R n v(x) 2 dx; We now prove Theorem.2. The proof is similar to that of Theorem..
8 8 BAISHENG YAN Proof of Theorem.2. Given w X div, let w ǫ = w ρ ǫ be the smooth approximation of w. Then w ǫ X div C (R 3 ; R 3 ). Define where (3.) (3.2) m ǫ (x, c) = (p ǫ (x, c), q ǫ (x, c), ), x R 3, c [, ], p ǫ (x, c) = q ǫ (x, c) = x3 c x3 c x2 wǫ(x 2, x 2, s)ds w ǫ(x, x 2, s)ds. Since divw ǫ =, one can easily verify that We now estimate w ǫ (x) = m ǫ (x, c), c [, ]. w 3 ǫ(x, t, c)dt; x3 q ǫ (x, c) 2 ( x 3 + ) 2 + wǫ(x, x 2, s) 2 ds. x 3 Integrating this inequality over the cube Q R () = {x R 3 x i < R, i =, 2, 3}, we obtain (3.3) q ǫ (x, c) 2 dx 2R(R + ) R 2 wǫ(x) 2 dx. 3 Q R () Next we write p ǫ (x, c) = g ǫ (x, c) f ǫ (x, c) with x = (x, x 2 ), where (3.4) g ǫ (x, c) = x3 c w 2 ǫ(x, x 2, s)ds; f ǫ (x, c) = x2 For g ǫ (x, c), we have the same estimate as q ǫ (x, c): (3.5) g ǫ (x, c) 2 dx 2R(R + ) R 2 wǫ(x) 2 2 dx. 3 Q R () w 3 ǫ(x, t, c)dt. For f ǫ (x, c), we easily estimate that (3.6) f ǫ (x, c) 2 dx 2R 2 x i <R wǫ(x 3, c) 2 dx = 2R 2 H ǫ (c). R 2 Note that H ǫ (c) = R wǫ(x 3, c) 2 dx L (R). It follows that 2 m{c R H ǫ (c) > α} H ǫ (c)dc = w α R α ǫ(x) 3 2 dx. R 3 Let E ǫ = {c [, ] H ǫ (c) α }, where α = R wǫ(x) 3 2 dx. Then it 3 follows that E ǫ for all ǫ. Therefore, as above, there exists a sequence ǫ k and a point c [, ] such that c k= E ǫ k ; that is, H ǫk (c ) wǫ(x) 3 2 dx k =, 2,. R 3
9 Hence by (3.6) (3.7) THE L 2 HODGE THEORY AND REPRESENTATION ON R n 9 Q R () Using this c we define a new sequence f ǫ (x, c ) 2 dx 4R 3 R 3 w 3 ǫ(x) 2 dx. u k (x) = m ǫk (x, c ). Then we have w ǫk = curlu k and, for all R, by (3.3)(3.6), (3.8) u k (x) 2 dx CR 3 w 2. Q R () By using diagonal subsequences, there exists a subsequence u kj and a function m L 2 loc (R3 ) such that u kj m weakly as k j on all cubes x i < R, R >. This field m must satisfy w = curlm L 2 (R 3 ; R 3 ); hence m M. Moreover, by (3.8), sup R R 3 m(x) 2 dx C w(x) 2 dx. R 3 This completes the proof. Q R () 4. Proof of Theorem.3 In this section, we prove Theorem.3. As above, let Y 2 be the closure of C (Rn ) in Y under the norm defined above. When extended by zero outside Ω, functions in H (Ω) belong to Y 2 for all bounded domains Ω R n. In what follows, let n 3 and 2 = 2n n 2. Let W = {f L 2 (R n ) f L 2 (R n ; R n )}. We prove Theorem.3 through several lemmas. Lemma 4.. Y 2 W. Moreover (4.) f C f L 2 (R n ) f Y 2. Proof. Let f Y 2. Then there exists a sequence f j C (Rn ) such that f j f as j. Therefore f j L 2 f L 2. By Sobolev GaliardoNirenberg inequality, f j L 2 (R n ) C f j L 2 (R n ) j. Hence f j g L 2 (R n ). Since f j f in L 2 (B R ) for all R >. We have f = g. Hence f W. Furthermore, by Hölder s inequality, f j L 2 (B R ) c n R 2 f j L 2 (B R ) C R2 f j L 2 (R n ). Hence, by taking limits as j, it follows that sup R R n+2 f 2 dx C f(x) 2 dx, B R R n which proves (4.).
10 BAISHENG YAN Lemma 4.2. W Y 2. Proof. Let f W. Define f j = fρ j, where ρ j W, (R n ) defined by ρ j (x) = on x j, ρ j (x) = on x 2j and ρ j (x) is linear in x for j x 2j. Then f j Y 2. It can be easily shown that lim f j f =, j which proves f Y 2 and hence W Y 2. Lemma 4.3. : Y 2 X Curl is surjective. Proof. Given any v X Curl, let ϕ R H (B R) be the function determined as in the minimization problem (2.) above with u = v. Then ϕ R v in X = L 2 (R 2 ; R 2 ) as R. Lemma 4. implies that {ϕ R } is a Cauchy sequence in Y 2 and hence its limit f belongs to Y 2 and satisfies f = v. This completes the proof. We first prove the following result. 5. Proof of Theorem.4 Lemma 5.. Let ρ(x) be defined as above. Then for all ϕ H (R 2 ), (5.) ϕ(x) ϕ ρ 2 ρ(x)dx β ϕ(x) 2 dx. R 2 R 2 Proof. First of all, by Poincaré s inequality, (5.2) ψ(x) (ψ) 2 dx C ψ(x) 2 dx B B for all ψ H (B ), where (ψ) is the average value of ψ on B ; that is, (ψ) = ψ(x)dx = 2 ψ(x)ρ(x) dx. π B { x <} Given ϕ H (R 2 ), let ψ(x) = ϕ( x ). Then ψ H (B x 2 ). Using the above Poincaré inequality for this ψ, after change of variable, one obtains that (5.3) ϕ(y) (ψ) 2 y 4 dy C ϕ(y) 2 dy, where (ψ) = π { y >} B ψ(x)dx = π { y >} { y >} ϕ(y) y 4 dy = 2 ϕ(x)ρ(x) dx. { x >} Combining (5.2) for ψ(x) = ϕ(x) with (5.3) we obtain (5.). Note that, for all f Y, (5.4) sup R R 4 f(x) 2 dx 2π B R R 2 f(x) 2 ρ(x)dx.
11 Let THE L 2 HODGE THEORY AND REPRESENTATION ON R n S = {ϕ ϕ ρ ϕ H (R 2 )}. Then the previous lemma and (5.4) imply sup R R 4 f 2 dx 2π f 2 ρ dx C f 2 dx f S. B R R 2 R 2 We have thus proved the following result. Proposition 5.2. Let Z be the closure of S in Y. Then f f L 2 for all f Z. Furthermore, for all f Z, f(x)ρ(x)dx =, f(x) 2 ρ(x)dx C f 2 dx, R 2 R 2 R 2 where C is a constant independent of f. Note that the weighted Sobolev estimates of type (5.) resemble the general ones studied in [6]. Finally, we prove the following result to complete the proof of Theorem.4. Proposition 5.3. : Z X Curl is surjective. Proof. The proof is similar to that of Lemma 4.3 above. Given any v X Curl, let ϕ R H (B R) be the function determined as in the minimization problem (2.) above with u = v. Let f R = ϕ R ϕ R ρ. Then f R S and f R = ϕ R v in X = L 2 (R 2 ; R 2 ) as R. Proposition 5.2 implies that {f R } is a Cauchy sequence in Y and hence its limit f belongs to the closure Z of S and satisfies f = v. This completes the proof. References [] J. Bourgain and H. Brezis, On the equation div Y = f and application to control of phases, Journal of Amer. Math. Soc., 6(2) (22), [2] L.C. Evans, Partial Differential Equations, A.M.S., Providence, 998. [3] D.Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., SpringerVerlag, Berlin, 984. [4] T. Iwaniec, pharmonic tensors and quasiregular mappings, Ann. Math., 36 (992), [5] T. Iwaniec and G. Martin, Quasiregular mappings in even dimensions, Acta Math., 7 (993), [6] C. Pérez, Sharp L p weighted Sobolev inequalities, Ann. de l institut Fourier, 45(3) (995), [7] E. Stein, Harmonic Analysis, Princeton University Press, Princeton, 993. Department of Mathematics, Michigan State University, East Lansing, MI address:
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