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 self-contained proofs to provide some representation results on curl-free and divergence-free 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 curl-free fields and coclosed forms become divergence-free fields. These fields also present a useful tool in studying many important physical and applied problems, such as the Maxwell equations, in particular, the electro-magnetics, 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, divergence-free, curl-free.

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 matrix-valued distribution A = (a ij ), Div A denotes the vector-valued 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 divergence-free and curl-free 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 well-known 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 side-length 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 curl-free or divergence-free 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 Sobolev-Gagliardo-Nirenberg 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 Sobolev-Gagliardo-Nirenberg inequality; instead, there is a John-Nirenberg-Trudinger type of BMO-estimates for functions with gradient in L 2 (R 2 ) (see [2, 7]). However, we shall try to avoid the BMO-estimates. 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- Galiardo-Nirenberg 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., Springer-Verlag, Berlin, 984. [4] T. Iwaniec, p-harmonic 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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