Integral estimates for transport densities - Math Berkeley - University of ...
INTEGRAL ESTIMATES FOR TRANSPORT DENSITIES L. DE PASCALE, L. C. EVANS, AND A. PRATELLI Abstract. We introduce some integration-by-parts methods that improve upon the Lp estimates on transport densitites from the recent paper by De Pascale–Pratelli [DP-P].
1. Introduction This paper provides some PDE methods that improve upon the Lp estimates on the “transport densities” in certain Monge–Kantorovich mass transfer problems, as derived in the earlier paper [DP-P] by the first and third authors. Our main estimate provides the bound σk Lq ≤ C (f Lq + 1)
(1)
for each 2 ≤ q < ∞, when u solves the quasilinear elliptic equation − div (σk Duk ) = f
(2)
σk := e 2 (|Duk |
(3)
for k
2
−1)
and k sufficiently large. The constant C in (1) depends on q, but not on the parameter k. This problem arises as an approximation of the fundamental transport (or continuity) equation for the Monge–Kantorovich mass transfer problem, as explained for instance in [E2]. In this interpretation, we seek an optimal rearrangement of the measure µ+ := f + dx into µ− := f − dy. In the limit k → ∞, we have uk → u, σk → a and the potential u solves − div (aDu) = f, |Du| ≤ 1, (4) |Du| = 1 where a > 0. We call a the transport density. It turns out that an optimal mass reallocation plan can be constructed using u and a. The paper [DP-P] by De Pascale and Pratelli studied how the integrability properties of f = f + − f − affect those of the transport density. They showed that (i) a ∈ L∞ if f ∈ L∞ , and (ii) a ∈ Lq− if f ∈ Lq , for 1 ≤ q < ∞ and each > 0. We introduce in this paper some PDE integration–by–parts methods to improve assertion (ii), by demonstrating a ∈ Lq
if f ∈ Lq , for 2 ≤ q < ∞.
We have tried, and failed, to extend our methods to include q = ∞. LCE’s research is supported in part by NSF Grant DMS-0070480 and by the Miller Institute for Basic Research in Science, UC Berkeley. 1
2
L. DE PASCALE, L. C. EVANS, AND A. PRATELLI
A PDE like (4) comes up also in the general formulation of Bouchitt´e and Buttazzo [B-B] for finding a distribution of a given amount of conductor to best dissipate heat. Then f represents a heat source and u the temperature of the system. The survey [E2] describes several more applications. 2. Approximation We will for simplicity take U = B 0 (0, R), the open ball with center 0 and radius R > 0. Hereafter we always suppose that f ∈ L1 (U ), with U f dx = 0. Denote by uk the solution of the nonlinear boundary–value problem − div (σk Duk ) = f in U (5) uk = 0 on ∂U, where we write σk := e 2 (|Duk | k
2
−1)
.
(6)
Observe that uk is the unique minimizer of the functional 1 k (|Dv|2 −1) Fk [v] := − f v dx e2 U k in W01,k . This approximation is suggested by the recent paper [E1]. Regularity theory (Cf. Marcellini [M]) implies that uk is smooth, provided f is. We want to study the limits of uk and σk as k → ∞, and begin with some uniform bounds. 1,q Lemma 2.1. Suppose that f ∈ L1 (U ). Then the sequence {uk }∞ k=1 is bounded in W0 (U ), for each 1 ≤ q < ∞. x2 −1
1
Proof. Observe first that x ≤ e 2 for x ≥ 0, and therefore that |Duk | ≤ σkk . Recalling then (5), (6), we deduce for k > n that |Duk |k+2 dx ≤ |Duk |2 σk dx = f uk dx ≤ Cuk L∞ ≤ CDuk Lk . U
Duk kLk
U
Duk k+2 + C. Lk+2
Note that ≤ We deduce for each k > q that
U
Hence Duk kLk ≤ C + CDuk Lk , and so Duk Lk ≤ C.
Duk Lq ≤ Duk Lk 1
kq
L k−q
≤ C.
We next indentify the Γ–limit of problem (5), (6) as k → ∞. For this, define − U f v dx if v ∈ C00,1 (U ), |Dv| ≤ 1 a.e. F [v] := +∞ otherwise. Theorem 2.2. As k goes to infinity, we have Γ
Fk −→ F. with respect to the uniform convergence of functions.
(7)
ESTIMATES FOR TRANSPORT DENSITIES
Proof. 1. Since the mapping u → f, u = U f u dx is linear, it is enough to prove 2 k 1 Γ Ek [v] := e 2 (|Dv| −1) dx −→ E[v], k U for 0 if v ∈ C00,1 (U ), |Dv| ≤ 1 a.e. E[v] := +∞ otherwise.
3
(8)
(9)
2. If E[v] < ∞, we clearly have E[v] = 0 = lim Ek [v]. k→∞
Suppose now that vk → v uniformly, and lim supk→∞ Ek [vk ] ≤ C < ∞. Fix an integer m and x2 −1
let k > m. Since e 2 ≥ x, we have for each open set V ⊆ U that
1/m 1/k m 1/m−1/k k |Dvk | dx ≤ |V | |Dvk | dx V
V
≤ |V |1/m−1/k k 1/k Ek (vk )1/k ≤ |V |1/m−1/k k 1/k C 1/k .
Passing to limits in k and recalling the lower semicontinuity of the Lm norm of the gradient, we discover
1/m |Dv|m dx ≤ |V |1/m . V
This inequality, valid for all V as above, implies that Dv is in L∞ , with |Dv| ≤ 1 almost everywhere. Consequently, E[v] = 0 ≤ lim inf Ek [vk ]. k→∞
Introduce next the vector fields Gk := σk Duk
(k = 1, . . . ).
Theorem 2.3. Suppose that for some 1 < q < ∞ we have the uniform bounds sup Gk Lq (U ;Rn ) < ∞. k
Define fk := − div (Gk ), and assume
q fk f weakly in L (U ) Gk G weakly in Lq (U ; Rn ), uk → u uniformly.
Then there exists a positive function a ∈ Lq such that G = aDu, |Du| = 1 a.e. on {a > 0}, and σk a weakly in Lq (U ). In particular, a = |G|.
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L. DE PASCALE, L. C. EVANS, AND A. PRATELLI
Proof. 1. First of all, note that − div G = f ; that is, G · Dψ dx = f ψ dx U
U
for all ψ ∈ C , ψ = 0 on ∂U . 1
Let us now fix 0 < λ < 1 and calculate:
|G| dx ≤ lim inf |Gk | dx = lim inf |Gk | dx + |Gk | dx k→∞ k→∞ U U {|Duk |2 >1−λ} {|Duk |2 ≤1−λ}
k √ 1 ≤ lim inf √ |Gk ||Duk | dx + e− 2 λ 1 − λ dx . k→∞ 1−λ U U When k goes to infinity, the last integral goes to 0. Notice also that |Gk ||Duk | dx = σk |Duk |2 dx = fk uk dx. U
Therefore
√
U
U
1−λ
|G| dx ≤ lim inf
fk uk dx =
k→∞
U
U
for each 0 < λ < 1, and consequently
G · Du dx
f u dx = U
U
|G| dx ≤
G · Du dx.
U
(10)
U
2. Reasoning now as in the proof of Theorem 2.2, we fix an integer m and let k > m. Then for each open set V ⊆ U
1/m 1/k+1 m 1/m−1/k+1 k+1 |Dvk | dx ≤ |V | |Dvk | dx V
V 1/k+1
≤ |V |1/m−1/k+1 ||Gk ||L1 Pass to limits in k to find
≤ |V |1/m−1/k+1 C 1/k+1 .
1/m |Dv|m dx
≤ |V |1/m ,
V
and therefore |Du| ≤ 1 almost everywhere. The first two assertions of the Theorem now follow from (10). a, let us fix ψ ∈ C0∞ and prove σk ψ dx → aψ dx.
3. To show also that σk
U
We write
U
U
σk |Duk |2 ψ dx +
σk ψ dx = U
Notice now that
A1 =
U
= U
σk (1 − |Duk |2 )ψ dx =: A1 + A2 . U
ψ Gk · Duk dx = Gk · D(uk ψ) dx − U fk uk ψ dx − uk Gk · Dψ dx. U
uk Gk · Dψ dx U
ESTIMATES FOR TRANSPORT DENSITIES
5
This expression converges as k → ∞ to f uψ dx − u G · Dψ dx = G · D(uψ) dx − u G · Dψ dx U U U U 2 = ψ G · Du dx = ψa|Du| dx = aψ dx. U
U
U
4. It remains to show that A2 → 0. If we write ϕk := |Duk |2 − 1, then k e 2 ϕk |ϕk | dx. |A2 | ≤ ψL∞ U −x 2
Since xe
≤ 1 for each x > 0, we have k|ϕk | k 1 |U | e 2 ϕk |ϕk | dx = k|ϕk |e− 2 dx ≤ → 0. k k {ϕk 1 there exists a constant cq > 0 such that ex(q−1) ≥ cq > 0 x for all x > 0. Consequently, k k k 2 e 2 ϕk |ϕk | dx = e 2 ϕk ϕk dx k {ϕk >0} 2 {ϕk >0} qk 2 2 2C q ϕ ≤ → 0. e 2 k dx = σkq dx ≤ cq k {ϕk >0} cq k U cq k This completes the proof that A2 → 0.
3. Estimates I The full calculations for our main estimate in §4 are fairly involved, and so for the reader’s convenience we provide in this section a simpler computation illustrating the main ideas. Suppose 2 ≤ q < ∞. Theorem 3.1. There exists a constant C, depending on q, but independent of k, such that
σkq dx ≤ C |f |q dx + 1 . (11) U
U
Proof. 1. To simplify notation, we hereafter in the proof do not write the subscripts k. Observe that since Du is bounded in each space Lq and u = 0 on ∂U , we have the bound |u| ≤ C for some constant C. 2. Multiply (5) by σ q−1 u and integrate by parts: q 2 q−1 q−1 σ |Du| + (q − 1)σ Du · Dσ u dx = σui (σ u)i dx = f σ q−1 u dx U
U
≤C
q1
|f |q dx U
U
σ q dx
1− q1
(12) .
U
Here and afterwards we write the subscript i to denote the partial derivative with respect to the variable xi .
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L. DE PASCALE, L. C. EVANS, AND A. PRATELLI
Notice that |Du|2 ≥ 1 if σ ≥ 1. Therefore
σ q dx ≤ C |f |q dx + σ q−1 |Du · Dσ| dx + 1 . U
U
(13)
U
3. Next, multiply (5) by −(σ q−1 uj )j : (σui )i (σ q−1 uj )j dx = − f (σ q−1 uj )j dx U U q−2 = fσ (−(σuj )j ) dx − f (q − 2)σ q−2 σj uj dx U U ≤C f 2 σ q−2 + |f |σ q−2 |Du · Dσ| dx.
(14)
U
The term on the left is A:=− σui (σ q−1 uj )ij dx + σui ν i (σ q−1 uj )j dHn−1 U ∂U q−1 = (σui )j (σ uj )i dx + σui ν i (σ q−1 uj )j − σui ν j (σ q−1 uj )i dHn−1 , U
(15)
∂U
where ν = (ν 1 , . . . , ν n ) is the unit outer normal to ∂U . The boundary integral is B := σ q (ui ν i ujj − ui ν j uij ) dHn−1 ∂U + (q − 1)σ q−1 (ui ν i uj σj − ui ν j σi uj ) dHn−1 .
(16)
∂U
The integrand of the last term equals 0, since σ = e 2 (|Du| −1) and so σj = kul ulj σ. Consider a point x0 ∈ ∂U ; without loss, we can take x0 = (0, . . . , R). Then ν = (0, . . . , 1) and Du = (0, . . . , un ), since u = 0 on ∂U . The integrand of the first term on the right hand side of (16) at x0 therefore equals k
2
σ q (∆u − unn )un .
(17)
Lastly, write x = (x1 , . . . , xn−1 ) and observe that u(x , R2 − |x |2 ) ≡ 0 for small x . We differentiate this identity twice and set x = 0, to compute ∆u − unn = n−1 R un at x0 . Hence n−1 σ q |Du|2 dHn−1 ≥ 0. B= R ∂U 4. Therefore
A= U =
(σui )i σ q−1 uj j dx ≥ (σuij + σj ui )(σ
q−1
(σui )j (σ q−1 uj )i dx U
uij + (q − 1)σ q−2 σi uj ) dx
U
σ q |D2 u|2 + (q − 1)σ q−2 |Du · Dσ|2 + qσ q−1 σj ui uij dx.
= U
(18)
ESTIMATES FOR TRANSPORT DENSITIES
Recall that σj = kul ulj σ. Hence (14) and (18) imply q σ q |D2 u|2 + (q − 1)σ q−2 |Du · Dσ|2 + σ q−2 |Dσ|2 dx k U ≤C f 2 σ q−2 + |f |σ q−2 |Du · Dσ| dx U q−1 ≤ σ q−2 |Du · Dσ|2 + C |f |2 σ q−2 dx; 2 U U and consequently
7
(19)
σ q−2 |Du · Dσ|2 dx ≤ C
U
|f |2 σ q−2 dx.
(20)
U
5. Combine (13),(20): q q σ dx ≤ C |f | dx + C σ q−1 |Du · Dσ| dx + C U U U 1 ≤C |f |q dx + σ q dx + C σ q−2 |Du · Dσ|2 dx + C 3 U U U 1 q q ≤C |f | dx + σ dx + C |f |2 σ q−2 dx + C 3 U U U 2 q ≤C |f | dx + σ q dx + C 3 U U
(21)
This gives (11). Remark. The boundary integral term B is in fact nonnegative for any convex, smooth domain replacing U = B(0, R): see for instance the similar calculations in §1.5 of Ladyzhenskaja [L]. 4. Estimates II In this section we derive our main integral estimate. ¯ ). Then there exist a constant C, Theorem 4.1. Assume that 2 ≤ q < ∞ and that f ∈ C ∞ (U depending only on q, and a constant K, depending only on f L∞ , such that
q q q σk |Duk | dx ≤ C |f | dx + 1 (22) U
U
for all k ≥ K. The proof is similar to that of Theorem 3.1, except that we must handle the additional term |Duk |q on the left. This makes our multipliers and estimates more intricate. Proof. 1. For notational simplicity we hereafter write σ and u in place of σk and uk . Since f is smooth, the same is true for u and σ. Observe also the bound |u| ≤ C. We record for later reference these consequences of (6): σi σj |Du|i = , ui uij = . kσ|Du| kσ
(23)
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L. DE PASCALE, L. C. EVANS, AND A. PRATELLI
2. We multiply the PDE (5) by σ q−1 |Du|q+1 u and integrate by parts, to find σDu · D σ q−1 |Du|q+1 u dx = σ q−1 |Du|q+1 uf dx. U
(24)
U
The right hand term in (24) is less than or equal to 1 q−1 q+1 C σ |Du| |f | dx ≤ σ q |Du|q+2 dx + 2q−1 C q 2 U {|f |≤
σ|Du| 2C }
|Du|2 |f |q dx. {|f |>
σ|Du| 2C }
But if σ|Du| < 2C|f |, then obviously σ|Du| ≤ 2Cf L∞ . Recalling (6), we see that this implies |Du| ≤ 2 provided k ≥ K, for some constant K depending only upon f L∞ . Therefore 1 σ q−1 |Du|q+1 uf dx ≤ σ q |Du|q+2 dx + C |f |q dx. (25) 2 U U U 3. We use (23) to evaluate the left hand term in (24): σDu · D σ q−1 |Du|q+1 u dx = σ q |Du|q+3 dx U U q−1 q+1 +(q − 1) σ |Du| u Dσ · Du dx + (q + 1) σ q u|Du|q Du · (D|Du|) dx U U = σ q |Du|q+3 dx U q+1 q−1 q+1 +(q − 1) σ |Du| u Dσ · Du dx + σ q−1 u|Du|q−1 Du · Dσ dx. k U U But σ ≥ 1 only if |Du| ≥ 1; and hence q q+2 σ |Du| dx ≤ σ q |Du|q+3 dx + C, U
(26)
(27)
U
since U is bounded. Combining (27), (26), (24) and (25), we deduce the inequality 1 q q+2 q q+2 σ |Du| dx ≤ C + σ |Du| dx + C |f |q dx 2 U U U C +C σ q−1 |Du|q+1 |Dσ · Du| dx + σ q−1 |Du|q−1 |Dσ · Du| dx. k U U Arguing as before (this means dividing the integrals in the set where |Dσ · Du| ≤ σ|Du|/C and in the rest of U ), we see that therefore σ q |Du|q+2 dx ≤ C + C |f |q dx + σ q |Du|q+2 dx + σ q |Du|q dx k U U U U (28) C2 C2 + σ q−2 |Du|q |Dσ · Du|2 dx + σ q−2 |Du|q−2 |Dσ · Du|2 dx U k U q q q q+2 for any > 0. Since U σ |Du| dx ≤ U σ |Du| dx + C, this implies our first main estimate: σ q |Du|q+2 dx ≤ C + C |f |q dx + C σ q−2 |Du|q |Dσ · Du|2 dx U U U (29) C q−2 q−2 2 + σ |Du| |Dσ · Du| dx. k U
ESTIMATES FOR TRANSPORT DENSITIES
9
4. The last two terms in (29) involving Dσ · Du are dangerous, since Dσ is of order k: we need another estimate to control them. Let us therefore continue by multiplying the PDE (5) by − div σ q−1 |Du|q Du and thereby deriving the identity div (σDu) div (σ q−1 |Du|q Du) dx = − f div (σ q−1 |Du|q Du) dx. (30) U
U
The term on the right equals f σ q−2 |Du|q (− div (σDu)) dx − f σDu · D σ q−2 |Du|q dx = |f |2 σ q−2 |Du|q dx U U U q−2 q − (q − 2) fσ |Du| Du · Dσ dx − q f σ q−1 |Du|q−1 Du · (D|Du|) dx. U
U
We again recall (23) and deduce − f div (σ q−1 |Du|q Du) dx ≤ |f |2 σ q−2 |Du|q dx U U q q−2 q + (q − 2) |f |σ |Du| |Du · Dσ| dx + |f |σ q−2 |Du|q−2 |Du · Dσ| dx. k U U The left hand term of (30) is A:=− σui (σ q−1 |Du|q uj )ij dx + σui ν i (σ q−1 |Du|q uj )j dHn−1 ∂U U q−1 q = (σui )j (σ |Du| uj )i dx U + σui ν i (σ q−1 |Du|q uj )j − σui ν j (σ q−1 |Du|q uj )i dHn−1 .
(31)
(32)
∂U
Call the boundary term B. Then, almost exactly as in step 3 of the previous proof, we can show that n−1 B= σ q |Du|q+2 dHn−1 ≥ 0. R ∂U Consequently, A= (σui )i σ q−1 |Du|q uj j dx ≥ (σui )j σ q−1 |Du|q uj i dx U U q = (σuij + σj ui ) σ q−1 |Du|q uij + (q − 1)σ q−2 |Du|q σi uj + σ q−2 |Du|q−2 σi uj dx k U q q−2 q q 2 2 q 2 q−2 q = σ |Du| |D u| + σ |Du| |Dσ| + (q − 1)σ |Du| |Dσ · Du|2 + k U q q + 2 σ q−2 |Du|q−2 |Dσ|2 + σ q−2 |Du|q−2 |Dσ · Du|2 dx. k k The first, the second and the fourth terms in the last expression are positive, and so we deduce q (q − 1) σ q−2 |Du|q |Dσ · Du|2 dx + σ q−2 |Du|q−2 |Dσ · Du|2 dx k U U (33) q−1 q ≤ div (σDu) div (σ |Du| Du) dx. U
10
L. DE PASCALE, L. C. EVANS, AND A. PRATELLI
Collecting (33), (30) and (31), we find 1 σ q−2 |Du|q |Dσ · Du|2 dx + σ q−2 |Du|q−2 |Dσ · Du|2 dx k U U 2 q−2 q ≤C |f | σ |Du| dx + C |f |σ q−2 |Du|q |Du · Dσ| dx U U C + |f |σ q−2 |Du|q−2 |Du · Dσ| dx. k U Take > 0 to be a small constant, which will be fixed later on. Then f 2 σ q−2 |Du|q dx ≤ σ q |Du|q+2 dx + C |f |q |Du|2 dx √ U U {|f |>σ|Du| } q q+2 ≤ σ |Du| dx + C |f |q dx, U
(34)
(35)
U
√ since |Du| ≤ 2 wherever |f | > σ|Du| , provided k ≥ K and K is large. Recalling (35), we can likewise estimate for each δ > 0 that |f |σ q−2 |Du|q |Dσ · Du| dx ≤ δ σ q−2 |Du|q |Dσ · Du|2 dx + C f 2 σ q−2 |Du|q dx U U U
q−2 q 2 q q+2 q ≤δ σ |Du| |Dσ · Du| dx + C σ |Du| dx + |f | dx . U
U
(36)
U
Similarly, |f |σ q−2 |Du|q−2 |Du · Dσ|dx ≤ δ σ q−2 |Du|q−2 |Dσ · Du|2 dx + C f 2 σ q−2 |Du|q−2 dx U U U
(37) q−2 q−2 2 q q q ≤δ σ |Du| |Dσ · Du| dx + C σ |Du| dx + C |f | dx . U
U
U
Since σ ≥ 1 only if |Du| ≥ 1, we have σ q |Du|q dx ≤ σ q |Du|q+2 dx + C, U
U
and therefore |f |σ q−2 |Du|q−2 |Du · Dσ| dx ≤ δ σ q−2 |Du|q−2 |Dσ · Du|2 dx U U
q q+2 q +C σ |Du| dx + |f | dx + 1 . U
(38)
U
Taking δ > 0 small, we then derive from (34), (35), (36) and (38) our second main inequality 1 q−2 q 2 σ |Du| |Dσ · Du| dx + σ q−2 |Du|q−2 |Dσ · Du|2 dx k U U (39) ≤ σ q |Du|q+2 dx + C |f |q + C. U
U
5. Putting together inequalities (29) and (39) and fixing > 0 small, we finally discover σ q |Du|q+2 dx ≤ C + C |f |q dx. U
As σ ≥ 1 only if |Du| ≥ 1, estimate (22) follows.
U
ESTIMATES FOR TRANSPORT DENSITIES
11
Theorem 4.1 concerns only smooth functions f . However, since the bound for the Lq norm of the transport density depends only upon the Lq norm of f , we can approximate: Theorem 4.2. For each 2 ≤ q < ∞ and f ∈ Lq (U ) the associated transport density a belongs to Lq (U ). Furthermore, there is a constant C, depending only upon n and U , such that aLq ≤ C (f Lq + 1) .
(40)
Proof. 1. Let us first define fj := f ∗ η1/j , the convolution of f with a standard mollifier. For each integer j, we then solve − div (σk,j Duk,j ) = fj in U uk,j = 0 on ∂U,
(41)
for σk,j := e 2 (|Duk,j | k
2
−1)
.
(42)
2. According to (22), we have the estimate
q σk,j |Duk,j |q dx ≤ C |fj |q dx + 1 ≤ C |f |q dx + 1 U
U
(43)
U
for all k greater than or equal to some constant K = k(j), depending only on the L∞ norm of fj . Now define σj := σk(j),j , uj := uk(j),j , Gj := σj Duj . Clearly fj → f in Lq . Furthermore, (43) implies that Gj is bounded in Lq . We may therefore assume upon reindexing that Gj
G
weakly in Lq (U ; Rn ).
Finally we may pass as necessary to a further subsequence to ensure uj converges uniformly to a limit u. Apply Theorem 2.3.
References [B-B]
G. Bouchitt´e and G. Buttazzo, Characterization of optimal shapes and masses through Monge– Kantorovich equation, Journal European Math. Soc., 3 (2001), 139–168. [B-B-S] G. Bouchitt´e, G. Buttazzo and P. Seppecher, Shape optimization solutions via Monge–Kantorovich equation, C. R. Acad. Sci. Paris, 324–I (1997), 1185–1191. [B-B-DP] G. Bouchitt´e, G. Buttazzo and L. De Pascale, A p-Laplacian approximation for some mass optimization problems, to appear on J.O.T.A. (Journal of Optimization Theory and Applications), (2002). [DP-P] L. De Pascale and A. Pratelli, Regularity properties for Monge transport density and for solutions of some shape optimization problems, Calculus of Variations and Partial Differential Equations, 14 (2002), 249–274. [E1] L. C. Evans, Some new PDE methods for weak KAM theory, to appear in Calculus of Variations and Partial Differential Equations. [E2] L. C. Evans, Partial differential equations and Monge-Kantorovich mass transfer (survey paper), Current Developments in Mathematics, 1997, International Press (1999), edited by S. T. Yau. [L] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, trans. by Richard Silverman, Gordon and Breach, (1969). [M] P. Marcellini, General growth conditions and regularity, in Variational methods for discontinuous structures, edited by Serapioni, Raul et al., Progress in Nonlinear Differential Equations and Applications #25, Birkhauser (1996), 111-118.
12
L. DE PASCALE, L. C. EVANS, AND A. PRATELLI
` di Pisa, 56126 Pisa, Italy (L. De Pascale) Dipartimento di Matematica Applicata, Universita (L. C. Evans) Department of Mathematics, University of California, Berkeley, CA 94720, USA (A. Pratelli) Scuola Normale Superiore, 56126 Pisa, Italy
1. Introduction This paper provides some PDE methods that improve upon the Lp estimates on the “transport densities” in certain Monge–Kantorovich mass transfer problems, as derived in the earlier paper [DP-P] by the first and third authors. Our main estimate provides the bound σk Lq ≤ C (f Lq + 1)
(1)
for each 2 ≤ q < ∞, when u solves the quasilinear elliptic equation − div (σk Duk ) = f
(2)
σk := e 2 (|Duk |
(3)
for k
2
−1)
and k sufficiently large. The constant C in (1) depends on q, but not on the parameter k. This problem arises as an approximation of the fundamental transport (or continuity) equation for the Monge–Kantorovich mass transfer problem, as explained for instance in [E2]. In this interpretation, we seek an optimal rearrangement of the measure µ+ := f + dx into µ− := f − dy. In the limit k → ∞, we have uk → u, σk → a and the potential u solves − div (aDu) = f, |Du| ≤ 1, (4) |Du| = 1 where a > 0. We call a the transport density. It turns out that an optimal mass reallocation plan can be constructed using u and a. The paper [DP-P] by De Pascale and Pratelli studied how the integrability properties of f = f + − f − affect those of the transport density. They showed that (i) a ∈ L∞ if f ∈ L∞ , and (ii) a ∈ Lq− if f ∈ Lq , for 1 ≤ q < ∞ and each > 0. We introduce in this paper some PDE integration–by–parts methods to improve assertion (ii), by demonstrating a ∈ Lq
if f ∈ Lq , for 2 ≤ q < ∞.
We have tried, and failed, to extend our methods to include q = ∞. LCE’s research is supported in part by NSF Grant DMS-0070480 and by the Miller Institute for Basic Research in Science, UC Berkeley. 1
2
L. DE PASCALE, L. C. EVANS, AND A. PRATELLI
A PDE like (4) comes up also in the general formulation of Bouchitt´e and Buttazzo [B-B] for finding a distribution of a given amount of conductor to best dissipate heat. Then f represents a heat source and u the temperature of the system. The survey [E2] describes several more applications. 2. Approximation We will for simplicity take U = B 0 (0, R), the open ball with center 0 and radius R > 0. Hereafter we always suppose that f ∈ L1 (U ), with U f dx = 0. Denote by uk the solution of the nonlinear boundary–value problem − div (σk Duk ) = f in U (5) uk = 0 on ∂U, where we write σk := e 2 (|Duk | k
2
−1)
.
(6)
Observe that uk is the unique minimizer of the functional 1 k (|Dv|2 −1) Fk [v] := − f v dx e2 U k in W01,k . This approximation is suggested by the recent paper [E1]. Regularity theory (Cf. Marcellini [M]) implies that uk is smooth, provided f is. We want to study the limits of uk and σk as k → ∞, and begin with some uniform bounds. 1,q Lemma 2.1. Suppose that f ∈ L1 (U ). Then the sequence {uk }∞ k=1 is bounded in W0 (U ), for each 1 ≤ q < ∞. x2 −1
1
Proof. Observe first that x ≤ e 2 for x ≥ 0, and therefore that |Duk | ≤ σkk . Recalling then (5), (6), we deduce for k > n that |Duk |k+2 dx ≤ |Duk |2 σk dx = f uk dx ≤ Cuk L∞ ≤ CDuk Lk . U
Duk kLk
U
Duk k+2 + C. Lk+2
Note that ≤ We deduce for each k > q that
U
Hence Duk kLk ≤ C + CDuk Lk , and so Duk Lk ≤ C.
Duk Lq ≤ Duk Lk 1
kq
L k−q
≤ C.
We next indentify the Γ–limit of problem (5), (6) as k → ∞. For this, define − U f v dx if v ∈ C00,1 (U ), |Dv| ≤ 1 a.e. F [v] := +∞ otherwise. Theorem 2.2. As k goes to infinity, we have Γ
Fk −→ F. with respect to the uniform convergence of functions.
(7)
ESTIMATES FOR TRANSPORT DENSITIES
Proof. 1. Since the mapping u → f, u = U f u dx is linear, it is enough to prove 2 k 1 Γ Ek [v] := e 2 (|Dv| −1) dx −→ E[v], k U for 0 if v ∈ C00,1 (U ), |Dv| ≤ 1 a.e. E[v] := +∞ otherwise.
3
(8)
(9)
2. If E[v] < ∞, we clearly have E[v] = 0 = lim Ek [v]. k→∞
Suppose now that vk → v uniformly, and lim supk→∞ Ek [vk ] ≤ C < ∞. Fix an integer m and x2 −1
let k > m. Since e 2 ≥ x, we have for each open set V ⊆ U that
1/m 1/k m 1/m−1/k k |Dvk | dx ≤ |V | |Dvk | dx V
V
≤ |V |1/m−1/k k 1/k Ek (vk )1/k ≤ |V |1/m−1/k k 1/k C 1/k .
Passing to limits in k and recalling the lower semicontinuity of the Lm norm of the gradient, we discover
1/m |Dv|m dx ≤ |V |1/m . V
This inequality, valid for all V as above, implies that Dv is in L∞ , with |Dv| ≤ 1 almost everywhere. Consequently, E[v] = 0 ≤ lim inf Ek [vk ]. k→∞
Introduce next the vector fields Gk := σk Duk
(k = 1, . . . ).
Theorem 2.3. Suppose that for some 1 < q < ∞ we have the uniform bounds sup Gk Lq (U ;Rn ) < ∞. k
Define fk := − div (Gk ), and assume
q fk f weakly in L (U ) Gk G weakly in Lq (U ; Rn ), uk → u uniformly.
Then there exists a positive function a ∈ Lq such that G = aDu, |Du| = 1 a.e. on {a > 0}, and σk a weakly in Lq (U ). In particular, a = |G|.
4
L. DE PASCALE, L. C. EVANS, AND A. PRATELLI
Proof. 1. First of all, note that − div G = f ; that is, G · Dψ dx = f ψ dx U
U
for all ψ ∈ C , ψ = 0 on ∂U . 1
Let us now fix 0 < λ < 1 and calculate:
|G| dx ≤ lim inf |Gk | dx = lim inf |Gk | dx + |Gk | dx k→∞ k→∞ U U {|Duk |2 >1−λ} {|Duk |2 ≤1−λ}
k √ 1 ≤ lim inf √ |Gk ||Duk | dx + e− 2 λ 1 − λ dx . k→∞ 1−λ U U When k goes to infinity, the last integral goes to 0. Notice also that |Gk ||Duk | dx = σk |Duk |2 dx = fk uk dx. U
Therefore
√
U
U
1−λ
|G| dx ≤ lim inf
fk uk dx =
k→∞
U
U
for each 0 < λ < 1, and consequently
G · Du dx
f u dx = U
U
|G| dx ≤
G · Du dx.
U
(10)
U
2. Reasoning now as in the proof of Theorem 2.2, we fix an integer m and let k > m. Then for each open set V ⊆ U
1/m 1/k+1 m 1/m−1/k+1 k+1 |Dvk | dx ≤ |V | |Dvk | dx V
V 1/k+1
≤ |V |1/m−1/k+1 ||Gk ||L1 Pass to limits in k to find
≤ |V |1/m−1/k+1 C 1/k+1 .
1/m |Dv|m dx
≤ |V |1/m ,
V
and therefore |Du| ≤ 1 almost everywhere. The first two assertions of the Theorem now follow from (10). a, let us fix ψ ∈ C0∞ and prove σk ψ dx → aψ dx.
3. To show also that σk
U
We write
U
U
σk |Duk |2 ψ dx +
σk ψ dx = U
Notice now that
A1 =
U
= U
σk (1 − |Duk |2 )ψ dx =: A1 + A2 . U
ψ Gk · Duk dx = Gk · D(uk ψ) dx − U fk uk ψ dx − uk Gk · Dψ dx. U
uk Gk · Dψ dx U
ESTIMATES FOR TRANSPORT DENSITIES
5
This expression converges as k → ∞ to f uψ dx − u G · Dψ dx = G · D(uψ) dx − u G · Dψ dx U U U U 2 = ψ G · Du dx = ψa|Du| dx = aψ dx. U
U
U
4. It remains to show that A2 → 0. If we write ϕk := |Duk |2 − 1, then k e 2 ϕk |ϕk | dx. |A2 | ≤ ψL∞ U −x 2
Since xe
≤ 1 for each x > 0, we have k|ϕk | k 1 |U | e 2 ϕk |ϕk | dx = k|ϕk |e− 2 dx ≤ → 0. k k {ϕk 1 there exists a constant cq > 0 such that ex(q−1) ≥ cq > 0 x for all x > 0. Consequently, k k k 2 e 2 ϕk |ϕk | dx = e 2 ϕk ϕk dx k {ϕk >0} 2 {ϕk >0} qk 2 2 2C q ϕ ≤ → 0. e 2 k dx = σkq dx ≤ cq k {ϕk >0} cq k U cq k This completes the proof that A2 → 0.
3. Estimates I The full calculations for our main estimate in §4 are fairly involved, and so for the reader’s convenience we provide in this section a simpler computation illustrating the main ideas. Suppose 2 ≤ q < ∞. Theorem 3.1. There exists a constant C, depending on q, but independent of k, such that
σkq dx ≤ C |f |q dx + 1 . (11) U
U
Proof. 1. To simplify notation, we hereafter in the proof do not write the subscripts k. Observe that since Du is bounded in each space Lq and u = 0 on ∂U , we have the bound |u| ≤ C for some constant C. 2. Multiply (5) by σ q−1 u and integrate by parts: q 2 q−1 q−1 σ |Du| + (q − 1)σ Du · Dσ u dx = σui (σ u)i dx = f σ q−1 u dx U
U
≤C
q1
|f |q dx U
U
σ q dx
1− q1
(12) .
U
Here and afterwards we write the subscript i to denote the partial derivative with respect to the variable xi .
6
L. DE PASCALE, L. C. EVANS, AND A. PRATELLI
Notice that |Du|2 ≥ 1 if σ ≥ 1. Therefore
σ q dx ≤ C |f |q dx + σ q−1 |Du · Dσ| dx + 1 . U
U
(13)
U
3. Next, multiply (5) by −(σ q−1 uj )j : (σui )i (σ q−1 uj )j dx = − f (σ q−1 uj )j dx U U q−2 = fσ (−(σuj )j ) dx − f (q − 2)σ q−2 σj uj dx U U ≤C f 2 σ q−2 + |f |σ q−2 |Du · Dσ| dx.
(14)
U
The term on the left is A:=− σui (σ q−1 uj )ij dx + σui ν i (σ q−1 uj )j dHn−1 U ∂U q−1 = (σui )j (σ uj )i dx + σui ν i (σ q−1 uj )j − σui ν j (σ q−1 uj )i dHn−1 , U
(15)
∂U
where ν = (ν 1 , . . . , ν n ) is the unit outer normal to ∂U . The boundary integral is B := σ q (ui ν i ujj − ui ν j uij ) dHn−1 ∂U + (q − 1)σ q−1 (ui ν i uj σj − ui ν j σi uj ) dHn−1 .
(16)
∂U
The integrand of the last term equals 0, since σ = e 2 (|Du| −1) and so σj = kul ulj σ. Consider a point x0 ∈ ∂U ; without loss, we can take x0 = (0, . . . , R). Then ν = (0, . . . , 1) and Du = (0, . . . , un ), since u = 0 on ∂U . The integrand of the first term on the right hand side of (16) at x0 therefore equals k
2
σ q (∆u − unn )un .
(17)
Lastly, write x = (x1 , . . . , xn−1 ) and observe that u(x , R2 − |x |2 ) ≡ 0 for small x . We differentiate this identity twice and set x = 0, to compute ∆u − unn = n−1 R un at x0 . Hence n−1 σ q |Du|2 dHn−1 ≥ 0. B= R ∂U 4. Therefore
A= U =
(σui )i σ q−1 uj j dx ≥ (σuij + σj ui )(σ
q−1
(σui )j (σ q−1 uj )i dx U
uij + (q − 1)σ q−2 σi uj ) dx
U
σ q |D2 u|2 + (q − 1)σ q−2 |Du · Dσ|2 + qσ q−1 σj ui uij dx.
= U
(18)
ESTIMATES FOR TRANSPORT DENSITIES
Recall that σj = kul ulj σ. Hence (14) and (18) imply q σ q |D2 u|2 + (q − 1)σ q−2 |Du · Dσ|2 + σ q−2 |Dσ|2 dx k U ≤C f 2 σ q−2 + |f |σ q−2 |Du · Dσ| dx U q−1 ≤ σ q−2 |Du · Dσ|2 + C |f |2 σ q−2 dx; 2 U U and consequently
7
(19)
σ q−2 |Du · Dσ|2 dx ≤ C
U
|f |2 σ q−2 dx.
(20)
U
5. Combine (13),(20): q q σ dx ≤ C |f | dx + C σ q−1 |Du · Dσ| dx + C U U U 1 ≤C |f |q dx + σ q dx + C σ q−2 |Du · Dσ|2 dx + C 3 U U U 1 q q ≤C |f | dx + σ dx + C |f |2 σ q−2 dx + C 3 U U U 2 q ≤C |f | dx + σ q dx + C 3 U U
(21)
This gives (11). Remark. The boundary integral term B is in fact nonnegative for any convex, smooth domain replacing U = B(0, R): see for instance the similar calculations in §1.5 of Ladyzhenskaja [L]. 4. Estimates II In this section we derive our main integral estimate. ¯ ). Then there exist a constant C, Theorem 4.1. Assume that 2 ≤ q < ∞ and that f ∈ C ∞ (U depending only on q, and a constant K, depending only on f L∞ , such that
q q q σk |Duk | dx ≤ C |f | dx + 1 (22) U
U
for all k ≥ K. The proof is similar to that of Theorem 3.1, except that we must handle the additional term |Duk |q on the left. This makes our multipliers and estimates more intricate. Proof. 1. For notational simplicity we hereafter write σ and u in place of σk and uk . Since f is smooth, the same is true for u and σ. Observe also the bound |u| ≤ C. We record for later reference these consequences of (6): σi σj |Du|i = , ui uij = . kσ|Du| kσ
(23)
8
L. DE PASCALE, L. C. EVANS, AND A. PRATELLI
2. We multiply the PDE (5) by σ q−1 |Du|q+1 u and integrate by parts, to find σDu · D σ q−1 |Du|q+1 u dx = σ q−1 |Du|q+1 uf dx. U
(24)
U
The right hand term in (24) is less than or equal to 1 q−1 q+1 C σ |Du| |f | dx ≤ σ q |Du|q+2 dx + 2q−1 C q 2 U {|f |≤
σ|Du| 2C }
|Du|2 |f |q dx. {|f |>
σ|Du| 2C }
But if σ|Du| < 2C|f |, then obviously σ|Du| ≤ 2Cf L∞ . Recalling (6), we see that this implies |Du| ≤ 2 provided k ≥ K, for some constant K depending only upon f L∞ . Therefore 1 σ q−1 |Du|q+1 uf dx ≤ σ q |Du|q+2 dx + C |f |q dx. (25) 2 U U U 3. We use (23) to evaluate the left hand term in (24): σDu · D σ q−1 |Du|q+1 u dx = σ q |Du|q+3 dx U U q−1 q+1 +(q − 1) σ |Du| u Dσ · Du dx + (q + 1) σ q u|Du|q Du · (D|Du|) dx U U = σ q |Du|q+3 dx U q+1 q−1 q+1 +(q − 1) σ |Du| u Dσ · Du dx + σ q−1 u|Du|q−1 Du · Dσ dx. k U U But σ ≥ 1 only if |Du| ≥ 1; and hence q q+2 σ |Du| dx ≤ σ q |Du|q+3 dx + C, U
(26)
(27)
U
since U is bounded. Combining (27), (26), (24) and (25), we deduce the inequality 1 q q+2 q q+2 σ |Du| dx ≤ C + σ |Du| dx + C |f |q dx 2 U U U C +C σ q−1 |Du|q+1 |Dσ · Du| dx + σ q−1 |Du|q−1 |Dσ · Du| dx. k U U Arguing as before (this means dividing the integrals in the set where |Dσ · Du| ≤ σ|Du|/C and in the rest of U ), we see that therefore σ q |Du|q+2 dx ≤ C + C |f |q dx + σ q |Du|q+2 dx + σ q |Du|q dx k U U U U (28) C2 C2 + σ q−2 |Du|q |Dσ · Du|2 dx + σ q−2 |Du|q−2 |Dσ · Du|2 dx U k U q q q q+2 for any > 0. Since U σ |Du| dx ≤ U σ |Du| dx + C, this implies our first main estimate: σ q |Du|q+2 dx ≤ C + C |f |q dx + C σ q−2 |Du|q |Dσ · Du|2 dx U U U (29) C q−2 q−2 2 + σ |Du| |Dσ · Du| dx. k U
ESTIMATES FOR TRANSPORT DENSITIES
9
4. The last two terms in (29) involving Dσ · Du are dangerous, since Dσ is of order k: we need another estimate to control them. Let us therefore continue by multiplying the PDE (5) by − div σ q−1 |Du|q Du and thereby deriving the identity div (σDu) div (σ q−1 |Du|q Du) dx = − f div (σ q−1 |Du|q Du) dx. (30) U
U
The term on the right equals f σ q−2 |Du|q (− div (σDu)) dx − f σDu · D σ q−2 |Du|q dx = |f |2 σ q−2 |Du|q dx U U U q−2 q − (q − 2) fσ |Du| Du · Dσ dx − q f σ q−1 |Du|q−1 Du · (D|Du|) dx. U
U
We again recall (23) and deduce − f div (σ q−1 |Du|q Du) dx ≤ |f |2 σ q−2 |Du|q dx U U q q−2 q + (q − 2) |f |σ |Du| |Du · Dσ| dx + |f |σ q−2 |Du|q−2 |Du · Dσ| dx. k U U The left hand term of (30) is A:=− σui (σ q−1 |Du|q uj )ij dx + σui ν i (σ q−1 |Du|q uj )j dHn−1 ∂U U q−1 q = (σui )j (σ |Du| uj )i dx U + σui ν i (σ q−1 |Du|q uj )j − σui ν j (σ q−1 |Du|q uj )i dHn−1 .
(31)
(32)
∂U
Call the boundary term B. Then, almost exactly as in step 3 of the previous proof, we can show that n−1 B= σ q |Du|q+2 dHn−1 ≥ 0. R ∂U Consequently, A= (σui )i σ q−1 |Du|q uj j dx ≥ (σui )j σ q−1 |Du|q uj i dx U U q = (σuij + σj ui ) σ q−1 |Du|q uij + (q − 1)σ q−2 |Du|q σi uj + σ q−2 |Du|q−2 σi uj dx k U q q−2 q q 2 2 q 2 q−2 q = σ |Du| |D u| + σ |Du| |Dσ| + (q − 1)σ |Du| |Dσ · Du|2 + k U q q + 2 σ q−2 |Du|q−2 |Dσ|2 + σ q−2 |Du|q−2 |Dσ · Du|2 dx. k k The first, the second and the fourth terms in the last expression are positive, and so we deduce q (q − 1) σ q−2 |Du|q |Dσ · Du|2 dx + σ q−2 |Du|q−2 |Dσ · Du|2 dx k U U (33) q−1 q ≤ div (σDu) div (σ |Du| Du) dx. U
10
L. DE PASCALE, L. C. EVANS, AND A. PRATELLI
Collecting (33), (30) and (31), we find 1 σ q−2 |Du|q |Dσ · Du|2 dx + σ q−2 |Du|q−2 |Dσ · Du|2 dx k U U 2 q−2 q ≤C |f | σ |Du| dx + C |f |σ q−2 |Du|q |Du · Dσ| dx U U C + |f |σ q−2 |Du|q−2 |Du · Dσ| dx. k U Take > 0 to be a small constant, which will be fixed later on. Then f 2 σ q−2 |Du|q dx ≤ σ q |Du|q+2 dx + C |f |q |Du|2 dx √ U U {|f |>σ|Du| } q q+2 ≤ σ |Du| dx + C |f |q dx, U
(34)
(35)
U
√ since |Du| ≤ 2 wherever |f | > σ|Du| , provided k ≥ K and K is large. Recalling (35), we can likewise estimate for each δ > 0 that |f |σ q−2 |Du|q |Dσ · Du| dx ≤ δ σ q−2 |Du|q |Dσ · Du|2 dx + C f 2 σ q−2 |Du|q dx U U U
q−2 q 2 q q+2 q ≤δ σ |Du| |Dσ · Du| dx + C σ |Du| dx + |f | dx . U
U
(36)
U
Similarly, |f |σ q−2 |Du|q−2 |Du · Dσ|dx ≤ δ σ q−2 |Du|q−2 |Dσ · Du|2 dx + C f 2 σ q−2 |Du|q−2 dx U U U
(37) q−2 q−2 2 q q q ≤δ σ |Du| |Dσ · Du| dx + C σ |Du| dx + C |f | dx . U
U
U
Since σ ≥ 1 only if |Du| ≥ 1, we have σ q |Du|q dx ≤ σ q |Du|q+2 dx + C, U
U
and therefore |f |σ q−2 |Du|q−2 |Du · Dσ| dx ≤ δ σ q−2 |Du|q−2 |Dσ · Du|2 dx U U
q q+2 q +C σ |Du| dx + |f | dx + 1 . U
(38)
U
Taking δ > 0 small, we then derive from (34), (35), (36) and (38) our second main inequality 1 q−2 q 2 σ |Du| |Dσ · Du| dx + σ q−2 |Du|q−2 |Dσ · Du|2 dx k U U (39) ≤ σ q |Du|q+2 dx + C |f |q + C. U
U
5. Putting together inequalities (29) and (39) and fixing > 0 small, we finally discover σ q |Du|q+2 dx ≤ C + C |f |q dx. U
As σ ≥ 1 only if |Du| ≥ 1, estimate (22) follows.
U
ESTIMATES FOR TRANSPORT DENSITIES
11
Theorem 4.1 concerns only smooth functions f . However, since the bound for the Lq norm of the transport density depends only upon the Lq norm of f , we can approximate: Theorem 4.2. For each 2 ≤ q < ∞ and f ∈ Lq (U ) the associated transport density a belongs to Lq (U ). Furthermore, there is a constant C, depending only upon n and U , such that aLq ≤ C (f Lq + 1) .
(40)
Proof. 1. Let us first define fj := f ∗ η1/j , the convolution of f with a standard mollifier. For each integer j, we then solve − div (σk,j Duk,j ) = fj in U uk,j = 0 on ∂U,
(41)
for σk,j := e 2 (|Duk,j | k
2
−1)
.
(42)
2. According to (22), we have the estimate
q σk,j |Duk,j |q dx ≤ C |fj |q dx + 1 ≤ C |f |q dx + 1 U
U
(43)
U
for all k greater than or equal to some constant K = k(j), depending only on the L∞ norm of fj . Now define σj := σk(j),j , uj := uk(j),j , Gj := σj Duj . Clearly fj → f in Lq . Furthermore, (43) implies that Gj is bounded in Lq . We may therefore assume upon reindexing that Gj
G
weakly in Lq (U ; Rn ).
Finally we may pass as necessary to a further subsequence to ensure uj converges uniformly to a limit u. Apply Theorem 2.3.
References [B-B]
G. Bouchitt´e and G. Buttazzo, Characterization of optimal shapes and masses through Monge– Kantorovich equation, Journal European Math. Soc., 3 (2001), 139–168. [B-B-S] G. Bouchitt´e, G. Buttazzo and P. Seppecher, Shape optimization solutions via Monge–Kantorovich equation, C. R. Acad. Sci. Paris, 324–I (1997), 1185–1191. [B-B-DP] G. Bouchitt´e, G. Buttazzo and L. De Pascale, A p-Laplacian approximation for some mass optimization problems, to appear on J.O.T.A. (Journal of Optimization Theory and Applications), (2002). [DP-P] L. De Pascale and A. Pratelli, Regularity properties for Monge transport density and for solutions of some shape optimization problems, Calculus of Variations and Partial Differential Equations, 14 (2002), 249–274. [E1] L. C. Evans, Some new PDE methods for weak KAM theory, to appear in Calculus of Variations and Partial Differential Equations. [E2] L. C. Evans, Partial differential equations and Monge-Kantorovich mass transfer (survey paper), Current Developments in Mathematics, 1997, International Press (1999), edited by S. T. Yau. [L] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, trans. by Richard Silverman, Gordon and Breach, (1969). [M] P. Marcellini, General growth conditions and regularity, in Variational methods for discontinuous structures, edited by Serapioni, Raul et al., Progress in Nonlinear Differential Equations and Applications #25, Birkhauser (1996), 111-118.
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L. DE PASCALE, L. C. EVANS, AND A. PRATELLI
` di Pisa, 56126 Pisa, Italy (L. De Pascale) Dipartimento di Matematica Applicata, Universita (L. C. Evans) Department of Mathematics, University of California, Berkeley, CA 94720, USA (A. Pratelli) Scuola Normale Superiore, 56126 Pisa, Italy
