Решения дифференциального неравенства с нуль-лагранжианом: повышающаяся интегрируемость и устранимость особенностей. II

In this paper we continue to study the properties of solutions v : V → R ^m , V ⊂ R ⁿ , of the following inequality

F (v ⁰ (x)) 6 KG(v (x)) + H (x) a.e. V                        (1)

constructed by means of a continuous function F : R ^{m x n} ^ R, a null Lagrangian G : R ^{m x n} ^ R, a measurable function H : V ^ R, and a constant K >  1. Here v 0 (x) denotes the differential of v at x ∈ V . Using the higher integrability theorem of the previous paper [8], we establish a result on removability of singularities for solutions to (1).

Many investigations have dealt with the problem of removable singularities for quasicon-farmal mappings and mappings with bounded distortion (for example, see [1, 2, 4], [9]–[29] and the bibliography therein). Painlev´e’s theorem, a classical result in complex function theory, states that sets of zero length are removable for bounded holomorphic functions. More precisely, if E is a closed subset of linear measure zero in a planar domain V and v is a bounded function holomorphic in V \ E , then v extends to a bounded holomorphic function of V . Observe that the class of planar mappings with 1-bounded distortion coincides with the class of holomrphic functions. The strongest removability conjecture, stated in [14] as the counterpart of Painlev´e’s theorem for mappings with bounded distortion, suggests that sets of Hausdorff а-measure zero, a 6 n/(K + 1) 6 n/2, are removable for bounded mappings with K -b ounded distortion in R ⁿ . In the case n = 2 this conjecture was verified

In this paper, using the Hodge decomposition theory developed by T. Iwaniec and G. Martin [13, 14, 15], we also obtain integral estimates for wedge products of closed differential forms (Theorem 2.3) and for minors of a Jacobian matrix (Theorem 2.1). These estimates are extensions of integral estimates derived in [15]. They have been used in the proof of the higher integrability theorem in [8].

Some results of this paper have been announced in [7].

This paper is organized as follows. In § 1 we establish a result on removability of singularities for solutions to (1). We derive integral estimates for wedge products of closed differential forms and for minors of a Jacobian matrix in § 2.

We use the notation and terms from [8].

It is clear that |E| = 0. Further, the higher integrability theorem [8, Theorem 2.1] gives the Caccioppoli-type estimate k^v0 IIls (V .Rmxn) 6 C ||lv 0 y°1 + |y|(e + e1 k H+)|Ls(V)                    (3)

for e > 0 and y G c°(V \ E), where the constant C = C(F,G, K, s) does not depend on the test function y or the mapping v. Let x G C°°(V) and E0 := E П suppy. Then E0 has zero s-capacity. Therefore there exists a sequence of functions (nj G c°(V))jeN such that 0 6 ηj 6 1, ηj = 1 on some neighbourhood of E0 , limj→∞ ηj = 0 almost everywhere in V , and liny ,° V |nj|s = 0. Put yj := (1 — nj)x G c°(V \ E) and vj := yjv G W01,s(V; Rm). Then the mappings vj are bounded in L∞ (V ; Rm) and converge to χv almost everywhere. We have vj0 = ϕjv0 + v ⊗ ϕ0j and ϕ0j = -χηj0 + (1 - ηj)χ0. Using (3), we obtain kvj IIls(V.Rmxn) 6 llyjv kLs(V.Rmxn) + llv 0 yj kLs(V.Rmxn)

• ^{6    (1 +} C ) (ll v ⁰ ^ j kL ^s (V;R ^{m x n} ) ⁺ ll^yj ^{(e + e} 1 k ^H + ) kL ^s (V) }

• ^{6    (1 +} C ) (llvllL ” (V \ E;R ^m ) k ^ j k L ^s (V;R ⁿ ) ⁺ ll^y j ^{(e + e} 1 k H +^ L s (V)}

• ^{6 (1 +} C ) (llvllL ” (V \ E;R m ) k X k L ” (V) k nj k L s (V ;R n )

• ⁺ 1Ы^” (V \ E;R m )k ^{(1 —} n j MIL (V ;R ⁿ ) ⁺ IK^{1 —} П ^{) | x | (e + e} 1 - k H +^L^V )) •

Passing to the limit over j , we get limsup ||vj |ls(v.Rmxn)

j →∞

• ^{6 (1} + C ) (k v k L ” (V \ E;R ^m ) k X ⁰ k L ^s (V ;R ⁿ ) + k X^(e + ^e 1 k H +^ L s (V)) • ⁽⁴⁾

• 2.    Integral Estimates

Therefore the sequence (v j ) j_eN is bounded in W ^1,s (V; R m ). Hence there exists its subsequence converged weakly in W ^1,s (V; R m ) to a mapping in this Sobolev space. Clearly, this limit coincides with χv almost everywhere in V .

Therefore xv G W₀ ^1,s (V;R m ) for all test functions x G c ° (V ). This yields v G W lO^ V; R m ). Since v is a solution to (1) almost everywhere in V, the higher integrability theorem [8, Theorem 2.1] implies v G W lOC (V;R m ). B

Remark 1.2. The assumption that v is bounded is of course rather more than we really need in the proof of Theorem 1.1. All that is required is that the sequence ( | v 0 y j | l s (v .R m x n ) ) j e N remains bounded as j ^ to. Thus, for instance, Theorem 1.1 can be extended to the case v ∈ L ^p ( V \ E ; R ^m ), p >  ns/ ( n - s ), if in addition we require the stronger restriction that the set E has zero r -capacity for r = sp/ ( p - s ). That this requirement is sufficient follows from Holder’s inequality applied to | v 0 y j | l s (v .R m x n ) instead of using the trivial bound |v| l ^ (v \ E.R m ) k y j I Il s (v ;R n ) used above.

In the proof of the higher integrability of solutions to (1) ([8, Theorem 2.1]) we have used the following theorem on integral estimates for minors of a Jacobian matrix. This theorem is an extension of T. Iwaniec and G. Martin’s result on integral estimates for Jacobians (cf. [15, Theorems 7.8.1 and 13.7.1]).

Theorem 2.1. Let n, m, k ∈ N with 2 6 k 6 min(m, n). Then for every distribution v = (vi, ...,vm) E D '(Rn; Rm) with v' E Lp(Rn; Rmxn), 1 6 p< to, and for every I = (i1,..., ik) E rm, J = (j1,... ,jk) E ГП we have the inequality b vl

p-k ^dv J

∂x I

6C

1 ¹ - k \И

with some constant C = C(k) depended only on k .

Remark 2.2. In the case k = n = m Theorem 2.1 coincides with T. Iwaniec and G. Martin’s result on integral estimates for Jacobians (see [15, Theorems 7.8.1 and 13.7.1]).

In the proof of Theorem 2.1 we need the following modification of T. Iwaniec and G. Martin’s result on integral estimates for wedge products of closed differential forms (cf. [15, Theorem 13.6.1]).

Let Λ ^l = Λ ^l (R ⁿ ), l ∈ N ∪ { 0 } , be the space of all l-exterior forms on R ⁿ . For I = (i 1 ,... ,i i ) E Г П we denote the l-exterior form dx i 1 Л • • • Л dx i l by dx j . We use the convention that dx j = 1 if 1 = 0. For ш E Л ¹ we have ш = V,,- _ri Y j dx i with some coefficients Y j E C.

I ^ ^ n

We put |ш| = pijeTi |YI|2)   • For p > 1 we denote by Lp(Rn;Лl) the space of differential l-forms on Rn with coefficients in Lp(Rn).

Theorem 2.3. Let n, k E N with 2 6 k 6 n. Consider P 1 ,...,P k ,E 1 ,...,E k E R

and 11,... ,1k E N such that 1 < pK < to,    + ... + -1 = 1, — 1 6 2ek p1            pk

—

6 ^{p κ} p κ

¹ , and

I := n — 1 1 — ... — 1 k >  0 . Let I = (i 1 ,..., i — ) E Г ^ . Suppose that (^ 1 ,..., ^ k ) be a k-tuple of closed differential forms with ^ _K E L ^{(1 - E k ) P k} (Rⁿ^ l ^K ) . Then

( _и ^Л-^Л .. ^Л dx j 6       .. 1                     1

J     | ^ 1 | ^{e i} . .. | ^ k | ^£ k               ^1 L ^{(1 - e}i^)pi ( R ⁿ ;Л ^l1 ) "'ll ^^к^ (^{1 - е}к ^{) p}k ( R ⁿ ;Л^l k ) ’

where e := max( | E 1 | ,..., | E k | ) and the constant C = C (p 1 ,... ,p k ) depends only on p 1 , ... ,p k .

Remark 2.4. In the case I = 0, i.e. dx — = 1, Theorem 2.3 coincides with [15, Theorem 13.6.1]. For proving Theorem 2.3 we use the Hodge decomposition technique developed in [13, 14, 15] and follow the proof of Theorem 13.6.1 in [15].

C Proof Of Theorem 2.3. Observe that (1 — e _k )p _K >  ^{Р к} 2+¹ > 1 and |e _k | < 1/2, к = 1,...,k. We have .^^ E L p^x (R ⁿ ;Л ^l к ). Denote by W ¹ ’ ^p (R ⁿ ;Л ^l ), 0 6 1 6 n, p >  1, the space of differential 1-forms on R n with coefficients in W ^1,p (R ⁿ ). We can consider the following Hodge decomposition in L ^{p κ} (R ⁿ ; Λ ^{l κ} ) ([14, Theorem 6.1], also see [15, § 10.6]):

I ^ ।   = ^daK + ^d * ^в к

ϕ κ ^{ε κ}

with some a _K E W ^{1 ,Р к} (R n ; Л ^{1 к - 1} ) and в _к E W ^{1 ,Р к} (R n ; Л ^{1 к +1} ). Here d is the exterior derivative, and d* is its formal adjoint, the coexterior derivative. The forms da _K and d * e _K , κ = 1, . . . , k, are uniquely determined and can be expressed by means of the Hodge projection operators

E : L ^p (R ⁿ ;Л ^l ) ^ dW ^(Rn;^ ^-1 ) and E * : L ^p (R ⁿ ;Л ^l ) ^ d * W 1’ ^p (R n ;Л ¹⁺¹ )

defined by [15, § 10.6, formulas (10.71) and (10.72)] for 1 < p < ∞ and 1 6 l 6 n- 1. Namely we have dax = E ( —^^ ) and d*вх = E* (        .                (8)

^κ          | ϕ κ | ^ε κ                   ^κ            | ϕ κ | ^ε κ

Applying [14, Theorem 6.1], we get the following bound for exact term:

kda K k L P K (R n ;Л к ) 6 С 1 (Р к ) k ^ K kL— K ) P K ( R n ;^ ) .                    (9)

By [15, § 10.6, formulas (10.73) and (10.74)] we have

KerE = {ϕ ∈ Lp(Rn; Λl) : d∗ϕ= 0} and

Ker E∗ = {ϕ ∈ Lp(Rn;Λl) : dϕ=0} for 1 < p < ∞ and 1 6 l 6 n - 1. Then E∗ (ϕκ) = 0. Therefore we can write d∗βκ as a commutator

d*e _K = E * ( ) -            .

| ϕ κ | ^{ε κ}        | E ^∗ (ϕ κ ) | ^{ε κ}

Applying [15, Theorem 13.2.1] (also see [13, Theorems 8.1 and 8.2]), we obtain

№в х kL P K (R n ;Л K ) 6 С 2 (Р к )|E k Ik ^ K k L - 1 - K )P K _(R n к ) ’                 (^)

Using (7), we have

/

^ 1 Л • • • Л ^ k Л dx | ^ 1 | ^{£ 1} - - - Il k | ^e k

j(da i + d*e i ) Л ••• Л (da k + d* e k ) Л dx I

j da- 1 Л • • • Л da k Л dx ^ +

B .

Since p1, - - - ,pk represents a Holder conjugate tuple, by Stokes’ formula via an approximation argument we obtain j da-1 Л • • • Л dak Л dx^ = 0-                             (12)

The integrand B is a sum of wedge products of the type ^1 Л^ • •Л ^k Л dxp where cK is either dακ or d βκ and at least one d∗βκ is always present, with at most 2k - 1 terms. Combining H¨older’s inequality with (9) and (10), we get j ^1 Л • • • Л ^k Л dxj 6 C3(k) ||^1 kLP1 (Rn;Al1 ) - - - IIW kLPk (Rn-Alk)

⁶ C⁴^, ^{- - - ,p} k kll^ ^{1 k} L (^{1 -} _E i⁾pi (R n -A li ) ^{- - -} №k ^k L (^{1 -} Ek)Pk (R n -A lk ) ^-

This with (11) and (12) yields (6). B

C Proof of Theorem 2.1. Let p _κ := k , ε _κ := ε := 1 - _k ^p , and l _κ := 1 for κ = 1 , . . . , k .

Then 1 < рк < to, p1! ++ p1k = 1, l := n - k = n - 11- lk > 0,(1 - Ek)Pk = P, and max(|E11,..., |Ek|) = |e| = |1 - p|. Let ^K := dvjK E L(1-£k)Pk(Жп;Л1к).  Let I =

(i1, - - -, i|) E ГП be the ordered l-tuple such that {i1, - - -, i^} = {1, - - -, n} \ {i1, - - -, ik}. We chose the sign sgn I such that sgn Idxi Л dx^ = dx1 Л • • • Л dxn.

When p lies outside the interval (k- ^ ¹ , 3k) the estimate is clear as (5) always holds with 1 in place C (k) | 1 - p | . In this case | 1 - p | >  k—- and inequality (5) holds with C (k) = ^ — 1 .

Suppose that k + 1 6 2p 6 3k. Then — 1 6 2e _K 6 p ^{K -1} and | e | 6 1/2. Applying p _κ

Theorem 2.3, we obtain d =

| dv j ₁ | ^ε . . . | dv j _k | ^ε

sgn Idv j 1 Л • • • Л dv j k Л dx j

| dv j ₁ | ^{ε 1} . . . | dv j _k | ^ε k

6 C i (k) | _£ |k dV j i k L^A i ) ... kdV j k k L - ₍ ^e R n _; A i ) 6 C i (k)E / |v 0 I p (13)

Using the elementary inequalities | g J | 6 | dv j 1 | ... | dv j _k | and | a — a ^{1 e} | 6 | e | for 0 6 a 6 1 and - 1 < ε < 1, we have

∂v J ∂x _I | v ⁰ | ^εk

∂v J

∂xI

| dv j ₁ | ^ε . . . | dv j _k | ^ε

Л/И_

| dv j ₁ | . . . | dv j _k |

| dv j ₁ | . . . | dv j _k | | v ⁰ | ^k

| dv j ₁ | . . . | dv j _k | | v ⁰ | ^k

1 — e

6 | ε || v ⁰ | ^p .

Combining this with (13), we obtain

| v ⁰ |

p - k ^∂v J ∂x I

∂v J             ∂v J

∂x I             ∂x I

-

| v ⁰ | ^εk       | dv j ₁ | ^ε . . . | dv j _k | ^ε

+

∂v J

∂x I

| dv j ₁ | ^ε . . . | dv j _k | ^ε

6 (C 1 (k) + 1) |E| У | v ⁰ | p . ▻

Acknowlegement. The author is greteful to A. P. Kopylov, Yu. G. Reshetnyak, A. S. Romanov, and S. K. Vodop’yanov for helpful discussions.