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

Автор: Егоров А.А.

Журнал: Владикавказский математический журнал @vmj-ru

Статья в выпуске: 4 т.16, 2014 года.

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Целью статьи является установление результата о затирании особенностей у решений дифференциального неравенства с нуль-лагранжианом. Также получены интегральные оценки для внешних произведений замкнутых дифференциальных форм и для миноров матрицы Якоби.

Короткий адрес: https://sciup.org/14318477

IDR: 14318477

Текст научной статьи 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 ) + llyj (e + e 1 k H + ) kL s (V) }

  • 6    (1 + C ) (llvllL (V \ E;R m ) k ^ j k L s (V;R n ) + lly 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 ) + IK1 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 ) jeN 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

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 - ei)pi ( R n l1 ) "'ll ^к^ (1 - ек ) pk ( R n 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 px (R n l к ). Denote by W 1 p (R n 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 к (R n ; Л 1 к - 1 ) and в к E W 1 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 l ) ^ dW ^(Rn;^ -1 ) and E * : L p (R n l ) ^ d * W 1’ p (R n 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 к ) 6 С 1 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 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 K ) 6 С 2 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.

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