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 \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 - ei)pi ( R n ;Л l1 ) "'ll ^к^ (1 - ек ) pk ( R n ;Лl 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 ;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.
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