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Владикавказский математический журнал 2014, Том 16, Выпуск 4, С. 41-48
WIK 517.957+517.548
SOLUTIONS OF THE DIFFERENTIAL INEQUALITY WITH A NULL LAGRANGIAN: HIGHER INTEGRABILITY AND REMOVABILITY OF SINGULARITIES. II1
A. A. Egorov
The aim of this paper is to establish a result on removability of singularities for solutions of the differential inequality with a null Lagrangian. Also, we obtain integral estimates for wedge products of closed differential forms and for minors of a Jacobian matrix.
Mathematics Subject Classification (2000): 30C65 (primary), 35F20, 35A15, 35B35, 26B25 (secondary).
Key words: null Lagrangian, removability of singularities, integral estimates, closed differential forms, minors of a Jacobian matrix.
Introduction
In this paper we continue to study the properties of solutions v: V ^ Rm, V C Rn, of the following inequality
F(v'(x)) < KG(v'(x)) + H(x) a.e. V (1)
constructed by means of a continuous function F: Rmxn ^ R, a null Lagrangian G: Rmxn ^ R, a measurable function H: V ^ R, and a constant K ^ 1. Here v'(x) denotes the differential of v at x e 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). Painleve'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 Painleve's theorem for mappings with bounded distortion, suggests that sets of Hausdorff a-measure zero, a ^ n/(K + 1) ^ n/2, are removable for bounded mappings with K-bounded distortion in Rn. In the case n = 2 this conjecture was verified
© 2014 Egorov A. A.
1 The study was supported by a grant from the Russian Foundation for Basic Research, project 14-0100768, the State Maintenance Program for the Leading Scientific Schools of the Russian Federation, grant № NSh-2263.2014.1, and the Integration Grant of the Siberian Division of the Russian Academy of Sciences, 2012, № 56.
by K. Astala [1] for a < 2/(K +1) and K. Astala, A. Clop, J. Mateu, J. Orobitg, I. Uriarte-Tuero [2] for a = 2/(K + 1) (also see [3]). The higher integrability results for mappings with bounded distortion are closely related to the removability problems. Caccioppoli-type estimates (one of the key ingredients in proofs of higher integrability results) can be used as the basic tool for proving removability theorems. More precisely, if there exists p(n, K) < n such that Caccioppoli-type estimates hold for p > p(n, K), then a close set E with the Hausdorff dimension dirndl?) < n — p(n, K) is removable for bounded mappings with K-bounded distortion in R™ (for example, see [13, 14, 15, 16]). In such way removability results have been established for the classes of mappings that are close to solutions of linear elliptic partial differential equations and for the classes of quasiregular mappings of several n-dimensional variables (for example, see [5, 6]). Mappings of these classes, as mappings with bounded distortion, can be considered as solutions to (1) with specific functions F, G, and H. Our removability result (Theorem 1.1) contain partially the known results on removability of singularities for mappings of these classes.
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].
1. Removability of Singularities
Using the higher integrability theorem from [8], we establish the following result on removability of singularities for solutions to (1).
Theorem 1.1 (Removability of singularities). Let n,m,k G N and t > k such that 2 ^ k ^ min{n,m}, and let V be a domain in Rn. Suppose that a continuous function F: RmXn ^ R satisfies
F(Z) ^ cf|Z|k, Z G RmX", (2)
with some constant cf > 0, a null Lagrangian G: RmXn ^ R is homogeneous of degree k, and a measurable function H: V —» R has H+ e L\oc{V). Fix K ^ 1. Let p = p(F, G, K) be the exponent from [8, Theorem 2.1]. For a closed subset E ofV with the Hausdorff dimension dimh(E) < n — p every bounded solution v e (V \ E; Rm) to (1) extends to a mapping of the class (V; Rm) which is defined over the whole domain V and also satisfies (1).
We follow the approach, developed in [14] (also see [15]), to prove Theorem 1.1. There are two key components in the proof. Firstly, the assumption on the size of the set E implies that E has zero s-capacity for an appropriate value of s. Secondly, the Cacciappoli-type estimate holds for this particular value of s. The definiton of s-capacity can be find in [9, 10, 15, 22, 23].
< Proof of Theorem 1.1. We have p < n - dimh{E). Let s e (p,n - dim^i?)). From [9, Ch. 3, Theorem 5.11] (also see [10, 15]), we obtain that the set E has zero s-capacity.
It is clear that |E| =0. Further, the higher integrability theorem [8, Theorem 2.1] gives the Caccioppoli-type estimate
llvv'||Ls(V;Kmxn) < C|||v ® v'| + |v|(e + e1-k) (3)
for e > 0 and v e C0°(V \ E), where the constant C = C(F, G, K, s) does not depend on the test function v or the mapping v. Let x e Cq°(V) and E' := E n suppThen E' has zero s-capacity. Therefore there exists a sequence of functions (nj e C0(V))jgN such that 0 ^ nj ^ 1, nj = 1 on some neighbourhood of E', limj^0 nj = 0 almost everywhere in V, and limj^ jv |nj|s = 0. Put Vj := (1 - nj)x e C°(V \ E) and Vj := Vjv e W01,s(V; ). Then the mappings Vj are bounded in L°(V; Rm) and converge to %v almost everywhere. We have vj = Vjv' + v ® vj and vj = —xnj + (1 — nj)x'. Using (3), we obtain
||vj ^L5(V;RmXn) ^ ||vjv' yLs(V;KmXn) + ||v ® vj yLs(V;KmXn)
< (1 + C ) II v ® ^j yLs(V;RmXn) + 11 Vj (e + e1-k H+)|ls(v )
^ (1 + C ^ ||v||L~ (V\E;Km)yvj Hls(v ;Mn) + yVj (e + ^^ H+)yLs (v ^ (1 + C^||v||L~(V\E;Km)yXyLTO(v) ynj yLs(v;!»)
+ I|v||l~ (v\E;Km) ll (1 — nj )x'IIls (V ;Rn) + ||(1 — nj )|x|(e + e1-k H+)IIls(v)).
Passing to the limit over j, we get limsup 11 v j ||_LS(V ;RmXn )
j^O
< (1 + C)(||v|L~(v\E;Rm)|x'lL«(v;K™) + ||x(e + e1-kH+)|Ls(V)). (4)
Therefore the sequence (vjjeN is bounded in W 1,s(V; Rm). Hence there exists its subsequence converged weakly in W 1,s(V; Rm) to a mapping in this Sobolev space. Clearly, this limit coincides with xv almost everywhere in V.
Therefore xv e W01,s(V; Rm) for all test functions x e C°(V). This yields v e W^V; Rm). Since v is a solution to (1) almost everywhere in V, the higher integrabil-ity theorem [8, Theorem 2.1] implies v e W^f(V; Rm). >
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 ® Vj ||ls(V;KmXn ))j€N remains bounded as j ^ to. Thus, for instance, Theorem 1.1 can be extended to the case v e Lp(V\E; Rm), 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 ® vjIIl
s (V;RmXn) instead of using the
trivial bound ||v||l °°(V\E;Mm) II Vj IlLs(V;Rn) used above.
2. Integral Estimates
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 ^ k ^ min(m, n). Then for every distribution v = (v1 ,...,vm) £ D '(Rn; Rm) with v' £ LP(Rn; Rmxn), 1 ^ p< x, and for every I = (i1,... ,ik) £ rm, J = (j1, ■ ■ ■, jk) £ rn we have the inequality
\v'rk^ < C 1 - f / |t>'|p (5)
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 Al = A1 (Rn), l £ N U {0}, be the space of all l-exterior forms on Rn. For I = (i1,... ,ii) £ rn we denote the l-exterior form dxil A • • • A dxby dxi. We use the convention that dxi = 1 if l = 0. For u £ A1 we have u = Y] T^ yidxi with some coefficients yi £ C.
T n
/ \ 1/2 l We put |u| = ( Y^igpi |yi|2) . For p ^ 1 we denote by LP(Rn; Al) the space of differential
l-forms on Rn with coefficients in LP(Rn).
Theorem 2.3. Let n,k £ N with 2 ^ k ^ n. Consider p1,...,pk,e1 ,...,ek £ R
and ¿i,..., lk £ N such that 1 < pH < oo, — + ... + — = 1, -1 ^ 2eH ^ an(j
Pk Pk
I := n — l1 — ... — lk ^ 0. Let I = ( i 1,..., i£ rln. Suppose that (ip1,... ,ipk) be a k-tuple of closed differential forms with £ L(1-£k)Pk (Rn;AlK). Then
f (pi A---A(pkAdXj ^ ri M 1-ei
where e := max(|e1|,..., |ek |) and the constant C = C (p1,... ,pk) depends only on p1,... ,pk.
Remark 2.4. In the case l = 0, i.e. dxj = 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].
< Proof of Theorem 2.3. Observe that (1 -eH)pH ^ ^^ > 1 and \eK\ < 1/2, x =l,...,k. We have ^^ G Lp-(Rra;A'"). Denote by W1'p(Mn;Al), 0 < I < n, p ^ 1, the space of differential l-forms on Rn with coefficients in W 1,P(Rn). We can consider the following Hodge decomposition in LPk(Rn;AlK) ([14, Theorem 6.1], also see [15, § 10.6]):
= daH + d* f3H (7)
|
with some aK £ W 1,Pk(R";A1k-1) and ßK £ Wx'Pk(R";A1k+1). Here d is the exterior derivative, and d* is its formal adjoint, the coexterior derivative. The forms daK and d* ßK, k = 1,... ,k, are uniquely determined and can be expressed by means of the Hodge projection operators
E: Lp(Rn; A1) ^ dW 1>p(Rn;A1-1) and E* : Lp(Rn;A¿) ^ d*W1'p (Rn;A1+1)
defined by [15, § 10.6, formulas (10.71) and (10.72)] for 1 < p < to and 1 ^ l ^ n — 1. Namely we have
da'=E(i¿f) »d d"ß'=EiiiF)' <8)
Applying [14, Theorem 6.1], we get the following bound for exact term:
||daK yLPK (Rn ;AiK ) ^ c1(pK )||vK )pk (R n;AiK ). (9)
By [15, § 10.6, formulas (10.73) and (10.74)] we have
Ker E = {v e Lp(Rn;A1) : d*v = 0}
and
Ker E* = {v e Lp(Rn;A1) : dv = 0}
for 1 < p < to and 1 ^ l ^ n — 1. Then E* (vK) = 0. Therefore we can write d*as a commutator
rl*R - F* ( \
Vivxi"J
Applying [15, Theorem 13.2.1] (also see [13, Theorems 8.1 and 8.2]), we obtain
||d*^xyLPK(Rn;aiK) ^ c2(pK)|eK||vKHL(ie-£*)Pk(Rn;AIk). (10)
Using (7), we have
f v1 A- • • A Vk A dx f f, ,
/ -i-1-i- = / (dai + d Pi) A • • • A (dak + cf/3k) A dxf
J |V1 |ei ... |VkP J 1
= J da1 A •••A dak A dxf + J B. (11)
Since p1;... ,pk represents a Holder conjugate tuple, by Stokes' formula via an approximation argument we obtain
J da1 A ■ ■ ■ A dak A dxf = 0. (12)
The integrand B is a sum of wedge products of the type A ■ ■ ■ A A dxf, where is either daK or and at least one is always present, with at most 2k — 1 terms. Combining Holder's inequality with (9) and (10), we get
j V>1 A ■ ■ ■ A ^k A dx7 < c3 (k)||^1 11LP1 (Rn;ali) . . . ^k yLPfc (Rn;alfc)
< c4 (p1 , . . . ,pk )e|v1 ||L-1-1£1)P1 (Rn;ali) . . . ||Vk yL-i-k )Pk (Rn ;alk).
This with (11) and (12) yields (6). >
< Proof of Theorem 2.1. Let pH := k, eH := e :=1 — and lH := 1 for x = 1,..., k.
Then 1 < pH < to, ± h-----h ^ = 1, 1 ■■= n - k = n - h-----lk ^ 0, (1 - eH)pH = p,
and max(|ei|,...,|ek|) = |e| = |l-f|. Let := dvjx G L(1-e-)P-(Rra;A'"). Let J = (i 1,..., if) e rn be the ordered ¿-tuple such that { i1,..., if} = {1,..., n} \ {i1,..., ik}. We chose the sign sgn I such that sgn /dxj A dxf = dx1 A ■ ■ ■ A dxn.
When p lies outside the interval the estimate is clear as (5) always holds with 1
in place C(k) 1 — In this case 1 — | ^ and inequality (5) holds with C(k) =
Suppose that k + 1 < 2p < 3k. Then -1 < 2eK < anci |e| ^ 1/2. Applying
Theorem 2.3, we obtain
df j dx i
Ijl |£ ••• |dvjk |£
f sgn Idj A ■ ■ ■ A A dxf
< Ci(k)|e| ||dvji HL-r^i) ••• ||dj ||]-(R„;AI) < Ci(k)e| |V |p. (13)
Using the elementary inequalities and —1 < e < 1, we have
dv j
dxj
^ |dvj11 • • • |dvjk| and |a — a1 e| ^ |e| for 0 ^ a ^ 1
dvj dx i
dvj dx i
dvj \„.f\p
1 1 l^iil ■ ■ ■ l^iJ _ (\dvjA---\dvju\
■ I Irln,. I U,nk I |„/|fc
|dvji| ••• |dvjk | |V|
|V |
1-e
Combining this with (13), we obtain
\vThP- <
OX/
dvj dx t
dvj dxr
|V |dVji |e ••• |e
+
dvj dxr
|dvjl|£ ••• |dvjk |£
< (Ci(k) + 1)|e| y >
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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Received April 3, 2013.
Egorov Alexander Anatol'evich Sobolev Institute of Mathematics, senior researcher
RUSSIA, 630090, Novosibirsk, Koptyug Avenue, 4; Novosibirsk State University, associate professor
RUSSIA, 630090, Novosibirsk, Pirogova Str., 2 E-mail yegorov@math.nsc.ru
РЕШЕНИЯ ДИФФЕРЕНЦИАЛЬНОГО НЕРАВЕНСТВА С НУЛЬ-ЛАГРАНЖИАНОМ: ПОВЫШАЮЩАЯСЯ ИНТЕГРИРУЕМОСТЬ И УСТРАНИМОСТЬ ОСОБЕННОСТЕЙ. II
Егоров А. А.
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.
Ключевые слова: нуль-лагранжиан, устранимость особенностей, интегральные оценки, внешнее произведение замкнутых дифференциальных форм, миноры матрицы Якоби.
