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Владикавказский математический журнал 2014, Том 16, Выпуск 3, С. 22-37
YffK 517.957:517.548
SOLUTIONS OF THE DIFFERENTIAL INEQUALITY WITH A NULL LAGRANGIAN: HIGHER INTEGRABILITY AND REMOVABILITY OF SINGULARITIES. I1
A. A. Egorov
Dedicated to Academician Yuri Grigor'evich Reshetnyak on the occasion of his 85th birthday
The aim of this paper is to derive the self-improving property of integrability for derivatives of solutions of the differential inequality with a null Lagrangian. More precisely, we prove that the solution of the Sobolev class with some Sobolev exponent slightly smaller than the natural one determined by the structural assumption on the involved null Lagrangian actually belongs to the Sobolev class with some Sobolev exponent slightly larger than this natural exponent. We also apply this property to improve Holder regularity and stability theorems of [19J.
Mathematics Subject Classification (2000): 30C65 (primary), 35F20, 35A15, 35B35, 26B25 (secondary).
Key words: null Lagrangian, higher integrability, self-improving regularity, Holder regularity, stability of classes of mappings.
Introduction
In this paper and [22] we establish higher integrability and removability 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. In the case H(x) = 0 this inequality has the form
F(v'(x)) < KG(v'(x)) a. e. V. (2)
In [19], the author has obtained some results on closure of sets of solutions to (2) with respect to the local convergence in the Lebesgue space and their Holder regularity (for example, see
© 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.
[19, Theorems 7 and 8]). Using these results, the author has proved the theorems ([19, Theorems 1 and 3-6]) on stability of the class of solutions to the equation
F(u'(x)) = G(u'(x)) a. e. x £ V (3)
(also see [16-18]).
Our main results are analogs of the well-known higher integrability and removability theorems for mappings with bounded distortion (quasiregular mappings). A mapping v £ W¿n(V; Rn) of an open set V C Rn is an (sense-preserving) mapping with K-bounded distortion, K ^ 1, if v satisfies the distortion inequality
|v'(x)|n < Kdet v'(x) a.e. V, (4)
where |v'(x)| is the operator norm of the matrix v'(x). If, in addition, v is topological, then v is K-quasiconformal. The distortion inequality is the particular case of (2) with the following functions F(v'(x)) = |v'(x)|n and G(v'(x)) = detv'(x). The theory of quasiconformal mappings and mappings with bounded distortion is the key part of modern geometric analysis which has many diverse applications (for example, see monographs [2, 6, 10, 30, 31, 32, 36, 41, 45, 46, 49, 52, 53, 58, 59, 60, 61, 62, 63, 68, 69] and the bibliography therein).
The higher integrability for planar mappings with bounded distortion was established by B. Bojarski [8, 9]. More precisely, he proved that there exists an exponent p(2,K) > 2 such that mappings with K-bounded distortion (a 'priori in W^2) belong to Wif for every s < p(2,K). F. W. Gehring [28] has extended this result to n-dimensional quasiconformal mappings. The existence of an exponent p(n, K) > n ^ 3 such that all n-dimensional mappings with K-bounded distortion lie in W^ for every s < p(n, K) was obtained by A. Elcart and N. G. Meyers [54] and, independently, by Yu. G. Reshetnyak [57] (also see [29, 50, 51, 58, 59, 60, 61, 62]). Moreover, Yu. G. Reshetnyak [57] has established that p(n,K) to as K ^ 1 (also see [33]). For n = 2 this result was obtained by O. Lehto [48]. The higher integrability result has a dual version. In two papers, T. Iwaniec and G. Martin [35] (for even dimensions) and T. Iwaniec [34] (for all dimensions) have proved that there exists an exponent 1 < p(n, K) < n such that if a mapping v £ Wwith some p > p(n, K) satisfies inequality (4), i. e. v is a weakly quasiregular mapping, then v belongs to W^f for every s < p(n, K) (also see [23, 36, 37] and for n = 2 the monograph [49]). Here the word "weakly" means that the Sobolev integrable exponent p of v may be smaller then the dimension n. In this case, det v' (x) need not be locally integrable. Thus the natural exponent for the distortion inequality is the dimension n. K. Astala [1] proved that p(2, K) = 2K/(K +1) and p(2,K) = 2K/(K — 1) are the sharp exponents for higher integrability of planar mappings with bounded distortion (also see [2]).
Also, higher integrability results have been established for the following mapping classes: the classes of mappings that are close to multidemensional holomorphic mappings [39, 40, 41]; the classes of mappings that are close to solutions of linear elliptic partial differential equations [7, 14, 41]; the classes of quasihomoteties [64, 65]; the classes of quasiregular mappings of several n-dimensional variables [12, 13, 14]; the classes of weakly (Ki,K2)-quasiregular mappings [24, 27, 67]; the classes of degenerate weakly (K1,K2)-quasiregular mappings [26]; the classes of weakly (K1, K2(x))-quasiregular mappings of several n-dimensional variables [25]; and a series of other calsses [41-43]. Mappings of these classes, as mappings with bounded distortion, can be considered as solutions to (1) with specific functions F, G, and H (some examples of such considerations can be found in [19, §2]). Our main higher integrability
result (Theorem 2.1) contain (either partially, or fully, or in an improved form) some of the known results on higher integrability for mappings of these classes. We also apply this result to improve the above-mentioned theorems on Holder regularity and stability. In this paper, as in [16-19], we develop approaches and methods used for investigations of mappings with bounded distortion to study properties of solutions to (1). Some results of this paper have been announced in [20, 21].
The main aim of this paper is to prove Theorem 2.1 on higher integrability of solutions to (1). Applying this theorem, in the next paper [22], we establish a result on removability of singularities for solutions to (1). Also, in [22], using the Hodge decomposition theory developed by T. Iwaniec and G. Martin [34-36], we derive integral estimates for minors of Jacobian matrixs. These estimates have independent interest. In this paper they are used in the proof of Theorem 2.1.
This paper is organized as follows. In § 1 we give the basic notation and terms. The main results are stated in § 2. In § 3 we expose some auxiliary lemmas. The proof of the higher integrability theorem (Theorem 2.1 is given in §4).
1. Notation and Terminology
Let A be a set in Rn. The topological boundary of A is denoted by dA. The diameter of A is defined as diam A := sup{|x — y| : x,y e A}. The outer Lebesgue measure of A is denoted by |A|. We use the symbol dim# A for the Hausdorff dimension of A.
The set Rmxn := j ( = ) M=i,...,m : e R, ^ = 1,... ,m, v = 1,... ,nj consists of all real m x n-matrices. We identify a matrix C = (C^v) e Rmxn with the linear mapping
v=1,...,n
(Ci ,...,Cm): Rn ^ Rm, where C„(x) := Zn=i C/*v Xv, » = 1,...,m, x = (xi ,...,Xn) e Rn The operator norm in Rmxn is defined as |C| := sup{|C(x)| : x e Rn, |x| < 1}. The number of k-tuples of ordered indices in rn := {I = (i1 , ...,ik) : 1 ^ i1 < ••• < ik ^ n, iK G {1 ,...,n}, >t = l,...,k} equals the binomial coefficient := ■ Given
x e Rn and I e rn, we put xj := (xi1 ,...,xik) e Rk. If I e rn and J e r^, then
(Zjlil ••• Zj1 ifc \
. .. . is the k x k-minors of the matrix C e Rmxn. For e > —k we
Zjfci1 ••• ^k'k ' put |C|e detJJ C = 0 at C = 0.
The Jacobian matrix of u = (u1,..., um): U C Rn ^ Rm at a point x e U is the matrix u'(x) ■= If / G Tkn and JGTkm then ^f(x) = S1'""^} (s) := det JlU'(x)
and diUfj,(x) := • • • > JsrM®)); I1 = 1 For h: U —> R we put h+(x) :=
sup(h(x), 0), x e U.
Let V be a real vector space equipped with a norm | ■ |. We say that a function $: V ^ R is positively homogeneous of degree p e R if $(tx) = tp$(x) for all t > 0 and x e V \ {0}. For e > —1 we put |x|ex = 0 at x = 0.
Following Ch. B. Morrey [55], we say that a continuous function F: Rmxn ^ R is quasi-convex, if
|B(0,1)|F(C) ^ y F(C + p'(x)) dx (5)
5(0,1)
for all ^ e C0°(B(0,1);Rm) and C e Rmxn Let p ^ 1. Following M. A. Sychev [66], we say that a quasiconvex function F is strictly p-quasiconvex if, for C e Rmxn and e, C > 0,
there is § = §(Z,e, C) > 0 such that, for each mapping ^ e Cq°(B(0, 1);Rm) satisfying ll^llLp(£(o)№xn) < C|B(0,1)|1/p, the condition /B(0>1) F(Z + p'(x)) dx < |B(0,1)|(F(Z) + §) implies |{x e B(0,1) : |^>'(x)| ^ e}| ^ e|B(0,1)|. Observe that in the mathematical literature the term strictly quasiconvexity is also used for another property (which is close but nonequivalent to ours) consisting in the fact that the strict inequality in the definition of quasiconvexity (5) is valid for nonzero mappings ^ (for example, see [38]). In this article we use the term in the sense of M. A. Sychev's definition [66]. In the case p > 1 the notion of strictly p-quasiconvexity for functions F of this article is equivalent to the notion of strictly closed p-quasiconvexity from J. Kristensen's article [44] which is defined in terms of the theory of gradient Young measures (see [44, Proposition 3.4]). Observe that we can replace the ball B(0,1) in the definitions of quasiconvexity and strictly p-quasiconvexity by an arbitrary bounded domain U with |dU| = 0 (for example, see [56]). A function G: Rmxn ^ R is a null Lagrangian if both functions G and —G are quasiconvex. The term "null Lagrangian" appeared due to the following fact: The Euler-Lagrange equation corresponding to the variational integral fu G(u'(x)) dx with null Lagrangian G holds identically for all admissible mappings u: U C Rn ^ Rm (see [4] and also [5, 36, 56]). The only the affine combinations of minors (called quasiaffine functions) are null Lagrangians [15, 47] (also see [3, 4, 5, 36, 55, 56]); i. e.
min{m,raj
G(Z)= Yo + £ E YJ/ detjj Z, Z e Rmxn, (6)
u m i-1 n
for some y0, yj/ e R.
2. Statement of the Main Results
The principal result on higher integrability of solutions to (1) states as follows
Theorem 2.1 (Higher integrability). Let n,m, k e N and t > k such that 2 ^ k ^ min{n, m}, and let V be an open set in Rn. Suppose that a continuous function F: Rmxn ^ R satisfies
F(Z) ^ CF|Z|k, Z e Rmxn, (7)
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). Then for K ^ 1 there exist two numbers p = p(F, G, K) and p = p(F, G, K) depended only F, G, and K with l^p<k<p^t such that for a given exponent p > p every solution v e ^¿'^(V; Rm) to (1) actually lies in W^S(V;R™) for all s e (p,p). Moreover, for a number e > 0, a vector b e Rm, and a test function ip e Cq°(F) we have the Caccioppoli-type inequality
(-)lLs(V;Rmxn) < C|||(v(.) — b) <X> p'(.)| + |^(.)|(£ + e1-kH+(.))|L,(V) (8)
with some constant C = C(F, G, K, s) > 0 depended only F, G, K, and s.
In the case H(x) = 0 a straightforward consequence of Theorem 2.1 is the following.
Corollary 2.2. Under the conditions of Theorem 2.1, there exist two numbers p^ = p (F,G, K) and p0 = p0(F,G, K) depended only F, G, and K with 1 ^ ^ < k < p0 such that for a given exponent p > pQ every solution v e W^tV; Rm) to (2) actually lies in Rm) for all s e (p0,Po)- Moreover, for a vector b e Rm and a test function
p G Cq°(V) we have the homogeneous Caccioppoli-type inequality
II^O' (OHL« (V ;«mxn) < Co||«) - ft) <8> (•)yLs (V ;KmXn)
(9)
with some constant C0 = C0(F, G, K, s) > 0 depended only F, G, K, and s.
Now we apply Theorem 2.1 to improve some results from [19].
Let n, m, k e N such that 2 ^ k ^ min{n, m}. We need the following hypothesis on continuous functions F: Rmxn ^ R and G: Rmxn ^ R (see [19]):
(H1) F is quasiconvex;
(H1') F is strictly k-quasiconvex;
(H2) G is a null Lagrangian;
(H3) F and G are positively homogeneous of degree k; (H4) sup{K ^ 0 : F(C) ^ KG(C), C e Rmxn} = 1; (H5) cF := inf{F(C) : C e Rmxn, |C| = 1} > 0;
(H6) dG := sup{J]Jerfc Jepfc Iyjt||xj|2 : x e Rn, |x| = 1} < kc^/(n — k) in the case k < n.
Here the coefficients yjj are taken from (6) for the null Lagrangian G. By (H3), the representation (6) for the null Lagrangian G consists only of k x k-minors; i.e.,
G(C)= E YJJ detjj C, C e Rmxn (10)
J erm,J ern
Since F is continuous, (H3) implies (7) with cF = c^.
The following theorem on Holder regularity of solutions to (1) is a straightforward consequence of Theorem 2.1 and [19, Theorem 8].
Theorem 2.3 (Holder regularity). Suppose F and G satisfy (H2)-(H6). Put K0 = to
kc0
for k = n and Ko = for k < n. Suppose that K e [1,-Ko) and 5 e (0,1) satisfy the
inequality
_I__(11)
kcF n — k + k5 v 7
Let p is the exponent from Theorem 2.1, p > p, and V be an open set in R™. Then every solution v e WjO'C^V; Rm) to (1) with H(x) = const satisfies the Holder condition with exponent 5 on each compact subset in V.
The next theorem on stability of the class of solutions to (3) is a straightforward consequence of Corollary 2.2 and [19, Theorem 4].
Theorem 2.4 (Stability in the C-norme). Suppose that F and G satisfy (H1)-(H6). Let K ^ 1, and let pQ denote the exponent from Corollary 2.2. Let V be a domain in R™, and let U be a compact subset in V. Then there is a function a(K) = aF)G)y;U(K) defined for 1 ^ K < K0 and such that limK^1 a(K) = a(1) = 0 and, for each mapping v e Wof (V; Rm), p > p , which satisfies inequality (2) there is a mapping u e Rm) which is a solution
to (3) such that
IIv — u||c(u) ^ a(K) diam v(V). (12)
The following theorem improves Theorem 2.4 in the case when the function F satisfies (H1'). In this case, in addition to the estimate (12) of proximity (in the C-norm) of solutions of inequality (1) to solutions to equation (3), we obtain proximity estimates (in the Lk-norm) for the derivatives of these mappings. The theorem is a straightforward consequence of Corollary 2.2 and [19, Theorem 6].
Theorem 2.4 (Stability in the Sobolev norme). Suppose that F and G satisfy (H1') and (H2)-(H6). Then the conclusion of Theorem 2.4 is valid together with (12) and the following inequality:
||v' — u'||Lk(^;Rmxn) ^ a(K) diam v(V). (13)
3. Auxiliary Lemmas
In the proof of the higher integrability theorem we need the following lemmas.
Lemma 3.1. Let n,m, k e N, 2 ^ k ^ min(m,n), p ^ 1, b e Rm, I = (i1,... ) e J = (j,..., ) e rm, and V C Rn be an open set. For a mapping v: V ^ Rm and a function p: V ^ R define the mappings w = (w1,..., wm): V ^ Rm and h = (h1,..., hm): V ^ Rm by the rules w(-) := v(-) — b and h(-) := p(-)w(-). If v and p are differentiable at x e V, then the following inequality
|p(x)v'(x)|p-k detj/(p(x)v'(x)) — |h'(x)|p-k detj/ h'(x)
< (2k + p)|w(x) <8> p'(x)|(|h'(x)| + |w(x) ® p'(x)|)p-1 (14)
holds.
< Observe that
We have
p(x)v'(x) = h'(x) — w(x) ® p'(x).
/ d/ hj — w^ d/ p^ detj/(h' — w ® p') = det .
\d/ j — wjfc d/ Pj Hence, detj/(h' — w ® p') = detJ/ h' — K=1 where
(15)
/
:= det
ô/ h
ji
\
d/ hjK-i
j d/ P d/hjK+i - j+i d/P
V d/hjk - j d/P /
Observe that
|AK1 < |d/ hji 1 . . . |d/ hjK-1 | Kk d/ P| |d/ hjK+1 - WK+I d/ P| • • • |d/ hjk - j d/ P|
< |WJ <g> d/p| |d/hj|K-1 |d/hj - WJ <g> d/p|k-K < |w <g> p'| |h'|K-1 |h' - w <X> P'|k-K•
Therefore,
k
| detj/(h' - pv') - detj/ h'| < |w <g> p'|h'|K-1 |h' - w <g> p'|k-K• (16)
K=1
Let us consider 2 cases. Case 1:
0 < |h'(x) - w(x) ® p'(x)| < |h'(x)|•
(17)
We have
I |h' - w <g> p'|p-k det J/(h' - w <g> p') - |h'|p-k det J/ h'|
< |h' - w <g> p'|p| detJ/(h' - w <g> p')| ||h' - w <g> p'|-k - |h'|-k| + |h' - w ® p'|p|h'|-k| detj/(h' - w <g> p') - detj/ h'|
+ |h'|-k| detj/ h'| ||h' - w <g> p'|p — |h'|p| • (18)
We estimate each term on the right-hand side of (18). We obviously have the inequalities
a-k - a-k < k(a2 - ai) a-k-1 (19)
and
aP — ai ^ p(a2 — a1) apP- (20)
for 0 < a1 ^ a2. Using (17) and (19), we get
||h' — w ® <|-k — |h'|"fc| = |h' — w ® </|-k — |h/|-k
< k(|h'| — |h' — w ® ^'|)|h' — w ® < k|w ® <| |h' — w ® <|-k-1. (21)
By Hadamard's inequality
| det ji(h — w ® </)| < |h' — w ® </|k. (22)
Combining (21) with (22), we have
|h' — w ® <|p| detJ1 (h — w ® <)| | |h' — w ® </|-k — |h'|-k|
< k|w ® <'| |h' — w ® </|P-1 < k|w ® <|(|h'| + |w ® <'|)p_1. (23)
Using (16) and (17), we obtain
|h/ — w ® <|p|h/|-k| det ji(h/ — w ® </) — detjj h/1
k
< |h/ — w ® <|p|h/|-k|w ® |h/|K-1|h/ — w ® </|k-K
K=1
k
< |w ® </| |h/ — w ® <?-1 ^(|h/ — w ® </|/|h/|)k-K+1
K=1
< k|w ® </| |h/ — w ® </|P-1 < k|w ® <|(|h/| + |w ® </|)P-1. (24) Using (17) and (20), we get
||h/ — w ® <|p — |h/|p| = |h/|p — |h/ — w ® <|p
< p(|h/1 — |h/ — w ® <|)|h/|p-1 < p|w ® </| |h/|p-1. (25)
By Hadamard's inequality
| detj/ h/| < |h/|k. (26)
Combining (25) with (26), we have
|h/|-k| detj/ h/1 ||h/ — w ® </|P — |h/|p| < p|w ® <1 |h/|p-1
< p|w ® <|(|h/| + |w ® </|)P-1. (27)
Using (15), (18), (23), (24), and (27), we obtain (14) in case 1. Case 2:
0 < |h/(x)| < |h/(x) — w(x) ® </(x)|. (28)
We have
||h - w 8 <T-k detjj (h - w 8 <0 - |h |p-k detjj h'|
< |h' - w ® <|-k| detj/(h - w <8 <)| ||h' - w <8 - |h|p|
+ |h' - w 8 p'|-k|h'|p| detjj(h' - w 8 p') - detjj h'|
+ |h'|p| detjj h'| ||h' - w <8 p'|-k -|h'|-k|. (29)
We estimate each term on the right-hand side of (29). Using (20) and (28), we get
||h - w 8 <¿7 - |h|p| = |h' - w 8 <¿7 - |h/|p
< p(|h' - w 8 <'| - |h'|)|h' - w 8 </|p-1 < p|w 8 <'| |h' - w 8 <'|p-1.
If we combain this with (22), we obtain
|h' - w 8 <'|-k| det j/(h - w 8 <')| | |h' - w 8 <T - |h'|p|
< p|w 8 <'| |h' - w 8 <T-1 < p|w 8 </|(|h/1 + |w 8 <|)p-1. (30)
Using (16) and (28), we get
|h/ - w 8 </|-k|h/|p| det J/(h/ - w 8 </) - detJ/ h/|
k
< |h/ - w 8 <|-k|h/|p|w 8 |h/|K-1|h/ - w 8 </|k-K
K=1
k
< |w 8 p'| |h'|p-1 ^(|h'|/|h' - w 8 p'|)K
K=1
< k|w 8 p'| |h'|p-1 < k|w 8 p'|(|h'| + |w 8 p'|)p-1. (31)
Using (19) and (28), we have
||h' - w 8 p'|-k - |h'|-k| = |h'|-k - |h' - w 8 p'|-k
< k(|h' - w 8 p'| - |h'|)|h'|-k-1 < k|w 8 p'| |h'|-k-1.
If we combain this with (26), we obtain
|h'|p| detjj h'| ||h' - w 8 p'|-k - |h'|-k| < k|w 8 p'| |h'|p-1
< k|w 8 p'|(|h'| + |w 8 p'|)p-1. (32)
Combining (15) and (29)-(32), we get (14) in case 2. >
Lemma 3.2. Suppose that a null Lagrangian G is homogeneous of degree k. Under the conditions of Lemma 3.1, we have
||p(x)v'(x)|p-kG(p(x)v'(x)) - |h'(x)|p-kG(h'(x))|
< C|w(x) 8 p'(x)|(|h'(x)| + |w(x) 8 p'(x)|)p-1 (33)
with some constant C = C(G,p) depended only on G and p.
< Using (10) and (14), we get
||p(x)v'(x)|p-kG(p(x)v'(x)) - |h'(x)|p-kG(h'(x))|
< E lYJiI ||p(x)v'(x)|p-k detj/(p(x)v'(x)) - |h'(x)|p-k detjj h'(x)| k
m'J v—n
< (2k + p)( E J M|w(x) <g> p'(x)|(|h'(x)| + |w(x) <g> p'(x)|)p-1. > V Jerm,/ern /
Remark 3.3. We can use C(G,p) = (2k + p) V Jerfc Ierfc |yj1 | as the constant in (33).
J er mer n
Here 7j/ are the coefficients from (10).
Also, we need the following version of Gehring's lemma (for example, see [36, Corollary 14.3.1]):
Lemma 3.4. Suppose f and g are non-negative functions of the class Lq(Rn), 1 < q < œ, and satisfy
(
\
i/q
1
V
|B (a,R)|
f q
<
A
{
|B(a, 2R)|
f +
1
|B (a, 2R)|
\
gq
B(a,2R) )
i/q
B(a,2R) \
for all balls B(a, R) C and some constant A > 0. Then the inequality
ft << C//.
holds with some exponent q' = q'(n, q, A) > q and some constant C = C(n, q, A) > 0 depended only n, q, and A.
4. Proof of the Higher Intagrability Theorem
We are now in a position to prove the higher intagrability theorem.
< Proof of Theorem 2.1. Let V C be an open set and 1 < p ^ t. A suitable range of Sobolev exponents p will be defined below (see (46), (47), and (48)).
Fix a solution v e Wl0'(f(V;Rm) to (1). Consider p e Cq°(V). We may assume that p is non-negative as otherwise we could consider |p| which has not effect on the required Caccioppoli-type inequality (8). Using (1) and F(Z) ^ cF|Z|k, we get
|v'(x)|k < c-1KG(v'(x)) + c-1H+(x) a.e. V. (34)
Multipling both sides by pp(x)|v'(x)|p-k and using k-homogeneity of G, we obtain
|p(x)v'(x)|p < c-1K|p(x)v'(x)|p-kG(p(x)v'(x))
+ c-1 pp(x)|v'(x)|p-kH+(x) a. e. V. (35)
Let e > 0. Put V1 := {x e V : |v'(x)| ^ e} and V2 := {x e V : |v'(x)| < e}. We have V1 n V2 = 0 and |V \ (V1 U V2)| = 0. Then
J |pv'|p = J |pv'|p + J |pv'|p. (36)
V Vi V2
We estimate each term on the right-hand side of (36). Using (35), we have
J |<v/|p < c-1K y |<v/|p-kG(<V) + c-1 y <p|v/|p-kH+. (37)
y'|p < c-1^ |pv'|p-kG(pv') + c-1y pp|v'|p-k, Vi Vi Vi
We estimate each term on the right-hand side of (37).
Using (10) and Hadamard's inequality, we have -G(Z) < |G(Z)| < C1(G)|Z|k, Z G Rmxn, with C1(G) := jgrfc jgrfc |YJJ|. Here yjj are the coefficients from (10). Then |Z|p-kG(Z) + C1(G)|Z|p ^ 0. Therefore, n
J |<v/|p-kG(<V) + C1 (G) J |<v/|p ^ 0.
Vi Vi
Consequently,
J |pv'|p-kG(pv') ^J |pv'|p-kG(pv') + j |pv'|p-kG(pv')
Vi Vi V2
+ C1(G^ |pv'|p = J |pv'|p-kG(pv') + C1(G^ |pv'|p. (38)
V2 V V2
We estimate each term on the right-hand side of (38).
Let b G Define the auxiliary mappings w: V ^ and h: ^ by the rules w(x) := v(x) - b, x G V, and
ip(x)w(x), x G V [0, x G \ V.
Then h G W 1p(Rn; Rm). We have pv' = h' - w 8 p'.
Successively using Lemma 3.2, (10), and [22, Theorem 2.1], we deduce
|<v/|p-kG(<v/) ^ J |h/|p-kG(h/) + C(G,p^(|h/1 + |w 8 <|)p-1|w 8 </|
V
E YJ/ / |h/|p-k detj/(h/) + C(G,p) /(|h1 + |w 8 </|)p-1|w 8 <1
V
< C2(G)|1 - p/k^ |h/|p + C(G,p^ y (|h/| + |w 8 <|)p-1|w 8 </|, (39)
where C2(G) := C(k)C1(G), C(k) is from [22, Theorem 2.1], and C(G,p) = (2k + p)C1(G) is from Remark 3.3.
Since |v'| < e on V2, we have
y |pv'|p < ej |pv'|p 1 p. (40)
V2 V2
Combining (38), (39), and (40), we get
J |pv'|p-kG(pv') < C2(G)|1 - p/k^ |h'|p
Vi
+ C(G,p^y (|h'| + |w <g> p'|)p-1 |w <g> p'| + eC1(G^ |pv'|p-1p. (41)
V V2
Since |v'| ^ e on V1, we have
y |pv'|P-1
Vi Vi
Using (37), (41), and (42), we obtain
J pV|p-kH+ < e1-^ |pv'|p-1pH+. (42)
|pv'|p < c-1KC2(G)|1 -p/k^ |h'|p
-1KC(G,p) y (|h'| + |w <g> p'|)p-1|w <g> p'|
V
+ ec-1 KC1(G^ |pv'|p-1 p + e1-kc-^ |pv'|p-1pH+. (43)
Vi
-1
V2 Vi
Now combining (36), (40), and (43), and using |pv'| ^ |h'| + |w ® p'|, we get
J |pv'|p < c-1KC2(G)|1 -p/k^ |h'|p
V
+c-1KC(G,p^y (|h'| + |w <g> p'|)p-1|w ® p'|
V
->-1 KC (G) + 1) / |pv'lp-1p + e1-kc-1 / |pv'lp-1/
+e(c-1KC1(G) + 1^ |pv'|p-1p + e1-kc-1 J |pv'|p-1pH+ (44)
V2 Vi
< c-1KC2(G)|1 -p/k|y |h'|p + [c-1K(C(G,p) + C1(G) + 1) + 1] X y (|h'| + |w <g> p'|)p-1 (|w <g> p'| + p(e + e1-kH+)).
V
Observe that |h'|p ^ 2p-1(|pv'|p + |w <g> p'|p). By (44), we get I |h'|p < 2p-^ y |pv'|p + 2p-^ y |w <g> p'|p < 2p-1c-1KC2(G)|1 - p/k^ |h'|p
VV
+ 2p-1 [c-1K (C (G,p) + C1 (G) + 1) + 2]
xj(|h'| + |w ® p'|)p-1(|w <g> p'| + p(e + e1-kH+)). (45)
V
Put
p = p(F,G,K) :=mî{p^l:2p-1CF1KC2(G)\l-p/k\ < l} (46)
and
p = p(F,G,K,t) :=sup {p^t:2p~1Cp1KC2(G)\l-p/k\ < l}. (47)
For
P G (p,p) (48)
we have 2p-1 c-1KC2(G)|1 -p/k| < 1. From (45) we derive
J |h/|p < Ca(F,G,K,p)y (|h/1 + |w 8 </|)p-1 (|w 8 </| + <(e + £1-kH+)), (49)
V
l-2P-1c-F1KC2{G)\l-p/k\
wWp r (F r K r,\ — 2 [cFiK(C(G,p)+C1(G)+l)+2]
where G3(F,G,K,p)
Successively using (|h'| + |w8p'|)p ^ 2p-1 (|h'|p + |w8p'|p), (49), and Holder's inequality, we obtain
y (|h'| + |w 8 p'|)p < 2p-1 I y |h'|p + J |w 8 p'|p
V V V
< C(F, G, K,p) J(|h'| + |w 8 p'|)p-1 (|w 8 p'| + p(e + e1-kH+))
V
<C{F,G,K,p) ^J(|/i'| + |w8</|)pj ^J(\w®ip'\ +cp(£ + £1-kH+)f where C(F, G, K,p) := 2p-1 (C3(F, G, K,p) + 1). Therefore,
ll|h'| + |w 8 p'|||lp(v) < C(F, G, K,p)|||w 8 p'| + p(e + e1-kH+)||LP(v). Using |pv' | ^ |h'| + |w 8 p' | and w = v - b, we get the Caccioppoli-type estimate
llpv'||lp(v;Rmxn) < C(F, G, K,p)|||(v - b) 8 p'| + p(e + e1-kH+)||lp(v). (50)
Of course now we observe that this inequality holds with p replaced by s for any s G (p,p), provided we know a priori that v G W10'cs(V; Rm).
Let S = {se (p,p) : v G IR"1)}. We have peS. Therefore, S / 0. For s G S
we have (8); the constant C = C(F, G, K, s) which depends continuously on s is finite in the range (p, p) but may blow up at the endpoints. If we combine this with the Sobolev embedding theorem, we obtain that S is relatively closed in (p, p). The theorem will be proved if we can show that S is open. Obviously, if s G S, then (p, s] C (p,p). We are therefore left only with the task of showing higher integrability of the differential. It is at this point that Gehring's lemma comes to the rescue. Using (50), we easily derive reverse Holder inequalities for v'.
Consider x0 G V and 0 < R < dist(x0, V)/3. Put BR := B(x0,R) and B2R := B(x0, 2R). Let n G C0°°(B2R) be a nonnegative function such that
7] = 1 on BR, O^t/^1 and |r/| < ^^ on B2r. (51)
Substituting cp by i] and b by vb2R '■= \b2r\ Jb2r v we
llnv' IIlp (B2R;«mxn) < C(F, G, K, p) ||| (v - vB2R) ® n'| + n(e + e1-kH+) .
Therefore,
J |nv'|p < C4(F,G,K,p)( J |(v - vb2r ) <£> n'|p + / n(e + e1-kH+/) ,
B2R B2R B2R
where C4(F,G,K,p) = (C(F,G,K,p))p. Successively using (51) and the Poincare-Sobolev inequality (for example, see [36, Theorem 4.10.3]), we obtain
f |p < R-P/ |v - VB2R |p + J + £1-kH+)p
,/Vo I c1-k H.)P
Br x b2r b2r
n-\-p
J + J (£ + £1~kH+)
B2R B2R
with some consants C5 (F, G, K,p) and C6 (F, G, K,p). Therefore,
n+p
br V V B2R 7 B2R 7
Hence, we have the reverse Holder inequality
n
1 /Vi^
|Br | J ^ |B2R |
br B2R
+ ( J-T J {C9(F,G,K,p)(e + e1-kH+)Y
|B2R |
B2R
n n+p
I H 7 ' t'J-'
Put q = > 1, / = and 5 = (Cg(F, G, K,p)(e + e1_fcii+))™+p. By Lemma 3.4 we
conclude that f is integrable with a power slightly larger than q. This in turn means that v' is integrable with a slightly higher power then p and so v G (V; Rm) for some p' > 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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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. I
Егоров А. А.
Целью настоящей статьи является установление свойства самоулучшающейся интегрируемости производных решений дифференциального неравенства с нуль-лагранжианом. Более точно, мы доказываем, что решение класса Соболева с показателем суммирумости, немного меньшим естественно определенного структурными предположениями на нуль-лагранжиан показателя, фактически принадлежит пространству Соболева с показателем суммируемости, немного большим естественного показателя. Мы также применяем это свойство, чтобы улучшить теоремы о гёльдеровой регулярности и об устойчивости из статьи [19].
Ключевые слова: нуль-лагранжиан, повышающаяся интегрирумость, самоулучшающаяся регулярность, гёльдерова регулярность, устойчивость классов отображений.
