DOI 10.1007/s10958-018-3884-z

Journal of Mathematical Sciences, Vol. 232, No. 4, July, 2018

STRONG PRECOMPACTNESS OF BOUNDED SEQUENCES UNDER NONLINEAR ULTRAPARABOLIC DIFFERENTIAL CONSTRAINTS E. Yu. Panov Novgorod State University 41, Bol’shaya St.-Peterburgskaya ul., Velikiy Novgorod 173003, Russia [email protected]

UDC 517.95

It is shown that bounded sequences satisfying nonlinear differential constraints, strongly precompact under an exact condition of nondegeneration of these conditions. The proof is based on new localization principles for ultraparabolic H-measures with continuous indices. Bibliography: 16 titles. Dedicated to the memory of Vasilii Vasil’evich Zhikov

1

Introduction

Let Ω be an open subset in Rn . We assume that ϕ(x, u) ∈ L2loc (Ω, C(R, Rn )) and B(x, u) ∈ L2loc (Ω, C(R, Symn )) are Carath´eodory vector and matrix in the domain Ω, where Symn stands for the linear space of symmetric n × n matrices. This means that ϕ(x, u) and B(x, u) are measurable with respect to the variables x ∈ Ω and continuous with respect to u ∈ R; moreover, αM (x) = max (|ϕ(x, u)| + |B(x, u)|) ∈ L2loc (Ω) |u|M

∀M > 0.

Here and below, we denote by |ξ| the Euclidean norm of a finite-dimensional vector. We also assume that the monotonicity condition holds: ∀x ∈ Ω, u, v ∈ R, u > v

B(x, u) − B(x, v)  0,

(1.1)

i.e., (B(x, u) − B(x, v))ξ · ξ  0 for all ξ ∈ Rn (here, ξ · η denotes the inner product of vectors ξ, η ∈ Rn ). We also assume that the matrices B(x, u)−B(x, v) degenerate on the linear subspace X ⊂ Rn : ∀ξ ∈ X (B(x, u) − B(x, 0))ξ = 0. Let uk (x) ∈ L∞ (Ω), k ∈ N, be a bounded sequence. We can assume, passing to a subsequence if necessary, that this sequence weakly converges to some function u(x) ∈ L∞ (Ω): uk  u weakly − ∗ in L∞ (Ω) as k → ∞. Translated from Problemy Matematicheskogo Analiza 93, 2018, pp. 101-118. c 2018 Springer Science+Business Media, LLC 1072-3374/18/2324-0516 

516

Suppose that the sequences of distributions div ϕ(x, sa,b (uk (x))) − D 2 · B(x, sa,b (uk (x))) are −1,−2 precompact in the anisotropic Sobolev space Wd,loc (Ω) (the exact definition of this space will be given below) for some d > 1 and all a, b ∈ R, a < b, where sa,b (u) = max(a, min(u, b)) are cut-off functions. Here, 2

D · B(x, sa,b (uk (x))) =

n  i,j=1

∂2 bij (x, sa,b (uk (x))), ∂xi ∂xj

where bij (x, u), i, j = 1, . . . , n, are entries of the matrix B(x, u). The main result of this paper is the following property of strong precompactness. Theorem 1.1. We assume that for almost all x ∈ Ω and all ξ ∈ X, ξ ∈ X ⊥ such that  ξ + ξ = 0 the functions λ → ξ · ϕ(x, λ), λ → B(x, λ)ξ · ξ are not constant in any neighborhood u(x). Then uk (x) → u(x) (strongly) in L1loc (Ω) as k → ∞. In the case X = Rn , Theorem 1.1 is proved in [1]. In [2], Theorem 1.1 is proved under more restrictive nondegenerate condition: For almost all x ∈ Ω and all ξ ∈ X, ξ ∈ X ⊥ , ξ+ ξ = 0, the functions λ → ξ · ϕ(x, λ) and λ → B(x, λ)ξ · ξ are not constant on nondegenerate intervals. To prove the strong precompactness properties, the theory of ultraparabolic H-measures with continuous indices was developed in [2] (cf. also [3]). In this paper, we use a version of ultraparabolic H-measures similar to that introduced in [1] for “hyperbolic” H-measures (corresponding to the case X = Rn ). We note that the nondegeneracy condition in Theorem 1.1 is essential. Indeed, let ϕ(x, u) = ϕ(u), B(x, u) = B(u), and ξ · ϕ(u) = const in the δ-neighborhood of the point u0 for ξ ∈ X, ξ = 0. Then the sequence uk (x) = u0 + δ sin(kξ · x)  u0 as k → ∞ and satisfies the condition div ϕ(uk ) − D 2 · B(uk ) = 0. It is obvious that this sequence does not contain strongly converging subsequences. In Section 2, we describe the main notions, in particular, the notion of an H-measure. In Section 3, we prove the localization principle and the main theorem (Theorem 1.1).

2

The Main Notions

We recall (cf. [4]–[6]) that a measure-valued function on Ω is a weakly measurable mapping x → νx from the set Ω to the space of probability Borel measures with compact support in R. The  weak measurability of νx means that for any continuous function f (λ) the function x→

f (λ)dνx (λ) is Lebesgue measurable in Ω.

A measure-valued function νx is called bounded if there exists M > 0 such that supp νx ⊂ [−M, M ] for almost all x ∈ Ω. We denote by νx ∞ the least value of M possessing this property. Finally, measure-valued functions of the form νx (λ) = δ(λ−u(x)), where δ(λ−u) is the Dirac measure at the point u, are called regular. We identify them with the corresponding functions u(x). Thus, the set MV(Ω) of bounded measure-valued functions on Ω contains the space L∞ (Ω). We note that νx ∞ = u ∞ for the regular measure-valued function νx (λ) = δ(λ − u(x)). Measure-valued functions naturally arise as the limits of bounded sequences in L∞ (Ω) in the sense of the following result [6]. 517

Theorem 2.1. Let uk (x) ∈ L∞ (Ω), k ∈ N, be a bounded sequence. Then there exists a subsequence ur = ukr (x) and a measure-valued function νx ∈ MV(Ω) such that  f (ur )  f (λ)dνx (λ) weakly − ∗ in L∞ (Ω) as r → ∞ (2.1) for all f (λ) ∈ C(R). Moreover, the limit measure-valued function νx is regular: νx (λ) = δ(λ − u(x)) if and only if ur (x) → u(x) in L1loc (Ω). We assume that X ⊂ Rn is a linear subspace, X ⊥ is its orthogonal complement, P1 and P2 are the orthogonal projections onto X and X ⊥ respectively. For ξ ∈ Rn we set ξ = P1 ξ  2 + |ξ|4 = 1 }. and ξ = P2 ξ, so that ξ ∈ X, ξ ∈ X ⊥ , ξ = ξ + ξ. Let SX = { ξ ∈ Rn | |ξ| Then SX is a compact smooth manifold of codimension 1 and coincides with the unit sphere S = {ξ ∈ Rn | |ξ| = 1 } if X = {0} or X = Rn . We define the projection πX : Rn \ {0} → SX by πX (ξ) =

ξ

 2 + |ξ|4 )1/2 (|ξ|

+

ξ

 2 + |ξ|4 )1/4 (|ξ|

We note that πX (ξ) = ξ/|ξ| if X = {0} or X = Rn . Let  F (u)(ξ) = e−2πiξ·x u(x)dx,

.

ξ ∈ Rn ,

be the Fourier transform extended as the unitary operator to the space L2 (Rn ). We denote by u → u, u ∈ C, the complex conjugation operation. We consider a sequence Uk (x) = (Uk1 (x), . . . , Ukl (x)) ∈ L2 (Ω, Rl ), l ∈ N, that weakly converges to zero. Proposition 2.1 (cf. [2, 3, 7]). There exists a family of complex Borel measures μ = {μij }li,j=1 on Ω × SX and a subsequence Ur (x) = Uk (x), k = kr , such that  F (Uri Φ1 )(ξ)F (Urj Φ2 )(ξ)ψ(πX (ξ)))dξ (2.2)

μij , Φ1 (x)Φ2 (x)ψ(ξ) = lim r→∞ Rn

for all Φ1 (x), Φ2 (x) ∈ C0 (Ω) and ψ(ξ) ∈ C(SX ). The family μ = {μij }li,j=1 is called the H-measure corresponding to the subsequence Ur (x). The notion of an H-measure in the homogeneous case X = Rn was introduced in [8] and independently in [9]. Later, parabolic H-measures (corresponding to the case dim X = 1) were defined in [10]. The notion of an H-measure was generalized in [11] (cf. also [12, 13]) to the case of “continuous” indices i, j. A similar generalization to the case of ultraparabolic H-measures was proposed in [2, 3]. Let uk (x) ∈ L∞ (Ω) be a bounded sequence converging weakly (in the sense of (2.1)) to a bounded measure-valued function νx ∈ MV(Ω). We consider the measures γxk (λ) = δ(λ − uk (x)) − νx (λ) and the corresponding distribution functions Uk (x, p) = γxk ((p, +∞)), u0 (x, p) = νx ((p, +∞)), p ∈ R. Let   E = E(νx ) = p0 ∈ R | u0 (x, p) → u0 (x, p0 ) in L1loc (Ω), p → p0 . As was shown in [11, Lemma 4], the complement E = R \ E is at most countable and if p ∈ E, then Uk (x, p) ∈ L∞ (Ω) for all k ∈ N and Uk (x, p)  0 weakly-∗ in L∞ (Ω) as k → ∞. The following result, similar to Proposition 2.1, is proved in [11] (cf. also [12]–[14]) in the case X = Rn and in [2, 3] in the general case of arbitrary X. 518

Proposition 2.2. 1. There exists a family of locally finite complex Borel measures {μpq }p,q∈E in Ω × SX and a subsequence Ur (x) = {Urp (x)}p∈E , Urp (x) = Ukp (x), k = kr , such that for all Φ1 (x), Φ2 (x) ∈ C0 (Ω) and ψ(ξ) ∈ C(SX )  F (Φ1 Urp )(ξ)F (Φ2 Urq )(ξ)ψ(πX (ξ))dξ. (2.3)

μpq , Φ1 (x)Φ2 (x)ψ(ξ) = lim r→∞ Rn

2. For p1 , . . . , pl ∈ E, ζ1 , . . . , ζl ∈ C l 

ζi ζj μpi pj  0.

i,j=1

The family of measures μ = {μpq }p,q∈E is referred to as the H-measure corresponding to the subsequence ur (x) = ukr (x).  ∈ C(Rn ) We note that the function ψ(πX (ξ)) in (2.3) can be replaced with the function ψ(ξ) which is equal to ψ(πX (ξ)) for large |ξ|. This fact follows from the compactness of the operator  − ψ(πX (ξ)) (cf. Lemma 2.4 below). with symbol ψ(ξ) We consider a countable dense set of indices D ⊂ E. Since |Urp (x)|  1, from (2.3) and the Plancherel identity we easily find that prΩ |μpq |  meas for all p, q ∈ E, where meas is the Lebesgue measure on Ω. In particular, the x-projection of the H-measure is absolutely continuous with respect to the Lebesgue measure and the Radon–Nikodym theorem allows us to obtain the representation (2.4) μpq = μpq x dx (disintegration of the H-measure), where μpq x is a family of complex finite Borel measures on   SX , p, q ∈ D, x ∈ Ω , Ω is a subset of Ω of full measure, and the relation (2.4) means that for all Φ(x, ξ) ∈ C0 (Ω × SX ) the function  x → μpq (ξ), Φ(x, ξ) = Φ(x, ξ)dμpq x x (ξ) SX

is Lebesgue measurable in Ω, bounded, and 

μpq x (ξ), Φ(x, ξ)dx.

μ , Φ(x, ξ) = pq

Ω

In the hyperbolic case X = Rn , this representation was obtained in [13] (cf. also [14, 1]). In the general case of an arbitrary X, the representation (2.4) was established in [2, 3]. Moreover, for a set Ω ⊂ Ω of full measure of the values of x, the measures μpq x are uniquely defined by the equality  pq F (Φm Urp )(ξ)F (Φm Urq )(ξ)ψ(πX (ξ))dξ (2.5)

μx , ψ(ξ) = lim lim m→∞ r→∞ Rn

for all ψ(ξ) ∈ C(SX ). Here, (Φm Urq )(y) = Φm (x − y)Urq (y), and the sequence Φm (x), m ∈ N, 2 n ∞ n is  defined by the relation |Φm (x)| = Km (x) = m K(mx) for K(x) ∈ C0 (R ), K(x)  0, K(x)dx = 1. 519

We note that the set Ω of full measure can be chosen in such a way that its points are Lebesgue points of functions u0 (x, p), u− 0 (x, p) = lim u0 (x, q) = νx ([p, +∞)) for all p ∈ D. q→p−

Since p ∈ D ⊂ E is a point of continuity of the mapping p → u0 (x, p) in L1loc (Ω), we have  u− 0 (x, p) = u0 (x, p) almost everywhere in Ω. By construction, x ∈ Ω is a common Lebesgue − point of the functions u0 (x, p) and u0 (x, p). Therefore, νx ({p}) = u− 0 (x, p) − u0 (x, p) = 0 ∀p ∈ D.

(2.6)

Remark 2.1. (a) Since the H-measure is absolutely continuous with respect to x, the identity (2.3) holds for Φ1 (x), Φ2 (x) ∈ L2 (Ω). Indeed, by Proposition 2.3, we can write this identity in the following form: for all Φ1 (x), Φ2 (x) ∈ C0 (Ω), ψ(ξ) ∈ C(SX ) 

 Φ1 (x)Φ2 (x) ψ(ξ), μpq x (ξ)dx Ω

= lim

r→∞ Rn

F (Φ1 Urp )(ξ)F (Φ2 Urq )(ξ)ψ(πX (ξ))dξ.

(2.7)

By assertion (i) of Lemma 2.1 below, both sides of this identity are continuous with respect to (Φ1 (x), Φ2 (x)) in L2 (Ω) × L2 (Ω) and, since C0 (Ω) is dense in L2 (Ω), we conclude that (2.7) is satisfied for any Φ1 (x), Φ2 (x) ∈ L2 (Ω). (b) If x ∈ Ω is a Lebesgue point of a function Φ(x) ∈ L2 (Ω), then from (2.5) it follows that  Φ(x) μpq x , ψ(ξ) = lim lim

m→∞ r→∞ Rn

F (ΦΦm Urp )(ξ)F (Φm Urq )(ξ)ψ(πX (ξ))dξ

(2.8)

for all ψ(ξ) ∈ C(SX ), where (ΦΦm Urp )(y) = Φ(y)Φm (x − y)Urp (y), (c)

l 

p pj

ζi ζj μxi

(Φm Urq )(y) = Φm (x − y)Urq (y);

 0 for x ∈ Ω and any family pi ∈ D, ζi ∈ C, i = 1, . . . , l. Indeed, this fact

i,j=1

pp

directly follows from (2.5) and Proposition 2.2(2). In particular, the matrix {μxi j (A)}li,j=1 is Hermitian and nonnegative definite for any Borel set A ⊂ SX . From this property (for l = 2, qp pq p1 = p, p2 = q) it follows that μpp x  0 and μx = μx for all p, q ∈ D. For a complex Borel measure μ ∈ M(SX ) we denote by |μ| its variation, i.e., the least nonnegative measure m on SX such that |μ(A)|  m(A) for all Borel sets A ⊂ SX . Denote by Var μ = |μ|(SX ) the total variation of a measure μ. We recall that Var μ is the standard norm in the Banach space M(SX ) of finite Borel measures on SX . As is known,        Var μ = sup f (λ)dμ(λ).   f (λ)∈C(SX ),f ∞ 1  SX

Lemma 2.1. The following estimates hold:  (i) Var μpq x  1 for all p, q ∈ D, x ∈ Ω , 

pq  1/2 for all p, p , q ∈ D, x ∈ Ω . (ii) Var (μpq x − μx )  2|u0 (x, p ) − u0 (x, p)|

520

Proof. From the Plancherel identity and estimate |Urp |  1 for all p ∈ E, r ∈ N       q p F (Φ U )(ξ)F (Φ U )(ξ)ψ(π (ξ))dξ   m r m r X   Rn

 ψ ∞ Φm Urp 2 |Φm Urq 2  ψ ∞ Φm 22 = ψ ∞ . By (2.5), for all ψ(ξ) ∈ C(SX )

| μpq x , ψ(ξ)|  ψ ∞ ,

which implies (i). Using the Plancherel identity again, we have        q p p F (Φm (Ur − Ur ))(ξ)F (Φm Ur )(ξ)ψ(πX (ξ))dξ     Rn





 ψ ∞ Φm (Urp − Urp ) 2 Φm Urq 2  ψ ∞ Φm (Urp − Urp ) 2 since

Φm Urq 2

(2.9)

 Φm 2 = 1. We note that  p Ur (y) = θ(λ − p)dγyr (λ) = θ(ur (y) − p) − u0 (y, p),

where

 θ(s) =

1,

s > 0,

0,

s  0,

denotes the Heaviside function. Since for all p ∈ D we have θ(ur (y) − p)  u0 (y, p) weakly-∗ in L∞ (Ω) as r → ∞, we see that 

|Urp (y) − Urp (y)|  |θ(ur (y) − p) − θ(ur (y) − p )| + |u0 (y, p) − u0 (y, p )| = sgn (p − p)(θ(ur (y) − p) − θ(ur (y) − p ) + u0 (y, p) − u0 (y, p ))  2 sgn (p − p)(u0 (y, p) − u0 (y, p )) = 2|u0 (y, p) − u0 (y, p )| 

weakly-∗ in L∞ (Ω) as r → ∞. Hence from the estimate |Urp (y) − Urp (y)|  2 it follows that    lim Φm (Urp − Urp ) 22 = lim Km (x − y)(Urp (y) − Urp (y))2 dy r→∞

r→∞

  2 lim

r→∞

Km (x −

y)|Urp (y)

Ω

−

 Urp (y)|dy

Ω

 4

Km (x − y)|u0 (y, p) − u0 (y, p )|dy.

Ω

Ω

are Lebesgue points of the function u0 (y, p) for all p ∈ D, from the above relation Since x ∈ it follows that  lim lim Φm (Urp − Urp ) 2  2|u0 (x, p) − u0 (x, p )|1/2 . (2.10) m→∞ r→∞

Taking into account (2.9) and (2.10), from (2.5) for all ψ(ξ) ∈ C(SX ) we get         q pq p p − μ , ψ(ξ)| = lim lim F (Φ (U − U )(ξ)F (Φ U )(ξ)ψ(π (ξ))dξ | μpq   r m m X r x x r m→∞ r→∞  Rn

 2 ψ ∞ |u0 (x, p) − u0 (x, p )|1/2 which implies (ii). 521

Since for x ∈ Ω the function u0 (x, ·) is continuous on D, we have |u0 (x, p) − u0 (x, p )| → 0 as p → p. We also note that |u0 (x, p) − u0 (x, p )| = νx ((min(p, p ), max(p, p ))).

(2.11)

The following lemma is similar to Lemma 3.1 in [1]. Lemma 2.2. For any p, q ∈ R, x ∈ Ω there exist unilateral limits in the space M(SX ) :  

(p , q  ) → (p, q), p , q  ∈ D, p > p, q  > q,

 

(p , q  ) → (p, q), p , q  ∈ D, p < p, q  < q.

μpx q → μpq+ x , μpx q → μpq− x ,

Moreover, Var μpq±  1 and for any finite tuples pi ∈ R, ζi ∈ C, i = 1, . . . , l, l 

p pj ±

ζi ζj μxi

 0.

(2.12)

i,j=1

Proof. Assume that x ∈ Ω , p, q ∈ R, p1 , q1 , p2 , q2 ∈ D, p2 > p1 > p, q2 > q1 > q. By pq Lemma 2.1(ii), the identities μqp x = μx (cf. Remark 2.1(c)), and (2.11), we have Var (μpx2 q2 − μpx1 q1 )  Var (μpx2 q2 − μpx1 q2 ) + Var (μqx2 p1 − μqx1 p1 )  2(νx ((p1 , p2 )))1/2 + 2(νx ((q1 , q2 )))1/2  2(νx ((p, p2 )))1/2 + 2(νx ((q, q2 )))1/2 → 0,

(p2 , q2 ) → (p, q).

(2.13)

 

By the Cauchy criterion, the limit μpq+ in M(SX ) of the family of measures μpx q exists as x (p , q  ) → (p, q), p , q  ∈ D, p > p, q  > q. Similarly, for all p1 , q1 , p2 , q2 ∈ D such that p2 < p1 < p, q2 < q1 < q Var (μpx2 q2 − μpx1 q1 )  2(νx ((p2 , p1 )))1/2 + 2(νx ((q2 , q1 )))1/2  2(νx ((p2 , p)))1/2 + 2(νx ((q2 , q)))1/2 → 0,

(p2 , q2 ) → (p, q),  

in M(SX ) of measures μpx q as (p , q  ) → (p, q), which implies the existence of the left limit μpq− x   p , q  ∈ D, p < p, q  < q. By Lemma 2.1 (i), Var μpx q  1 which implies Var μpq±  1 as x p → p±, q  → q±. Finally, if pi ∈ D, ζi ∈ C, i = 1, . . . , l, then (cf. Remark 2.1(c)) l 

p pj

ζi ζj μxi

 0.

i,j=1

Passing to the limit as pi → pi ±, we obtain (2.12). Corollary 2.1. Let p, q ∈ R, x ∈ Ω . Then for any Borel set A ⊂ SX pp+ qq+ 1/2 pp− qq− 1/2 |μpq+ , |μpq− . x |(A)  (μx (A)μx (A)) x |(A)  (μx (A)μx (A))

522

(2.14)

Proof. From (2.12) with l = 2, p1 = p, p2 = q it follows that the matrix M=

pp+

μx (A) μpq+ x (A) qq+ μqp+ x (A) μx (A)

is Hermitean and nonnegative definite. In particular, μpp+ (A)μqq+ (A) − |μpq+ (A)|2 = det M  0. By the Young inequality, for all c > 0 c 1 |μpq+ (A)|  (μpp+ (A)μqq+ (A))1/2  μpp+ (A) + μqq+ (A). 2 2c Since

1 c pp+ μ + μqq+ is a nonnegative Borel measure, from the last relation we get 2 2c c 1 |μpq+ |(A)  μpp+ (A) + μqq+ (A) 2 2c

∀c > 0

and, consequently, |μpq+ |(A)  inf

c>0

c 2

μpp+ (A) +

1 qq+ μ (A) = (μpp+ (A)μqq+ (A))1/2 . 2c

The inequality |μpq− |(A)  (μpp− (A)μqq− (A))1/2 is proved in a similar way. Remark 2.2. By the continuity μpq x with respect to the variables p, q ∈ D (which follows from the estimate (2.13)), for all p ∈ D = lim μpq± x

 



lim μpx q = lim μpq in M(SX ). x 

q →q± p →p±

Similarly, for q ∈ D

q →q±



μpq± = lim μxp q x p →p±

in M(SX ).

= μpq If p, q ∈ D, then μpq± x x . pq± The following property connecting the H-measure μ± x = {μx }p,q∈R and the measure-valued function (the Young measure) νx holds.

Lemma 2.3. Let x ∈ Ω , and let [a, b] be the smallest segment containing supp νx . If p0 ∈ (a, b) and S+ , S− ⊂ SX are Borel sets such that μxp0 p0 + (SX \ S+ ) = μxp0 p0 − (SX \ S− ) = 0,

(2.15)

then S+ ∩ S− = ∅. In particular, supp μpx0 p0 + ∩ supp μxp0 p0 − = ∅. 523

Proof. First of all, we note that x ∈ Ω is a Lebesgue point of the function u0 (·, p) for all p in the dense set D. Consequently, the distribution function u0 (x, λ) = νx ((λ, +∞)) of the measure νx is uniquely defined by the relation u0 (x, λ) = sup u0 (x, p). In particular, the p∈D,p>λ

measure νx ∈ M(SX ) is well defined for all x ∈ Ω . Let Borel sets S± ⊂ SX satisfy the condition (2.15). We need to prove that S+ ∩ S− = ∅. Assume the opposite. Let S+ ∩ S− = ∅. We consider the complements C± = SX \ S± . Then SX = C+ ∪ C− , μpx0 p0 + (C+ ) = μxp0 p0 − (C− ) = 0. If p, q ∈ D, p < p0 < q, then, in view of Corollary 2.1 and Remark 2.2, pq pq pq pp qq 1/2 qq 1/2 Var μpq + (μpp x = |μx |(SX )  |μx |(C+ ) + |μx |(C− )  (μx (C+ )μx (C+ )) x (C− )μx (C− )) 1/2 1/2  (μqq + (μpp , x (C+ )) x (C− ))

since Var μpp x  1 for all p ∈ D. The obtained estimate implies p0 p0 + lim Var μpq (C+ ))1/2 + (μpx0 p0 − (C− ))1/2 = 0. x  (μx

lim

p→p0 − q→p0 +

Thus, μpq x → 0 in M(SX ) as p → p0 −, q → p0 +. On the other hand, in view of (2.5), μpq x (SX )

(2.16)



= lim lim

m→∞ r→∞ Rn

F (Φm Ur (·, p))(ξ)F (Φm Ur (·, q))(ξ)dξ



= lim lim

m→∞ r→∞ Rn

Ur (y, p)Ur (y, q)Km (x − y)dy.

(2.17)

We recall that Ur (y, λ) = θ(ur (y) − λ) − u0 (y, λ). Since Ur (·, p)  0 as r → ∞ for all p ∈ D and (θ(ur (y) − p) − 1)θ(ur (y) − q) ≡ 0, we find   lim Ur (y, p)Ur (y, q)Km (x − y)dy = lim (Ur (y, p) − 1)Ur (y, q)Km (x − y)dy r→∞ Rn

r→∞ Rn



= lim

r→∞ Rn

(θ(ur (y) − p) − 1 − u0 (y, p))(θ(ur (y) − q) − u0 (y, q))Km (x − y)dy



= lim

r→∞ Rn

[(1 − θ(ur (y) − p))u0 (y, q) − u0 (y, p)(θ(ur (y) − q) − u0 (y, q))]Km (x − y)dy



(1 − u0 (y, p))u0 (y, q)Km (x − y)dy.

= Rn

Passing to the limit as m → ∞, we get  lim lim Ur (y, p)Ur (y, q)Km (x − y)dy m→∞ r→∞ Rn



= lim

m→∞ Rn

524

(1 − u0 (y, p))u0 (y, q)Km (x − y)dy = (1 − u0 (x, p))u0 (x, q).

We took into account that x is a common Lebesgue point of the functions u0 (y, p) and u0 (y, q). Using (2.16) and (2.17), we find lim μpq x (SX )

0 = lim

p→p0 − q→p0 +

lim (1 − u0 (x, p))u0 (x, q) = νx ((−∞, p0 ))νx ((p0 , +∞)) > 0

= lim

p→p0 − q→p0 +

since a < p0 < b and [a, b] is the smallest segment containing supp νx . We arrive at a contradiction. Thus, S+ ∩ S− = ∅, which is required. Corollary 2.2. supp μpp+ ∩ supp μpp− = ∅ for all p ∈ (a, b). In particular, μpp± = 0 for x x x all p ∈ (a, b). are zero. Indeed, by Lemma 2.2, the representaConversely, for p ∈ [a, b] the measures μpp± x tion (2.5) and the relations (Ur (y, p ))2 = θ(ur (y) − p ) − 2θ(ur (y) − p )u0 (y, p ) + (u0 (y, p ))2  u0 (y, p ) − (u0 (y, p ))2 weakly-∗ in L∞ (Ω) as r → ∞, we have  pp± μx (SX ) = lim lim lim |F (Φm Ur (·, p ))(ξ)|2 dξ  Dp →p± m→∞ r→∞ Rn



=

lim

lim lim

lim 

lim

Dp →p± m→∞ r→∞ Rn



=

Dp →p± m→∞ Rn

(Ur (y, p ))2 Km (x − y)dy

(1 − u0 (y, p ))u0 (y, p )Km (x − y)dy =

lim

Dp →p±

(1 − u0 (x, p ))u0 (x, p ) = 0.

We took into account that lim

(1 − u0 (x, p ))u0 (x, p ) = νx ((−∞, p])νx ((p, +∞)),

lim

(1 − u0 (x, p ))u0 (x, p ) = νx ((−∞, p))νx ([p, +∞)),

Dp →p+ Dp →p−

and both these limit values are zero for p ∈ [a, b]. We note that these relations also imply = 0 for p ∈ / [a, b) and μpp− = 0 for p ∈ / (a, b]. In particular, if p ∈ / (a, b), then at least one μpp+ x x pp± of the measures μx is zero. The following assertion is well known (cf., for example, [2, Lemma 6] ). Lemma 2.4. We assume that the sequence Ur (x) is bounded in L2 (Rn ) ∩ L1 (Rn ) and weakly converges to zero, whereas a(ξ) is a bounded function on Rn such that a(ξ) → 0 as |ξ| → ∞. Then a(ξ)F (Ur )(ξ) → 0 in L2 (Rn ) as r → ∞. We choose x ∈ Ω , p0 ∈ R, |p0 | < M , where M > sup ur ∞ . Let H+ , H− ⊂ Rn be the smallest linear subspaces containing supp μxp0 p0 + , supp μxp0 p0 − respectively. We consider the finite-dimensional Euclidean space G = Rn ⊕ Symn equipped with the inner product Fj = (vj , Aj ) ∈ G, j = 1, 2. F1 · F2 = v1 · v2 + Tr (A1 A 2 ), 525

The Euclidean norm of the vector F = (v, A) ∈ G is given by the standard equality |F | = (F · F )1/2 = (|v|2 + |A|2 )1/2 , where |A|2 = Tr (AA ). Let F (x, λ) = (f (x, λ), Q(x, λ)) ∈ L2loc (Ω, C(R, G)) be the G-valued Carath´eodory vector with components fi (x, λ), Qij (x, λ), i, j = 1, . . . , n. By assumption, ∀M > 0 F (x, ·) M,∞ = max |F (x, λ)|G  αM (x) ∈ L2loc (Ω). |λ|M

(2.18)

Since the space C(R, G) is separable with respect to the standard locally convex topology generated by seminorms · M,∞ , taking into account the Pettis theorem [15, Chapter 3], we find that the mapping x → F (x, ·) ∈ C(R, G) is strongly measurable. In particular (cf. [15, Chapter 3]), the set ΩF of common Lebesgue points of the mappings F (x, ·) and |F (x, ·)|2 has full measure. For x ∈ ΩF  lim

m→∞

Km (x − y) F (x, ·) − F (y, ·) 2M,∞ dy = 0 ∀M > 0

(2.19)

which easily follows from the equality |F (x, ·) − F (y, ·)|2G = |F (x, ·)|2G − |F (y, ·)|2G − 2F (y, ·) · (F (x, ·) − F (y, ·)). It is clear that each point x ∈ ΩF is a Lebesgue point of all G-valued functions x → F (x, λ), λ ∈ R. As above, we set γxr (λ) = δ(λ − ur (x)) − νx (λ),  Ir (fp )(y) = fp (y, λ)dγyr (λ) ∈ L2loc (Ω, Rn ),  Ir (Qp )(y) =

Qp (y, λ)dγyr (λ) ∈ L2loc (Ω, Symn ),

where fp (y, λ) = f (y, max(p, λ)) and Qp (y, λ) = Q(y, max(p, λ)). Denote G± = { (v, A) ∈ G | v · ξ = Aξ · ξ = 0 ∀ ξ ∈ H± }. Proposition 2.3. Assume that x ∈ Ω ∩ ΩF , F (x, λ) ∈ G± and ρ(ξ) ∈ C ∞ (Rn ) is a function  2 + |ξ|4  2. Then for all  2 + |ξ|4  1, ρ(ξ) = 1 for |ξ| such that 0  ρ(ξ)  1 and ρ(ξ) = 0 for |ξ| ψ(ξ) ∈ C(SX )   iξ · F (Φm Ir (fp ))(ξ) + F (Φm Ir (Qp ))(ξ)ξ · ξ  lim lim lim   2 + |ξ|4 )1/2 Dp→p0 ± m→∞ r→∞ (|ξ| Rn

× where i2 = −1, Φm = Φm (x − y) =

 

 F (Φm Urp )(ξ)ρ(ξ)ψ(πX (ξ))dξ  

= 0,

Km (x − y) and Ir (fp ), Ir (Qp ) are functions of y ∈ Ω.

Proof. First of all, we note that for p  M  |(Ir (fp )(y), Ir (Qp )(y))|G  |F (y, max(p, λ))|G d|γyr |(λ)  2αM (y). 526

(2.20)

We recall that M > sup ur ∞ . It is easy to prove that the function a(ξ) =

ρ(ξ)|ξ| 2  (|ξ| + |ξ|4 )1/2

satisfies the assumptions of Lemma 2.4 and the sequence Φm Ir (fp )(y), r ∈ N, is bounded in L2 (Rn ) and in L1 (Rn ) (since supp Φm is compact) and weakly converges to zero (in view of the weak convergence ur → νx as r → ∞ in the sense of (2.1)). Therefore, in view of Lemma 2.4, for every m ∈ N  lim

r→∞ Rn

ξ · F (Φm Ir (fp ))(ξ) F (Φm Urp )(ξ)ρ(ξ)ψ(πX (ξ))dξ = 0.  2 + |ξ|4 )1/2 (|ξ|

(2.21)

By (2.21),   iξ · F (Φm Ir (fp ))(ξ) + F (Φm Ir (Qp ))(ξ)ξ · ξ  lim  F (Φm Urp )(ξ)ρ(ξ)ψ(πX (ξ))dξ 2 4 1/2 r→∞   (|ξ| + |ξ| ) Rn

 − Rn

  iξ · F (Φm Ir (fp ))(ξ) + F (Φm Ir (Qp ))(ξ)ξ · ξ  p F (Φm Ur )(ξ)ρ(ξ)ψ(πX (ξ))dξ  = 0. 1/2 2 4   (|ξ| + |ξ| )

(2.22)

 p (λ) = Qp (x, λ), Assume that fp (λ) = fp (x, λ), Q Ir (fp )(y) =



 p )(y) = I r (Q



fp (λ)dγyr (λ),  p (λ)dγyr (λ). Q

Then  p ))|G |(Ir (fp ) − Ir (fp ), Ir (Qp ) − Ir (Q   |F (y, max(p, λ)) − F (x, max(p, λ))|G d|γyr |(λ)  2 F (y, ·) − F (x, ·) M,∞ . From this estimate, the Plancherel identity, and (2.19) it follows that    p )))(ξ)ξ · ξ iξ · F (Φm (Ir (fp ) − Ir (fp )))(ξ) + F (Φm (Ir (Qp ) − Ir (Q  lim lim  m→∞ r→∞   2 + |ξ|4 )1/2 (|ξ| Rn

×

 

 F (Φm Urp )(ξ)ρ(ξ)ψ(πX (ξ))dξ  

  2 ψ ∞ lim

m→∞

1/2 F (y, ·) − F (x, ·) 2M,∞ Km (x − y)dy

=0

Rn

527

 ξ ⊗ ξ)|G = (|ξ|  2 + |ξ|4 )1/2 ). By this relation and (2.22), it (we also took into account that |(ξ, suffices to prove that     p ))(ξ)ξ · ξ iξ · F (Φm Ir (fp ))(ξ) + F (Ir (Φm Q  lim lim  lim  2 + |ξ|4 )1/2 Dp→p0 ± m→∞ r→∞  (|ξ| Rn

(2.23)    p × F (Φm Ur )(ξ)ρ(ξ)ψ(πX (ξ))dξ  = 0. 

(2.24)

 The vector-valued function F (λ) = (f(λ), Q(λ)) = (f (x, λ), Q(x, λ)) is continuous and is independent of y. Therefore, for any ε > 0 there exists a vector-valued function F0 (λ) = (f0 (λ), Q0 (λ)) of the form k  F0 (λ) = vj θ(λ − pj ), j=1

where vj = (fj , Qj ) ∈ G± and pj ∈ D \ {p0 }, such that F − F0 L∞ ([−M,M ])  ε. Let f0p (λ) = f0 (max(p, λ)), Q0p (λ) = Q0 (max(p, λ)), F0p (λ) = (f0 (max(p, λ)), Q0 (max(p, λ))) = F0 (max(p, λ)). Then it is obvious that F0p (λ) =

k 

vj θ(max(p, λ) − pj ),

j=1

Fp − F0p L∞ ([−M,M ])  F − F0 L∞ ([−M,M ])  ε. Using the Plancherel identity and the relation  p ) − Ir (Q0p )))(ξ)ξ · ξ|2 |iξ · F (Φm (Ir (fp ) − Ir (f0p )))(ξ) + F (Φm (Ir (Q  2 + |ξ|4 ),  p ) − Ir (Q0p )))(ξ)|2 )(|ξ|  (|F (Φm (Ir (fp ) − Ir (f0p )))(ξ)|2 + |F (Φm (Ir (Q we find

    p ))(ξ)ξ · ξ iξ · F (Φm Ir (fp ))(ξ) + F (Φm Ir (Q  F (Φm Urp )(ξ)ρ(ξ)ψ(πX (ξ))dξ  1/2 2 4   (|ξ| + |ξ| ) Rn

 − Rn

  iξ · F (Φm Ir (f0p ))(ξ) + F (Φm Ir (Q0p ))(ξ)ξ · ξ  F (Φm Urp )(ξ)ρ(ξ)ψ(πX (ξ))dξ   2 + |ξ|4 )1/2  (|ξ|

 ψ ∞ Φm Ir (Fp − F0p ) 2  2ε ψ ∞ Φm 2 = 2ε ψ ∞ . 528

(2.25)

Taking into account that

 dγyr (λ) = 0,

we have Ir (F0p )(y) =

  k

vj θ(max(p, λ) − pj )dγyr (λ) =

j=1



p

vj Ur j (y),

pj p

and (2.8) implies the limit relation   ξ · F (Φm Ir (f0p ))(ξ) + F (Φm Ir (Q0p ))(ξ)ξ · ξ F (Φm Urp )(ξ)ρ(ξ)ψ(πX (ξ))dξ lim lim 1/2 2 4 m→∞ r→∞  (|ξ| + |ξ| ) n =



R

p p

μxj , (ifj · ξ + Qj ξ · ξ)ψ(ξ).

(2.26)

pj p

Here, we take into account that ρ(ξ) = 1 for large |ξ| (cf. Lemma 2.4). From (2.26), Remark 2.2, and the conditions pj = p0 , j = 1, . . . , k, it follows that   ξ · F (Φm Ir (f0p ))(ξ) + F (Φm Ir (Q0p ))(ξ)ξ · ξ lim lim lim p→p0 ± m→∞ r→∞  2 + |ξ|4 )1/2 (|ξ| n R

× F (Φm Urp )(ξ)ρ(ξ)ψ(πX (ξ))dξ =



p p0 ±

μxj

, (ifj · ξ + Qj ξ · ξ)ψ(ξ) = 0

(2.27)

pj >p0 p p ±

since supp μxj 0 ⊂ supp μxp0 p0 ± ⊂ H± in view of Corollary 2.1, whereas vj = (fj , Qj ) ∈ G± . This relation, together with (2.25), leads to the following:    p ))(ξ)ξ · ξ iξ · F (Φm Ir (fp ))(ξ) + F (Φm Ir (Q  lim lim lim  p→p0 ± m→∞ r→∞  2 + |ξ|4 )1/2 (|ξ| n R

   × F (Φm Urp )(ξ)ρ(ξ)ψ(πX (ξ))dξ   2ε ψ ∞ . 

Since ε > 0 is arbitrary, we obtain (2.24). The required assertion is proved. Proposition 2.3 is proved in a similar way if fp (y, λ) = f (y, min(p, λ)),

Qp (y, λ) = Q(y, min(p, λ)).

Assume that H ⊂ Rn is some of the spaces H+ , H− and G0 ⊂ G is the corresponding  = P1 (H) and space G+ , G− . Denote by P , P the orthogonal projections onto the spaces H H = P2 (H) respectively. It is easy to verify that the linear operator (v, A) → (P v, P AP ) is the operator of orthogonal projection from G onto G⊥ 0. The following two lemmas are proved in [2, 3]. We outline their proof. Lemma 2.5. Let {ξk | k = 1, . . . , l} ⊂ H be a basis for H. Then there exists a positive constant C such that for any v ∈ Rn , Q ∈ Symn |(v1 , Q1 )|G  C max |iv · ξk + Qξ k · ξ k |, k=1,...,l

(2.28)

where v1 = Pv, Q1 = P QP . 529

 Proof. It is clear that (v1 , Q1 ) ∈ G⊥ 0 and p(v1 , Q1 ) = max |iv · ξk +Qξ k ·ξ k | is the seminorm k=1,...,l

in If p(v1 , Q1 ) = 0, then v · ξ = Qξ · ξ = 0 for all ξ ∈ H. Consequently, v1 · ξ = v · Pξ = 0, Q1 ξ · ξ = QP ξ · P ξ = 0 for all ξ ∈ Rn . Therefore, (v1 , Q1 ) = 0 and hence p is a norm. Since norms in the finite-dimensional space G⊥ 0 are equivalent, we obtain (2.28). G⊥ 0.

Denote μ = μpx0 p0 + if H = H+ and μ = μxp0 p0 − if H = H− . We note that μ ∈ M(SX ) is a nonnegative finite Borel measure on SX . Lemma 2.6. There exist functions ψk (ξ) ∈ C(SX ), k = 1, . . . , l = dim H, and a constant C > 0 such that, in the notation of Lemma 2.5, for all v ∈ Rn , Q ∈ Symn such that Q  0 |(v1 , Q1 )|G  C max | μ, (iv · ξ + Qξ · ξ)ψk (ξ)|. k=1,...,l

(2.29)

Proof. Since H is the smallest linear subspace containing supp μ, we have 

 ψ(ξ)ξdμ(ξ) | ψ(ξ) ∈ C(SX )

H= SX

since the orthogonal complements of these linear spaces coincide. Therefore, we can choose ψk (ξ) ∈ C(SX ), k = 1, . . . , l, such that  ψk (ξ)  0,

ψk (ξ)dμ(ξ) = 1 ∀k = 1, . . . , l, SX

and the family

 ξk =

ψk (ξ)ξdμ(ξ),

k = 1, . . . , l,

SX

is a basis for the space H. It is clear that for all k = 1, . . . , l v · ξk =



 k (ξ)dμ(ξ). v · ξψ

(2.30)

SX

By condition, Q  0 and, consequently, the quadratic form Qξ · ξ is convex. By the Jensen inequality, for all k = 1, . . . , l  Qξ k · ξ k = Q SX

 ψk (ξ)ξdμ(ξ) ·

 ψk (ξ)ξdμ(ξ) 

SX

Qξ · ξψk (ξ)dμ(ξ).

(2.31)

SX

By Lemma 2.5 and the fact that |ia + b| is increasing with respect to the variable b  0, from (2.30) and (2.31) we obtain (2.29).

530

3

Localization Principle and Strong Precompactness Property We define an anisotropic Sobolev space Wd−1,−2 of distributions u(x) such that  2 + |ξ|4 )−1/2 F (u)(ξ) = F (v)(ξ), (1 + |ξ|

v = v(x) ∈ Ld (Rn ).

This space is a Banach space equipped with the norm u = v d . As is known (cf., for example, [2]), the following continuous embeddings take place: Wd−1 ⊂ Wd−1,−2 ⊂ Wd−2 .

(3.1)

−1,−2 (Ω) of distributions u(x) such that uf (x) belongs to We introduce the local space Wd,loc

−1,−2 Wd−1,−2 for all f (x) ∈ C0∞ (Ω). The space Wd,loc (Ω) is a locally convex space with topology generated by the family of seminorms u → uf W −1,−2 , f (x) ∈ C0∞ (Ω). In a similar way, we d

−1,−2 −1 −2 −1 −2 introduce locally convex spaces Wd,loc (Ω) and Wd,loc (Ω). By (3.1), Wd,loc ⊂ Wd,loc ⊂ Wd,loc , and these embeddings are continuous. We consider a bounded sequence uk ∈ L∞ (Ω) that weakly-∗ converges to u(x) and assume that for some d > 1 and any a, b ∈ R, a < b, the sequence of distributions

div ϕ(x, sa,b (uk (x))) − D 2 · B(x, sa,b (uk (x)))

−1,−2 is precompact in Wd,loc (Ω).

(3.2)

We recall that sa,b (u) = max(a, min(u, b)) is a cut-off function. We choose a subsequence ur (x) = ukr (x) that weakly converges to a bounded measurevalued function νx in the sense of the relation (2.1) and such that the ultraparabolic H-measure μpq = μpq x dx, p, q ∈ D, is well defined. According to (2.5) and Lemma 2.2, we introduce the  measure μpq± x , x ∈ Ω , p, q ∈ R. We define the set of full measure Ωϕ,B of common Lebesgue points of the mappings x → F (x, ·) = (ϕ(x, ·), B(x, ·)) ∈ C(R, G), x → |F (x, ·)|2G ∈ C(R). We fix x in the set of full measure Ω = Ω ∩ Ωϕ,B . Under the made assumptions, the following localization principle holds. Theorem 3.1. Let p0 ∈ R, and let H± be the linear spans of supp μpx0 p0 ± . Then there exists δ > 0 such that (3.3) (ϕ(x, λ) − ϕ(x, p0 )) · ξ = (B(x, λ) − B(x, p0 ))ξ · ξ = 0 for all ξ ∈ H+ , λ ∈ [p0 , p0 + δ] and all ξ ∈ H− , λ ∈ [p0 − δ, p0 ]. Proof. By (3.2) and weak convergence ur (y) → νy for all q, p ∈ D, q > p > p0 ,   −1,−2 r r 2 (Ω) Lp (y) = divy ϕ(y, sp,q (λ))dγy (λ) − D · B(y, sp,q (λ))dγyr (λ) → 0 in Wd,loc

(3.4)

as r → ∞. We recall that γyr (λ) = δ(λ − ur (y)) − νy (λ). We fix q ∈ D and for p ∈ (p0 , q) denote fp (y, λ) = ϕ(y, sp,q (λ)) and Qp (y, λ) = B(y, sp,q (λ)). By (3.4), div Ir (fp )(y) − D 2 · −1,−2 Ir (Qp )(y) → 0 in Wd,loc (Ω) as r → ∞. Let Φ(y) ∈ C0∞ (Ω). We introduce the vector Jr (y, p) = 2Ir (Qp )(y)∇Φ(y) with components (Jr )k (y, p) = 2

n 

Φyl (Ir (Qp )(y))kl (y),

k = 1, . . . , n,

l=1

531

and consider the sequence Lr = divy (ΦIr (fp )(y) + Jr (y, p)) − D 2 · (ΦIr (Qp )(y)) = Φ(y)Lpr (y) + Ir (fp )(y) · ∇Φ(y) + D 2 Φ(y) · Ir (Qp )(y). The last two terms on the right-hand side of this identity weakly converge to zero in L2 (Rn ) as r → ∞ and, consequently, strongly converge in Wd−1 ⊂ Wd−1,−2 (we can assume that d  2 ). Then, as it follows from (3.4), Lr → 0 in Wd−1,−2 as r → ∞. Applying the Fourier transform to  2 + |ξ|4 )−1/2 , where the function ρ(ξ) is introduced this relation and then multiplying by ρ(ξ)(|ξ| in Proposition 2.3, we get ρ(ξ)(2πiξ · F (ΦIr (fp ) + Jr (·, p))(ξ) + 4π 2 F (ΦIr (Qp ))(ξ)ξ · ξ) = F (lr )(ξ),  2 + |ξ|4 )1/2 (|ξ|

(3.5)

2 + where lr → 0 in Ld (Rn ) as r → ∞. Here, we took into account that the function ρ(ξ)(1 + |ξ|  2 + |ξ|4 )−1/2 is a Fourier multiplier on Ld (cf. [2, Proposition 6]). |ξ|4 )1/2 (|ξ|  Let ψ(ξ) ∈ C n (SX ). Then F (ΦUrp )(ξ)ψ(πX (ξ)) = F (hr ), where hr is bounded in Ld (Rn ),  d = d/(d − 1) since ψ(πX (ξ)) is a Fourier multiplier in Ld (again, in view of [2, Proposition 6]). From this relation and (3.5) we find  2πiξ · F (ΦIr (fp ) + Jr (·, p))(ξ) + 4π 2 F (ΦIr (Qp ))(ξ)ξ · ξ  2 + |ξ|4 )1/2 (|ξ| Rn

 ×

F (ΦUrp )(ξ)ρ(ξ)ψ(πX (ξ))dξ

lr (x)hr (x)dx → 0,

=

r → ∞.

(3.6)

Rn

We note that the sequences Φ(y)Ir (fp )(y) and Jr (·, p) are bounded in L2 ∩ L1 and weakly converge to zero. By Lemma 2.4, ρ(ξ)ξ · F (ΦIr (fp ) + Jr (·, p))(ξ) → 0 in L2 (Rn ) as r → ∞.  2 + |ξ|4 )1/2 (|ξ| Furthermore, ξ · F (Jr (·, p))(ξ) = 2 F (Ir (Qp ))(ξ)ξ =





e−2πiξ·y Ir (Qp )ξ · ∇y Φ(y)dy = 0,

(3.7)

(3.8)

Rn

 =0 e−2πiξ·y Ir (Qp )ξdy

(3.9)

Rn

since for all ξ ∈ X Ir (Qp )(y)ξ =



 r (λ) = B(y, sp,q (λ))ξdγ y



 r (λ) = 0. (B(y, sp,q (λ)) − B(y, 0))ξdγ y

Taking into account the relations (3.7), (3.8), (3.9) and the boundedness of the sequence F (ΦUrp )(ξ) in L2 (Rn ), from (3.6) we get  2πiξ · F (ΦIr (fp ))(ξ) + 4π 2 F (ΦIr (Qp ))(ξ)ξ · ξ F (ΦUrp )(ξ)ρ(ξ)ψ(πX (ξ))dξ → 0 (3.10) 1/2 2 4  (|ξ| + |ξ| ) Rn

532

as r → ∞. Assume that f1p (y, λ) = Pfp (y, λ),

Q1p (y, λ) = P Qp (y, λ)P ,

f2p (y, λ) = fp (y, λ) − f1p (y, λ),

Q2p (y, λ) = Qp (y, λ) − Q1p (y, λ)

so that (f1p (y, λ), Q1p (y, λ)) ∈ G⊥ + and (f2p (y, λ), Q2p (y, λ)) ∈ G+ . By Proposition 2.3 applied to f (x, λ) = ϕ(x, min(q, λ)) − P ϕ(x, min(q, λ)), Q(x, λ) = B(x, min(q, λ)) − P B(x, min(q, λ))P , we get

  2πiξ · F (Φm Ir (f2p ))(ξ) + 4π 2 F (Φm Ir (Q2p ))(ξ)ξ · ξ  lim lim lim  p→p0 + m→∞ r→∞   2 + |ξ|4 )1/2 (|ξ| Rn

   ×F (Φm Urp )(ξ)ρ(ξ)ψ(πX (ξ))dξ  = 0.  Hence from (3.10) it follows that   2πiξ · F (Φm Ir (f1p ))(ξ) + 4π 2 F (Φm Ir (Q1p ))(ξ)ξ · ξ  lim lim lim  p→p0 + m→∞ r→∞   2 + |ξ|4 )1/2 (|ξ| n R

   × F (Φm Urp )(ξ)ρ(ξ)ψ(πX (ξ))dξ  = 0. 

(3.11)

 sp,q (λ)), where ϕ(y,  sp,q (λ)) and Q1p (y, λ) = B(y,  λ) = P ϕ(y, λ) We note that f1p (y, λ) = ϕ(y,  and B(y, λ) = P B(y, λ)P . It is easy to compute that  p) + (ϕ(y,  q) − ϕ(y,  p))θ(λ − p) − (ϕ(y,  q) − ϕ(y,  λ))χ(λ), ϕ(y,  sp,q (λ)) = ϕ(y,  sp,q (λ)) = B(y,  p) + (B(y,  q) − B(y,  p))θ(λ − p) − (B(y,  q) − B(y,  λ))χ(λ), B(y, where χ(λ) = θ(λ − p) − θ(λ − q). Therefore,



Ir (f1p )(y) = Urp (y)(ϕ(y,  q) − ϕ(y,  p)) − Ir (Q1p )(y) =

 q) Urp (y)(B(y,

 p)) − − B(y,

(ϕ(y,  q) − ϕ(y,  λ))χ(λ)dγyr (λ), 

 q) − B(y,  λ))χ(λ)dγ r (λ). (B(y, y

(3.12) (3.13)

By (2.8) and Lemma 2.2, we have   2  + |ξ|4 )−1/2 (2πiξ · F (Φm U p (ϕ(·,  p))(ξ) |ξ| lim lim lim r  q) − ϕ(·, p→p0 + m→∞ r→∞ Rn

  q) − B(·,  p)))(ξ)ξ · ξ F (Φm Urp )(ξ)ρ(ξ)ψ(πX (ξ))dξ + 4π 2 F (Φm Urp (B(·,  q) − B(x,  p0 ))ξ · ξ)ψ(ξ). = μxp0 p0 + , (2πi(ϕ(x,  q) − ϕ(x,  p0 )) · ξ + 4π 2 (B(x,

(3.14) 533

From (3.11) and (3.12)–(3.14) it follows that  q) − B(x,  p0 ))ξ · ξ)ψ(ξ)| | μpx0 p0 + , (2πi(ϕ(x,  q) − ϕ(x,  p0 )) · ξ + 4π 2 (B(x,   iξ · F (Φm Vrp,q )(ξ) + F (Φm Hrp,q )(ξ)ξ · ξ   lim lim lim  p→p0 + m→∞ r→∞   2 + |ξ|4 )1/2 (|ξ| Rn

   × F (Φm Urp )(ξ)ρ(ξ)ψ(πX (ξ))dξ , 

(3.15)

where  Vrp,q (y) Hrp,q (y)

(ϕ(y,  q) − ϕ(y,  λ))χ(λ)dγyr (λ) ∈ Rn ,

= 2π = 4π

2



 q) − B(y,  λ))χ(λ)dγ r (λ) ∈ Symn . (B(y, y

Since C n (SX ) is dense in C(SX ), it is clear that (3.15) holds for ψ(ξ) ∈ C(SX ). It is easy to see that    ρ(ξ)(iξ · F (Φ V p,q )(ξ) + F (Φ H p,q )(ξ)ξ · ξ)    r r m m    2 + |ξ|4 )1/2   (|ξ| 

( F (Φm Vrp,q ) 22

+

F (Φm Hrp,q ) 22 )1/2

2

= ( Φm Vrp,q 22 + Φm Hrp,q 22 )1/2 .

(3.16)

Since |Urp |  1, using the Bunyakowsky inequality and (3.16), from (3.15) we find  q) − B(x,  p0 ))ξ · ξ)ψ(ξ)| | μpx0 p0 + , (2πi(ϕ(x,  q) − ϕ(x,  p0 )) · ξ + 4π 2 (B(x,  ψ ∞ lim

lim lim ( Φm Vrp,q 22 + Φm Hrp,q 22 )1/2 .

(3.17)

p→p0 + m→∞ r→∞

 λ)) ∈ G⊥   Denote F(x, λ) = (2π ϕ(x,  λ), 4π 2 B(x, + and set Mq (y) = max |F (y, q) − F (y, λ)|G . λ∈[p0 ,q]

We note that

 (Vrp,q (y), Hrp,q (y)) =

(F(y, q) − F (y, λ))χ(λ)dγyr (λ).

From this identity we obtain the estimate |(Vrp,q (y), Hrp,q (y))|2G

 

 Mq (y)

|χ(λ)|d|γyr |(λ)

2

 (Mq (y)(θ(ur (y) − p) − θ(ur (y) − q) + u0 (y, p) − u0 (y, q)))2  2(Mq (y))2 (θ(ur (y) − p) − θ(ur (y) − q) + u0 (y, p) − u0 (y, q))  4(Mq (y))2 (u0 (y, p) − u0 (y, q)), 534

r → ∞,

which implies lim

lim lim ( Φm Vrp,q 22 + Φm Hrp,q 22 )1/2

p→p0 + m→∞ r→∞

  2 lim

m→∞

1/2 2

Km (x − y)(Mq (y)) (u0 (y, p0 ) − u0 (y, q))dy Rn

= 2Mq (x)(u0 (x, p0 ) − u0 (x, q))1/2 ,

(3.18)

where we used the property that x ∈ Ω is a common Lebesgue point of the functions (Mq (y))2 ∈ L1loc (Ω), u0 (y, p) − u0 (y, q) ∈ L∞ (Ω) for all p, q ∈ D. By (3.18) in (3.17),  q) − B(x,  p0 ))ξ · ξ)ψ(ξ)|  q) − ϕ(x,  p0 )) · ξ + 4π 2 (B(x, | μpx0 p0 + , (2πi(ϕ(x,  2 ψ ∞ Mq (x)(u0 (x, p0 ) − u0 (x, q))1/2 .

(3.19)

By Lemma 2.6, it is possible to choose functions ψk (ξ) ∈ C(SX ), k = 1, . . . , l, such that for some positive constants C  q) − ϕ(x,  p0 )) · ξ |F(x, q) − F (x, p0 )|G  C max | μxp0 p0 + , (2πi(ϕ(x, k=1,...,l

 q) − B(x,  p0 ))ξ · ξ)ψk (ξ)|. + 4π 2 (B(x, By (3.19),

|F (x, q) − F(x, p0 )|G  cMq (x)ω(q),

(3.20)

where c is a positive constant and ω(q) = (u0 (x, p0 ) − u0 (x, q))1/2 → 0 as q → p0 +. We choose δ > 0 such that 2cω(p0 + δ)  ε < 1. By (3.20), for all q ∈ [p0 , p0 + δ] ∩ D ε max |F (x, q) − F (x, λ)|G , |F(x, q) − F (x, p0 )|G  2 λ∈[p0 ,q]

(3.21)

and since the vector-valued function F (x, q) is continuous with respect to q and D is everywhere dense, the estimate (3.21) holds for all q ∈ [p0 , p0 + δ]. By this estimate, it is easy to see that F (x, q) = F (x, p0 ) for all q ∈ [p0 , p0 + δ]. Indeed, assume that for q  ∈ [p0 , p0 + δ] |F(x, q  ) − F (x, p0 )|G =

max

λ∈[p0 ,p0 +δ]

|F(x, λ) − F (x, p0 )|G .

Then for λ ∈ [p0 , q  ] |F (x, q  ) − F (x, λ)|G  |F (x, λ) − F (x, p0 )|G + |F (x, q  ) − F(x, p0 )|G  2|F(x, q  ) − F (x, p0 )|G , max |F (x, q  ) − F (x, λ)|G  2|F (x, q  ) − F (x, p0 )|G .

λ∈[p0 ,q  ]

From (3.21) with q = q  we find |F (x, q  ) − F (x, p0 )|G  ε|F (x, q  ) − F (x, p0 )|G . 535

Since ε < 1, from this relation we get |F (x, q  ) − F(x, p0 )|G =

max

λ∈[p0 ,p0 +δ]

|F (x, λ) − F (x, p0 )|G = 0.

Thus, F (x, λ) = F(x, p0 ) for all λ ∈ [p0 , p0 + δ], which is equivalent to the identity (3.3) for all ξ ∈ H+ , λ ∈ [p0 , p0 + δ]. The case ξ ∈ H− , λ ∈ [p0 − δ, p0 ] is treated in a similar way. Namely, we choose p, q ∈ D such that q < p < p0 and argue as above (with obvious technical modifications). Thus, we obtain the following relation similar to (3.19):  q) − B(x,  p0 ))ξ · ξ)ψ(ξ)|  q) − ϕ(x,  p0 )) · ξ + 4π 2 (B(x, | μpx0 p0 − , (2πi(ϕ(x,  2 ψ ∞ Mq (x)(u0 (x, q) − u0 (x, p0 ))1/2 ,

(3.22)

 λ) = P B(y, λ)P , but P and P are the orthogonal projections where ϕ(y,  λ) = Pϕ(y, λ) and B(y, on the subspaces P1 (H− ) and P2 (H− ) respectively. From (3.22) we obtain (3.20) with Mq (x) = max |F(y, q) − F (y, λ)|G , λ∈[q,p0 ]

1/2 ω(q) = (u0 (x, q) − u− = (νx ((q, p0 )))1/2 . 0 (x, p0 ))

Choosing δ > 0 such that 2cω(p0 − δ)  ε < 1 and arguing as above, we find F (x, λ) = F (x, p0 ) for all λ ∈ [p0 −δ, p0 ], which is equivalent to the validity of (3.3) for all ξ ∈ H− , λ ∈ [p0 −δ, p0 ]. Now, we are ready to prove the main result of this paper. Proof of Theorem 1.1. Assume that x ∈ Ω and [a, b] is the smallest segment containing supp νx . We note that, in the case of a < b,  u(x) = λdνx (λ) ∈ (a, b) ∩ and, as follows from Theorem 3.1 and Corollary 2.2, there is a nonzero vector ξ ∈ supp μpp+ x supp μpp− x , where p = u(x) such that (ϕ(x, λ) − ϕ(x, p0 )) · ξ = (B(x, λ) − B(x, p0 ))ξ · ξ = 0 in some neighborhood of p = u(x); here, ξ + ξ = ξ = 0. But this fact contradicts the nondegeneracy condition in Theorem 1.1, which means that a = b = u(x) and, consequently, νx (λ) = δ(λ − u(x)) for all x ∈ Ω . Since Ω ⊂ Ω is a set of full measure, the measure-valued function νx is regular. By Theorem 2.1, the sequence ur strongly converges to u(x). Since the limit function u(x) is independent of the choice of ur (with the above properties), for the original sequence we have uk → u as k → ∞ in L1loc (Ω). Remark 3.1. Since there are cut-off functions in the assumption (3.2), the boundedness condition on the sequence uk can be omitted. We can assume that uk , k ∈ N, is a sequence of measurable functions such that p(uk )  p(u) as k → ∞, where p(u) is a continuous and strictly increasing mapping from the two-point compactification [−∞, +∞] of the line R to the segment [−1, 1] (for example, p(u) = π2 arctan u). Here, u(x) ∈ [−∞, +∞] can take infinite 536

values. It is easy to show that, under the assumptions of Theorem 1.1 (imposed only on finite u(x)) p(uk ) → p(u) strongly (in L1loc (Ω)) as k → ∞. We also note that the condition (3.2) is always valid for sequences of entropy (in the sense of Kruzhkov [16]) solutions (more general, quasisolutions) to the equation (cf. [2, 3]) div ϕ(x, u) − D 2 · B(x, u) = ψ(x, u).

Acknowledgments The work is supported by the Ministry of Education and Science of the Russian Federation (project No. 1.445.2016/1.4) and the Russian Foundation for Basic Research (project No. 1801-00258-a).

References 1.

E. Yu. Panov, “On a condition of strong precompactness and the decay of periodic entropy solutions to scalar conservation laws,” Netw. Heterog. Media 11, No. 2, 349–367 (2016).

2.

E. Yu. Panov, “Ultra-parabolic equations with rough coefficients. Entropy solutions and strong precompactness property,” J. Math. Sci., New York 159, No. 2, 180–228 (2009).

3.

H. Holden, K. H. Karlsen, D. Mitrovi´c, E. Yu. Panov, “Strong compactness of approximate solutions to degenerate elliptic-hyperbolic equations with discontinuous flux function,” Acta Math. Sci., Ser. B, Engl. Ed. 29, No. 6, 1573–1612 (2009).

4.

B. Dacorogna, Weak Continuity and Weak Lower Semicontinuity of Nonlinear Functionals, Springer, Berlin (1982).

5.

R. J. DiPerna, “Measure-valued solutions to conservation laws,” Arch. Ration. Mech. Anal. 88, 207–225 (1985).

6.

L. Tartar, “Compensated compactness and applications to partial differential equations,” Res. Notes Math. 39, 136–212 (1979).

7.

E. Yu. Panov, “Ultra-parabolic H-measures and compensated compactness,” Ann. Inst. Henri Poincar´e, Anal. Non Lineaire 28, No. 1, 47–62 (2011).

8.

L. Tartar, “H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations,” Proc. R. Soc. Edinb., Sect. A 115, No. 3-4, 193–230 (1990).

9.

P. G´erard, “Microlocal defect measures,” Commun. Partial Differ. Equations 16, No. 11, 1761–1794 (1991).

10.

N. Antoni´c and M. Lazar, “H-measures and variants applied to parabolic equations,” J. Math. Anal. Appl. 343, 207–225 (2008).

11.

E. Yu. Panov, “On sequences of measure-valued solutions of a first-order quasilinear equation,” Russ. Acad. Sci., Sb., Math. 81, No. 1, 211–227 (1995).

12.

E. Yu. Panov, “On strong precompactness of bounded sets of measure-valued solutions of a first order quasilinear equation,” Sb. Math. 186, No. 5, 729–740 (1995). 537

13.

E. Yu. Panov, “Property of strong precompactness for bounded sets of measure-valued solutions of a first-order quasilinear equation,” Sb. Math. 190, No. 3, 427–446 (1999).

14.

E. Yu. Panov, “Existence and strong pre-compactness properties for entropy solutions of a first-order quasilinear equation with discontinuous flux,” Arch. Rational Mech. Anal. 195, No. 2, 643–673 (2010).

15.

E. Hille and R. S. Phillips, Functional Analysis and Semigroups, Am. Math. Soc., Providence, RI (1957).

16.

S. N. Kruzhkov, “First order quasilinear equations in several independent variables,” Math. USSR, Sb. 10, 217–243 (1970).

Submitted on December 14, 2017

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