STRONG COMPACTNESS OF APPROXIMATE SOLUTIONS TO DEGENERATE ELLIPTIC-HYPERBOLIC EQUATIONS WITH DISCONTINUOUS FLUX FUNCTION

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1 STRONG COMPACTNESS OF APPROXIMATE SOLUTIONS TO DEGENERATE ELLIPTIC-HYPERBOLIC EQUATIONS WITH DISCONTINUOUS FLUX FUNCTION HELGE HOLDEN, KENNETH H. KARLSEN, DARKO MITROVIC, AND EVGUENI YU. PANOV Abstract. Under a non-degeneracy condition on the nonlinearities we show that sequences of approximate entropy solutions of mixed elliptic-hyperbolic equations are strongly precompact in the general case of a Caratheodory flux vector. The proofs are based on deriving localization principles for H-measures associated to sequences of measure-valued functions. This main result implies existence of solutions to degenerate parabolic convection-diffusion equations with discontinuous flux. Moreover, it provides a framework in which one can prove convergence of various types of approximate solutions, such as those generated by the vanishing viscosity method and numerical schemes. 1. Introduction Let be an open subset of. In the domain we consider the quasilinear elliptic equation div x ϕ(x, u) D 2 B(u) + ψ(x, u) = 0, (1) where D 2 B(u) = x 2 ix j b ij (u) (we use the conventional rule of summation over repeated indexes), B(u) = {b ij (u)} n i,j=1 is a symmetric matrix. We shall assume that this matrix is only continuous: b ij (u) C(R), i, j = 1,..., n. In this case the ellipticity of (1) is understood in the following sense B(u 1 ) B(u 2 ) 0, u 1, u 2 R, u 1 > u 2, (2) that is, for all ξ we have (B(u 1 ) B(u 2 ))ξ ξ 0 (here u v denotes the scalar product of vectors u, v ). We suppose that ϕ(x, u) = (ϕ 1 (x, u),..., ϕ n (x, u)) is a Caratheodory vector (i.e., it is continuous with respect to u and measurable with respect to x) such that the functions α M (x) = max ϕ(x, u) u M L2 loc() (3) for all M > 0 (here and below stands for the Euclidean norm of a finitedimensional vector). We also assume that for all p P, where P R is a set of full measure, the distribution div x ϕ(x, p) = γ p M loc (), (4) Date: October 29, Mathematics Subject Classification. Primary 35J70, 35K65; Secondary 35L65,35B25. Key words and phrases. Degenerate hyperbolic-elliptic equation, degenerate convectiondiffusion equation, conservation law, discontinuous flux, approximate solutions, compactness. This work was supported by the Research Council of Norway through the projects Nonlinear Problems in Mathematical Analysis; Waves In Fluids and Solids; Outstanding Young Investigators Award (KHK), and the Russian Foundation for Basic Research (grant No a) and DFG project No. 436 RUS 113/895/0-1 (EYuP). This article was written as part of the the international research program on Nonlinear Partial Differential Equations at the Centre for Advanced Study at the Norwegian Academy of Science and Letters in Oslo during the academic year

2 2 H. HOLDEN, K. H. KARLSEN, D. MITROVIC, AND E. YU. PANOV where M loc () denotes the space of locally finite Borel measures on equipped with the standard locally convex topology generated by semi-norms p Φ (µ) = Var (Φµ), with Φ = Φ(x) C 0 (). The function ψ(x, u) is assumed to be a Caratheodory function on R, and β M (x) = max u M ψ(x, u) L1 loc() for all M > 0. (5) Let γ p = γp r + γp s be the decomposition of the γ p into the sum of regular and singular measures, so that γp r = ω p (x)dx, ω p L 1 loc (), and γs p is a singular measure (supported on a set of zero Lebesgue measure). We denote by γp s the variation of the measure γp, s which is a non-negative locally finite Borel measure on. As usual we denote 1, u > 0, sign(u) = 1, u < 0, 0, u = 0. Now, we introduce a notion of entropy solution of (1). Definition 1. A measurable function u(x) on is called an entropy solution of equation (1) if ϕ i (x, u(x)), b ij (u(x)), ψ(x, u(x)) L 1 loc (), i, j = 1,..., n, and for almost all p P the Kružkov-type entropy inequality (see [10]) div x (sign(u(x) p)(ϕ(x, u(x)) ϕ(x, p))) (6) D 2 (sign(u(x) p)(b(u(x)) B(p))) + sign(u(x) p)[ω p (x) + ψ(x, u(x))] γ s p 0 holds in the sense of distributions on (in the space D ()); that is, for all nonnegative functions f(x) C0 () [ sign(u(x) p) (ϕ(x, u(x)) ϕ(x, p)) f(x) + (B(u(x)) B(p)) D 2 f ] (ω p (x) + ψ(x, u(x)))f(x) dx + f(x)d γp (x) s 0. We use the notation D 2 f for the matrix { 2 x ix j f} n i,j=1 and P Q = TrP Q = n p ij q ij i,j=1 denotes scalar product of symmetric matrices P = {p ij } n i,j=1, Q = {q ij} n i,j=1. In particular, (B(u(x)) B(p)) D 2 f = (b ij (u) b ij (p)) 2 x ix j f. In the case when the second-order term is absent (B(u) 0) our definition extends the notion of the entropy solution for first-order balance laws introduced for the case of one space variable in [6, 8], see also [7] for one-dimensional degenerate convection-diffusion equations. We also notice that we do not require u(x) to be a weak solution of (1). If u(x) L () and γp s = 0 for p P, then any entropy solution u(x) satisfies (1) in D (), i.e., u(x) is a weak solution of (1). Indeed, this follows from (6) with p > u and p < u. But in general, entropy solutions are not weak solutions, even in the case when the singular measures γp s are absent. For instance, as is easily verified, u(x) = sign x x 1/2 is an entropy solution of the first-order equation (xu 2 ) x = 0 on the line = R, but it does not satisfy this equation in D (R). We assume that equation (1) is non-degenerate in the following sense:

3 DEGENERATE EQUATIONS WITH DISCONTINUOUS COEFFICIENTS 3 Definition 2. Equation (1) is said to be non-degenerate if for almost all x for all ξ, ξ 0 the functions λ ξ ϕ(x, λ), λ B(λ)ξ ξ are not constant simultaneously on non-degenerate intervals. In this paper, we establish the strong precompactness property for sequences of entropy solutions. This result generalizes previous results of [12, 13, 14, 15, 17] to the case of quasi-linear elliptic equations. Theorem 3. Suppose that u k, k N, is a sequence of entropy solutions of the non-degenerate equation (1) such that ϕ(x, u k (x)) + ψ(x, u k (x)) + B(u k (x)) + m(u k (x)) is bounded in L 1 loc (), where m(u) is a nonnegative super-linear function1. Then there exists a subsequence of u k, which converges in L 1 loc () to an entropy solution u(x) of (1). We use here and everywhere below the notation B for the Euclidean norm of a symmetric matrix B, that is B 2 = B B. More generally, we establish the strong precompactness of approximate sequences u k (x) for non-degenerate equation (1). The only assumption we need is that the sequence of distributions div x ϕ(x, s a,b (u k (x))) D 2 B(s a,b (u k (x))) is precompact in the Sobolev space W 1 d,loc () for some d > 1, for each a, b R, a < b (see relation (78) below). Throughout this paper we use s a,b (u) to denote the cut-off function a, u < a, s a,b (u) = max(a, min(u, b)) = u, a u b, b, u > b. Observe that the non-degeneracy condition is essential for the statement of Theorem 3. In the case of the equation divϕ(u) D 2 B(u) = 0 this condition is necessary for strong precompactness property. For instance, if ξ ϕ(u) and B(u)ξ ξ are constant on the segment [a, b] with ξ, ξ 0 then the sequence u k (x) = [a + b + (b a) sin(kξ x)]/2 of entropy solutions does not contain strongly convergent subsequences. We also stress that for sequences of weak solutions (without additional entropy constraints) the statement of Theorem 3 does not hold. For example, the sequence u k = sign sin kx consists of weak solutions for the Burgers equation u t + (u 2 ) x = 0 (as well as for the corresponding stationary equation (u 2 ) x = 0) and converges only weakly, while the non-degeneracy condition is evidently satisfied. Theorem 3 will be proved in the last section. The proof is based on general localization properties for ultra-parabolic H-measures corresponding to bounded sequences of measure-valued functions. It also follows from these properties the strong convergence of various approximate solutions for equation (1). We describe below one useful approximation procedure. We assume for simplicity that ψ(x, u) 0, b ij (u) C 1 (R), i, j = 1,..., n. As shown in [17], there exists a sequence ϕ m (x, u) C ( R) such that ϕ m (x, u) ϕ(x, u) in L 2 loc (, C(R, Rn )) while for each p P, div x ϕ m (x, p) = γpr(x) m + γps(x), m where γpr(x) m ω p (x) in L 1 loc (), γm ps(x) γ s p weakly in M loc (). 1 A nonnegative super-linear function m satisfies m(u)/u as u.

4 4 H. HOLDEN, K. H. KARLSEN, D. MITROVIC, AND E. YU. PANOV By the ellipticity assumption A(u) = B (u) 0, we can choose a sequence of smooth symmetric matrices A m (u) = {a m ij (u)}n i,j=1 such that A m ε m I, ε m > 0 (here I is the identity matrix), and for each M > 0 ε 1/2 m Then we have the limit relation max A m(u) A(u) 0. u M (A m (u) A(u))(A m (u)) 1/2 0 in C(). Moreover, passing to a subsequence of A m if necessary, we may achieve that for each M > 0 and every compact K ( max (A m(u) A(u))(A m (u)) 1/2 = o I m (K, M + 1) 1/2), (7) u M where M I m (K, M) = 1 + div x ϕ m (x, p) dpdx. K M In general, the sequence I m (K, M) may tend to infinity as m. We consider the approximate equation div x [ϕ m (x, u) A m (u) u] = 0 (8) and suppose that u = u m (x) is a bounded weak solution of (8) (for instance, we can take u = u m (x) being a weak solution to the Dirichlet problem with a bounded data at ). This means (see [11, Chapter 4]) that u L () W2,loc 1 (), where W2,loc 1 () is the Sobolev space consisting of functions whose generalized derivatives are in L 2 loc (), and the following standard integral identity is satisfied: For all f = f(x) C0() 1 we have [ϕ m (x, u(x)) A m (u) u(x)] f(x)dx = 0. (9) We also assume that the sequence u m is bounded in L (). Under the above assumptions we establish the strong convergence of the approximations. Theorem 4. Suppose that equation (1) is non-degenerate. Then the sequence u m (x) u(x) in L 1 loc (), where u = u(x) is an entropy and a distributional solution of (1). We remark that Theorem 4 allows to establish the existence of entropy solutions of boundary value problems for equation (1) (as well as initial or initial boundary value problems for evolutionary equations of the kind (1)). For example, in [17] we use approximations and the strong precompactness property in order to prove the existence of entropy solutions to the Cauchy problem for an evolutionary hyperbolic equation with discontinuous multidimensional flux. This extends results of [9], where the two-dimensional case is treated by the compensated compactness method. We also remark that another approach to prove the strong precompactness property for equation (1) based on the kinetic formulation and averaging lemmas was developed in [21]. But this approach can be applied only when the flux ϕ = ϕ(u) does not depend on x, and when the flux vector as well as the diffusion matrix are sufficiently regular. In Sections 2, 3 we describe the main concepts, in particular the concept of measure-valued functions, and introduce a notion of the H-measure. Most of the statements in the sections are taken from [16]. For completeness we also reproduce

5 DEGENERATE EQUATIONS WITH DISCONTINUOUS COEFFICIENTS 5 the proofs of these statements. In [16] we considered the strong pre-compactness property for the general ultra-parabolic equation divϕ(x, u) D 2 B(x, u) + ψ(x, u) = 0, (10) where it is assumed that B(x, u) is a Caratheodory matrix-valued function, which satisfies the ellipticity condition sign(u 1 u 2 )(B(x, u 1 ) B(x, u 2 )) 0, and degenerates on a fixed subspace X (that is, X ker(b(x, u) B(x, u 0 ))). We have more complicated situation in (1) since the diffusion matrix B = B(u), u R, degenerates on a subspace X = X(u) depending on u R. Still, since the matrix B = B(p), p R, is continuous, we will be able to reduce our investigation on the behavior of the corresponding H-measure in a neighborhood of a fixed point p 0 R (see the statement of Theorem 25). Therefore, we will be able to use techniques from [16] (of course, in a rather nontrivial manner). Observe that results analogous to Theorems 3 and 4 were proved in [16] for equation (10) under the stronger non-degeneracy assumption: For almost all x and for all ξ X, ξ X such that ξ 0, ξ 0, the functions λ ξ ϕ(x, λ), λ B(x, λ) ξ ξ are not constant on non-degenerate intervals. Here X denotes the orthogonal complement to the subspace X. In Section 4 we prove the localization property for the above defined H-measures corresponding to sequences of measure-valued functions. Finally, in the last Section 5, these results are applied to prove our main theorems. 2. Main concepts Recall (see [2, 3, 22]) that a measure-valued function on is a weakly measurable map x ν x of the set into the space of probability Borel measures with compact support in R. The weak measurability of ν x means that for each continuous function f(λ) the function x f(λ)dν x (λ) is Lebesgue measurable on. Remark 5. If ν x is a measure-valued function, then, as was shown in [13], the functions g(λ)dν x (λ) are measurable in for all bounded Borel functions g(λ). More generally, if f(x, λ) is a Caratheodory function and g(λ) is a bounded Borel function then the function f(x, λ)g(λ)dν x (λ) is measurable. This follows from the fact that any Caratheodory function is strongly measurable as a map x f(x, ) C(R) (see [5, Ch. 2]) and, therefore, is a pointwise limit of step functions f m (x, λ) = i g mi(x)h mi (λ) with measurable functions g mi (x) and continuous h mi (λ) so that for x f m (x, ) f(x, ) in C(R). A measure-valued function ν x is said to be bounded if there exists M > 0 such that supp ν x [ M, M] for almost all x. We denote by ν x the smallest value of M with this property. Finally, measure-valued functions of the form ν x (λ) = δ(λ u(x)), where δ(λ u) is the Dirac measure concentrated at u are said to be regular; we identify them with the corresponding functions u(x). Thus, the set MV () of bounded measure-valued functions on contains the space L (). Note that for a regular measure-valued function ν x (λ) = δ(λ u(x)) the value ν x = u. Extending the concept of boundedness in L () to measure-valued functions, we shall say that a subset A of MV () is bounded if sup νx A ν x <. Below we define the weak and the strong convergence of sequences of measurevalued functions. Definition 6. Let ν k x MV (), k N, and let ν x MV (). Then

6 6 H. HOLDEN, K. H. KARLSEN, D. MITROVIC, AND E. YU. PANOV (1) the sequence νx k converges weakly to ν x if for each f(λ) C(R), f(λ)dνx(λ) k f(λ)dν x (λ) weak star in L (); k (2) the sequence νx k converges to ν x strongly if for each f(λ) C(R), f(λ)dνx(λ) k f(λ)dν x (λ) in L 1 k loc(). The next result was proved in [22] for regular functions ν k x. The proof can be easily extended to the general case, as was done in [13]. Theorem T. Let ν k x, k N, be a bounded sequence of measure-valued functions. Then there exist a subsequence ν r x = ν k x, k = k r, and a measure-valued function ν x MV () such that ν r x ν x weakly as r. Theorem T shows that bounded sets of measure-valued functions are weakly precompact. If u k (x) L () is a bounded sequence, treated as a sequence of regular measure-valued functions, and u k (x) converges weakly to a measure-valued function ν x then ν x is regular, ν x (λ) = δ(λ u(x)), if and only if u k (x) u(x) in L 1 loc () (see [22]). Obviously, if u k(x) converges to ν x strongly then u k (x) u(x) = λdν x (λ) in L 1 loc () and then ν x(λ) = δ(λ u(x)). We shall study the strong precompactness property using Tartar s technique of H-measures. Let F (u)(ξ) = e 2πiξ x u(x)dx, ξ, be the Fourier transform extended as unitary operator on the space u(x) L 2 ( ), and let S = S n 1 = {ξ : ξ = 1} be the unit sphere in. Denote complex conjugation of u C by u. The concept of H-measure associated to a sequence of vector-valued functions bounded in L 2 () was introduced by Tartar [23] and Gerárd [4] on the basis of the following result. For a fixed l N, let U k (x) = ( U 1 k (x),..., U l k (x)) L 2 (, R l ) be a sequence weakly convergent to the zero vector as k. Proposition 7 ([23, Thm. 1.1]). There is a family of complex Borel measures µ = { µ ij} l on S and a subsequence U i,j=1 r(x) = U k (x), k = k r, such that µ ij, Φ 1 (x)φ 2 (x)ψ(ξ) = lim for all Φ 1 (x), Φ 2 (x) C 0 () and ψ(ξ) C(S). F (U i rφ 1 )(ξ)f (U j r Φ 2 )(ξ)ψ ( ξ ξ ) dξ, (11) The family µ = { µ ij} l i,j=1 is called the H-measure associated to U r(x). Here, we shall need more general variant of the H measures developed in [16] and based on the concept of the parabolic H-measures recently introduced in [1]. Suppose that X is a linear subspace, X is its orthogonal complement, P 1, P 2 are orthogonal projections on X, X, respectively. For ξ, we write ξ = P 1 ξ, ξ = P2 ξ, so that ξ X, ξ X, ξ = ξ + ξ. Let { } S X = ξ : ξ 2 + ξ 4 = 1. Then S X is a compact smooth manifold of codimension 1. In the case when X = {0} or X = it coincides with the unit sphere S = {ξ : ξ = 1}. Let us define

7 DEGENERATE EQUATIONS WITH DISCONTINUOUS COEFFICIENTS 7 the projection π X : \ {0} S X by ξ π X (ξ) = ( ξ 2 + ξ 4 ) + ξ 1/2 ( ξ 2 + ξ 4 ). 1/4 Observe that in the case when X = {0} or X =, We denote p(ξ) = π X (ξ) = ξ/ ξ. ( ξ 2 + ξ 4) 1/4. The following useful property of the projection holds: Lemma 8 ([16, Lemma 1]). Let ξ, η, max(p(ξ), p(η)) 1. Then π X (ξ) π X (η) 6 ξ η max(p(ξ), p(η)). Proof. We define for ξ, α > 0 ξ α = α 2 ξ + α ξ. Observe that for all α > 0 π X (ξ α ) = π X (ξ). Without lose of generality we may suppose that p(ξ) p(η), and, in particular, p(ξ) 1. Remark that π X (ξ) = ξ α, π X (η) = η β, where α = 1/p(ξ), β = 1/p(η). Therefore, π X (ξ) π X (η) = ξ α η β (12) ξ α η α + η α η β = ( α 4 ξ η 2 + α 2 ξ η 2) 1/2 + ( (β 2 α 2 ) 2 η 2 + (β α) 2 η 2) 1/2 α ξ η + (β α) ( (β + α) 2 η 2 + η 2) 1/2. Here we take into account that α 1 and therefore α 4 α 2. Since we have the estimate (β + α) 2 4β 2 = 4( η 2 + η 4 ) 1/2 4/ η, (β + α) 2 η 2 + η 2 4( η + η 2 ) 4 ( 2( η 2 + η 4 ) ) 1/2 6(p(η)) 2. (13) Concerning the term β α, we estimate it as follows β α = = p(ξ) p(η) p(ξ)p(η) ξ 2 η 2 + ξ 4 η 4 p(ξ)p(η)(p(ξ) + p(η))((p(ξ)) 2 + (p(η)) 2 ) ( ξ + η ) ξ η + ( ξ + η )( ξ 2 + η 2 ) ξ η p(ξ)p(η)(p(ξ) + p(η))((p(ξ)) 2 + (p(η)) 2 ) ξ + η + ( ξ + η )( ξ 2 + η 2 ) p(ξ)p(η)(p(ξ) + p(η))((p(ξ)) 2 + (p(η)) 2 ξ η ) (p(ξ))2 + (p(η)) 2 + (p(ξ) + p(η))((p(ξ)) 2 + (p(η)) 2 ) p(ξ)p(η)(p(ξ) + p(η))((p(ξ)) 2 + (p(η)) 2 ξ η ) 1 + p(ξ) + p(η) ξ η p(ξ) + p(η) p(ξ)p(η) 2 ξ η p(ξ)p(η). (14) Here we use that ξ (p(ξ)) 2, ξ p(ξ), η (p(η)) 2, η p(η), and that p(ξ)+p(η) 1. Now it follows from (12), (13), (14) that π X (ξ) π X (η) ξ η p(ξ) ξ η p(ξ) 6 ξ η p(ξ) = 6 ξ η max(p(ξ), p(η)),

8 8 H. HOLDEN, K. H. KARLSEN, D. MITROVIC, AND E. YU. PANOV as was to be proved. Let b(x) C 0 ( ), a(z) C(S X ). Then we can define pseudo-differential operators B, A with symbols b(x), a(π X (ξ)), respectively. These operators are multiplication operators Bu(x) = b(x)u(x), F (Au)(ξ) = a(π X (ξ))f (u)(ξ). Obviously, the operators B, A are welldefined and bounded in L 2. As was proved in [23], in the case when S X = S, π X (ξ) = ξ/ ξ the commutator [A, B] = AB BA is a compact operator. Using the assertion of Lemma 8 one can easily extend this result to the general case (when dim X = 1 this was done in [1]). For completeness we give below the details for the general setting. Lemma 9 ([16, Lemma 2]). The operator [A, B] is compact in L 2. Proof. We can find sequences a k (z) C (S X ), b k (x) C ( ), k N, of symbols with the following properties: F (b k )(ξ) C0 ( ) and, as k, a k (z) a(z), b k (x) b(x) uniformly on S X,, respectively. Then the sequences of the operators A k, B k with symbols a k (π X (ξ)), b k (x) converge as k to the operators A, B, respectively (in the operator norm). Therefore, [A k, B k ] [A, B] and it is sufficient to prove that the operators [A k, B k ] are compact for all k N (then [A, B] is also compact operator as a limit of compact operators). Let u = u(x) L 2 ( ). Then by the well-known property F (bu)(ξ) = F (b) F (u)(ξ) = F (b)(ξ η)f (u)(η)dη, we obtain F ([A k, B k ]u)(ξ) = F (A k B k u)(ξ) F (B k A k u)(ξ) = a k (π X (ξ))f (b k u)(ξ) F (b k A k u)(ξ) = (a k (π X (ξ)) a k (π X (η)))f (b k )(ξ η)f (u)(η)dη. We have to prove that the integral operator Kv(ξ) = k(ξ, η)v(η)dη with the kernel k(ξ, η) = (a k (π X (ξ)) a k (π X (η)))f (b k )(ξ η) is compact on L 2 ( ). Since a k C (S X ), Lemma 8 implies ξ η a k (π X (ξ)) a k (π X (η)) C max(p(ξ), p(η)), for max(p(ξ), p(η)) 1. Thus for all ξ, η such that max(p(ξ), p(η)) > m > 1 and ξ η supp F (b k ) a k (π X (ξ)) a k (π X (η)) C ξ η. (15) m Let χ m (ξ, η) be the indicator function of { (ξ, η) R 2n : max(p(ξ), p(η)) m }, and k m (ξ, η) = χ(ξ, η) (a k (π X (ξ)) a k (π X (η))) F (b k )(ξ η), r m (ξ, η) = (1 χ(ξ, η)) (a k (π X (ξ)) a k (π X (η))) F (b k )(ξ η). Then k(ξ, η) = k m (ξ, η) + r m (ξ, η) and K = K m + R m, where K m, R m are integral operators with the kernels k m (ξ, η), r m (ξ, η), respectively. Since the function k m (ξ, η) is bounded and compactly supported, the operator K m is a Hilbert Schmidt operator, which is compact. On the other hand, in view of (15), R m v(ξ) C (ξ η)f (b k )(ξ η) v(η) dη = C m R m [ ξf (b k) v ](ξ) n

9 DEGENERATE EQUATIONS WITH DISCONTINUOUS COEFFICIENTS 9 and, by Young s inequality, R m v 2 C m ξf (b k) 1 v 2, v L 2 ( ). Therefore, R m const/m and R m 0 as m. We conclude that K m K and therefore K is a compact operator, as a limit of compact operators. This completes the proof. Now we fix a space X. An ultra-parabolic H-measure µ ij, i, j = 1,..., l, corresponding to a sequence U r (x) L 2 (, R l ) is defined on S X by the relation similar to (11), namely, for all Φ 1 (x), Φ 2 (x) C 0 (), ψ(ξ) C(S X ), µ ij, Φ 1 (x)φ 2 (x)ψ(ξ) = lim F (Φ 1 Ur)(ξ)F i (Φ 2 Ur j )(ξ)ψ(π X (ξ))dξ. (16) The existence of an H-measure µ ij is proved exactly in the same way as in [23], using Lemma 2. This H-measure satisfies the same properties as the usual H-measure µ pq (corresponding to the case X = {0} or X = ). The concept of an H-measure was extended in [13] (see also [14, 15]) to sequences of measure-valued functions. A similar extension can be provided for ultra-parabolic H-measures. We study the properties of such H-measures in the next section. 3. Ultra-parabolic H-measures corresponding to bounded sequences of measure-valued functions Let νx k MV () be a bounded sequence of measure-valued functions weakly convergent to a measure-valued function νx 0 MV (). For x and p R we introduce the distribution functions u k (x, p) = ν k x((p, + )), u 0 (x, p) = ν 0 x((p, + )). Then, as mentioned in Remark 5, for k N {0} and p R, the functions u k (x, p) are measurable in x ; thus, u k (x, p) L () and 0 u k (x, p) 1. Let { } E = E(νx) 0 = p 0 R : u 0 (x, p) u 0 (x, p 0 ) in L 1 p p0 loc(). We have the following result, whose proof can be found in [13]. Lemma 10. The complement Ē = R \ E is at most countable and if p E then u k (x, p) u 0 (x, p) weak star in L (). k By Lemma 10, as k, U p k (x) := u k(x, p) u 0 (x, p) 0 weak star in L (), for p E. Let X be a linear subspace of. The next result, similar to Proposition 7, was also established in [13] in the case X =. The general case of arbitrary X was proved exactly in the same way. Proposition 11. (1) There exists a family of locally finite complex Borel measures {µ pq } p,q E in S X and a subsequence U r (x) = {Ur p (x)} p E, Ur p (x) = U p k (x), k = k r, such that for all Φ 1 (x), Φ 2 (x) C 0 (), ψ(ξ) C(S X ), µ pq, Φ 1 (x)φ 2 (x)ψ(ξ) = lim F (Φ 1 Ur p )(ξ)f (Φ 2 Ur q )(ξ)ψ(π X (ξ))dξ. (17) (2) The correspondence (p, q) µ pq is a continuous map from E E into the space M loc ( S). We call the family of measures {µ pq } p,q E the ultra-parabolic H-measure corresponding to the subsequence ν r x = ν k x, k = k r.

10 10 H. HOLDEN, K. H. KARLSEN, D. MITROVIC, AND E. YU. PANOV Remark 12. We can replace the function ψ(π X (ξ)) in relation (17) (and in (16)) by a function ψ(ξ) C( ), which equals ψ(π X (ξ)) for large ξ. Indeed, since Ur q 0 weak star in L (), we have F (Φ 2 Ur q )(ξ) 0 pointwise and in L 2 loc (Rn ) (in view of the bound F (Φ 2 Ur q )(ξ) Φ 2 Ur q 1 const). Taking into account that χ(ξ) = ψ(ξ) ψ(π X (ξ)) is bounded and has a compact support, we conclude This implies that Therefore lim F (Φ 2 U q r )(ξ)χ(ξ) 0 in L 2 ( ). lim F (Φ 1 Ur p )(ξ)f (Φ 2 Ur q )(ξ)χ(ξ)dξ = 0. F (Φ 1 Ur p )(ξ)f (Φ 2 Ur q )(ξ) ψ(ξ)dξ = lim as required. F (Φ 1 U p r )(ξ)f (Φ 2 U q r )(ξ)ψ(π X (ξ))dξ = µ pq, Φ 1 (x)φ 2 (x)ψ(ξ), We point out the following important properties of an H-measure. Lemma 13 ([16, Lemma 4]). (i) µ pp 0 for each p E; (ii) µ pq = µ qp for all p, q E; (iii) for p 1,..., p l E, g 1,..., g l C 0 ( S X ), the matrix A with components a ij = µ pipj, g i g j, i, j = 1,..., l, is Hermitian and positive-definite. Proof. We begin by proving (iii). First, let the functions g i = g i (x, ξ) be finite sums of functions of the form Φ(x)ψ(ξ), where Φ(x) C 0 () and ψ(ξ) C(S X ). Then it follows from (17) that a ij = lim Hr(ξ)H i r j (ξ)dξ, (18) where Hr(ξ) i = F (g i (, π X (ξ))ur pi )(ξ). Hence, setting m g i (x, ξ) = g(x, ξ) = Φ k (x)ψ k (ξ), we obtain H i r(ξ) = m k=1 k=1 F (Φ k U pi r )(ξ)ψ k (π X (ξ)). It immediately follows from (18) that a ji = a ij, i, j = 1,..., l, which shows that A is a Hermitian matrix. Furthermore, for α 1,..., α l C, we have l l a ij α i α j = lim H r (ξ) 2 dξ 0, H r (ξ) = Hr(ξ)α i i, i,j=1 which means that A is positive-definite. In the general case when g i C 0 ( S X ), one carries out the proof of (iii) by approximating the functions g i, i = 1,..., l, in the uniform norm by finite sums of functions of the form Φ(x)ψ(ξ). Assertions (i) and (ii) are easy consequences of (iii). Indeed, setting l = 1, p 1 = p and g 1 = g, we obtain the relation µ pp, g 2 0, which holds for all g C 0 ( S X ), thus showing that µ pp is real and non-negative. To prove (ii) i=1

11 DEGENERATE EQUATIONS WITH DISCONTINUOUS COEFFICIENTS 11 we represent an arbitrary function g = g(x, ξ) with compact support in the form g = g 1 g 2. Let l = 2, p 1 = p and p 2 = q. In view of (iii), µ pq, g = µ pq, g 1 g 2 = µ qp, g 2 g 1 = µ qp, g = µ qp, g and µ pq = µ qp. The proof is complete. We consider now a countable dense index subset D E. Proposition 14 ([16, Proposition 3]). There exists a family of complex finite Borel measures µ pq x on S X with p, q D, x, where is a subset of of full measure, such that µ pq = µ pq x dx, that is, for all Φ(x, ξ) C 0 ( S X ) the function x µ pq x (ξ), Φ(x, ξ) = Φ(x, ξ)dµ pq x (ξ) S X is Lebesgue measurable on, bounded, and µ pq, Φ(x, ξ) = µ pq x (ξ), Φ(x, ξ) dx. Moreover, Var µ pq x 1 for all p, q D. Proof. We claim that pr Var µ pq meas for p, q E, where meas is the Lebesgue measure on. Assume first that p = q. By Lemma 13, the measure µ pp is nonnegative. Next, in view of (17) with Φ 1 (x) = Φ 2 (x) = Φ(x) C 0 () and ψ(ξ) 1, µ pp, Φ(x) 2 = lim F (ΦUr p )(ξ)f (ΦUr p )(ξ)dξ R n = lim Ur p (x) 2 Φ(x) 2 dx Φ(x) 2 dx (we use here Plancherel s equality and the estimate Ur p (x) 1). Thus, we see that that pr µ pp meas. Let p, q E, A be a bounded open subset of, and g = g(x, ξ) C 0 (A S X ), g 1. Let also g 1 = g/ g (we set g 1 = 0 for g = 0) and g 2 = g. Then g 1, g 2 C 0 (A S X ), g = g 1 g 2, g 1 2 = g 2 2 = g, and the matrix ( µ pp, g µ pq ), g µ pq, g µ qq, g is positive-definite by Lemma 13; in particular, µ pq, g ( µ pp, g µ qq, g ) 1/2 (µ pp (A S X )µ qq (A S X )) 1/2 meas(a). We take into account the inequalities pr µ pp meas and pr µ qq meas to obtain the last estimate. Since g can be an arbitrary function in C 0 (A S X ), g 1, we obtain the inequality Var µ pq (A S X ) meas(a). The measure µ pq is regular, therefore this estimate holds for all Borel subsets A of and It follows from (19) that for all ψ(ξ) C(S X ) we have pr Var µ pq meas. (19) Var pr (ψ(ξ)µ pq (x, ξ)) ψ pr Var µ pq ψ meas. (20) In view of (20) the measures pr (ψ(ξ)µ pq (x, ξ)) are absolutely continuous with respect to the Lebesgue measure, and the Radon Nikodym theorem shows that pr (ψ(ξ)µ pq (x, ξ)) = h pq ψ (x) meas, where the densities h pq ψ (x) are measurable on and, as seen from (20), h pq ψ (x) ψ. (21)

12 12 H. HOLDEN, K. H. KARLSEN, D. MITROVIC, AND E. YU. PANOV We now choose a non-negative function K(x) C0 ( ) with support in the unit ball such that K(x)dx = 1 and set K m (x) = m n K(mx) for m N. Clearly, the sequence of K m converges in D ( ) to the Dirac δ-function. Let B lim c m be a generalized Banach limit on the space l of bounded sequences c = {c m } m N, i.e., L(c) = B lim c m is a linear functional on l with the property: lim c m L(c) lim c m (for convergent sequences c = {c m }, L(c) = lim c m). For any complex sequence c m = a m + ib m, the Banach limits are defined by complexification: B lim c m = L(a) + il(b), where a = {a m }, b = {b m } are real and imaginary parts, respectively, of the sequence c = {c m }. Modifying the densities h pq ψ (x) on subsets of measure zero, for instance, replacing them by the functions B lim h pq ψ (y)k m(x y)dy (obviously, the value h pq ψ ), we shall assume that for all x h pq ψ h pq ψ (x) does not change for any Lebesgue point x of the function (x) = B lim h pq ψ (y)k m(x y)dy. (22) Let be the set of common Lebesgue points of the functions h pq ψ (x), u 0(x, p) = ν 0 x((p, + )), u 0 (x, p) = ν0 x([p, + )) = lim q p u 0(x, q), where p, q D and ψ belongs to some countable dense subset F of C(S X ). The family of (p, q, ψ) is countable, therefore is of full measure. The dependence of h pq ψ on ψ, regarded as a map from C(S X) into L (), is clearly linear and continuous (in view of (21)), therefore it follows from the density of F in C(S X ) that x is a Lebesgue point of the functions h pq ψ (x) for all ψ(ξ) C(S X ) and p, q D (here we also take (22) into account). For p, q D and x the equality l(ψ) = h pq ψ (x) defines a continuous linear functional in C(S X ); moreover, l 1 in view of (21). By the Riesz Markov theorem this functional can be defined by integration with respect to some complex Borel measure µ pq x (ξ) in S X and Var µ pq x = l 1. Hence h pq ψ (x) = µpq x (ξ), ψ = ψ(ξ)dµ pq x (ξ), ψ(ξ) C(S X ). (23) S X Equality (23) shows that the functions x S ψ(ξ)dµpq x (ξ) are bounded and measurable for all ψ(ξ) C(S X ). Next, for Φ(x) C 0 () and ψ(ξ) C(S X ) we have ( ) Φ(x)ψ(ξ)dµ pq x (ξ) dx = Φ(x)h pq ψ (x)dx = Φ(x)dpr (ψ(ξ)µ pq ) S X (24) = Φ(x)ψ(ξ)dµ pq (x, ξ). S X Approximating an arbitrary function Φ(x, ξ) C 0 ( S X ) in the uniform norm by linear combinations of functions of the form Φ(x)ψ(ξ) we derive from (24) that the integral S X Φ(x, ξ)dµ pq x (ξ) is Lebesgue-measurable with respect to x, bounded, and ( ) Φ(x, ξ)dµ pq x (ξ) dx = Φ(x, ξ)dµ pq (x, ξ), S X S X

13 DEGENERATE EQUATIONS WITH DISCONTINUOUS COEFFICIENTS 13 that is, µ pq = µ pq x dx. Recall that Var µ pq x 1. The proof is complete. The assumption that x are Lebesgue points of the functions u 0 (x, p), u 0 (x, p) for all p D, will be used later. Observe that since p D E is a continuity point of the map p u 0 (x, p) in L 1 loc (), then u 0 (x, p) = u 0(x, p) a.e. in. By construction x is a common Lebesgue point of the functions u 0 (x, p), u 0 (x, p), therefore ν 0 x({p}) = u 0 (x, p) u 0(x, p) = 0, p D. (25) Remark 15. (a) Since the H-measure is absolutely continuous with respect to x- variables (17) is satisfied for Φ 1 (x), Φ 2 (x) L 2 (). Indeed, by Proposition 14 we can rewrite this identity in the following form: For all Φ 1 (x), Φ 2 (x) C 0 (), ψ(ξ) C(S X ) Φ 1 (x)φ 2 (x) ψ(ξ), µ pq x (ξ) dx = lim F (Φ 1 Ur p )(ξ)f (Φ 2 Ur q )(ξ)ψ(π X (ξ))dξ. (26) Both sides of this identity are continuous with respect to (Φ 1 (x), Φ 2 (x)) in L 2 () L 2 () and since C 0 () is dense in L 2 () we conclude that (26) is satisfied for each Φ 1 (x), Φ 2 (x) L 2 (); (b) if x is a Lebesgue point of a function Φ(x) L 2 (), then Φ(x) µ pq x, ψ(ξ) = lim F (ΦΦ m Ur p )(ξ)f (Φ m Ur q )(ξ)ψ(π X (ξ))dξ (27) for all ψ(ξ) C(S X ), where (ΦΦ m U p r )(y) = Φ(y)Φ m (x y)u p r (y), (Φ m U q r )(y) = Φ m (x y)u q r (y), and Φ m (x y) = K m (x y), the sequence K m is defined in the proof of Proposition 14. Indeed, it follows from (26) that lim F (ΦΦ m Ur p )(ξ)f (Φ m Ur q )(ξ)ψ(π X (ξ))dξ = Now, since x is a Lebesgue point of the functions h pq ψ h pq ψ (y)φ(y)k m(x y)dy. (28) (y) and Φ(y), and the function h pq ψ (y) is bounded, x is also a Lebesgue point for the product of these functions. Therefore, h pq ψ (y)φ(y)k m(x y)dy = Φ(x)h pq ψ (x) = Φ(x) µpq x, ψ(ξ), lim and (27) follows from (28) in the limit as m ; (c) for x and each family p i D, ψ i (ξ) C(S X ), i = 1,..., l, the matrix µ pipj x, ψ i ψ j, i, j = 1,..., l, is positive definite. Indeed, as follows from Lemma 13(iii), for α 1,..., α l C l µ p ip j x, ψ i ψ j αi α j i,j=1 = lim l i,j=1 µ pipj (y, ξ), Φ m (x y)ψ i (ξ)φ m (x y)ψ j (ξ) α i α j 0. Taking in the above property l = 2, p 1 = p, p 2 = q, ψ 1 (ξ) = ψ(ξ)/ ψ(ξ) (ψ 1 = 0 for ψ = 0) and ψ 2 (ξ) = ψ(ξ), ψ(ξ) C(S X ), we obtain, as in the proof of ( µ pp x, ψ µ pq ) x, ψ Proposition 14, that the matrix µ pq x, ψ µ qq is positive definite. In x, ψ particular, µ pq x, ψ ( µ pp x, ψ µ qq x, ψ ) 1/2,

14 14 H. HOLDEN, K. H. KARLSEN, D. MITROVIC, AND E. YU. PANOV and this easily implies that for any Borel set A S X Var µ pq x (A) (µ pp x (A)µ qq x (A)) 1/2. (29) Denote by θ(λ) the Heaviside function: { 1, λ > 0, θ(λ) = 0, λ 0. Below we shall frequently use the following simple estimate. Lemma 16 ([16, Lemma 5]). Let p 0, p D, χ(λ) = θ(λ p 0 ) θ(λ p), V r (y) = χ(λ) d(νy(λ) r + νy(λ)), 0 Φ(y) L 2 (), x is a Lebesgue point of (Φ(y)) 2. Then lim lim Φ m(x y)φ(y)v r (y) 2 2 Φ(x) u 0 (x, p 0 ) u 0 (x, p) 1/2 Proof. It is clear that V r (y) = u r (y, p 0 ) u r (y, p) + u 0 (y, p 0 ) u 0 (y, p) = sign(p p 0 )(u r (y, p 0 ) u r (y, p) + u 0 (y, p 0 ) u 0 (y, p)) 2 and, in particular, (V r (y)) 2 2V r (y). Therefore, 0. p p0 Φ m (x y)φ(y)v r (y) sign(p p 0 ) (Φ(y)) 2 K m (x y)(u r (y, p 0 ) u r (y, p) + u 0 (y, p 0 ) u 0 (y, p))dy. Since p 0, p D E, u r (y, p 0 ) u r (y, p) u 0 (y, p 0 ) u 0 (y, p) as r weak star in L () and we derive from the above inequality that lim Φ m(x y)φ(y)v r (y) sign(p p 0 ) (Φ(y)) 2 K m (x y)(u 0 (y, p 0 ) u 0 (y, p))dy. Now, passing to the limit as m and taking into account that x is a Lebesgue point of the bounded function u 0 (y, p 0 ) u 0 (y, p) as well as the function (Φ(y)) 2 (therefore, x is a Lebesgue point of the product of these functions), we find lim lim Φ m(x y)φ(y)v r (y) 2 2 4(Φ(x)) 2 u 0 (x, p 0 ) u 0 (x, p). This implies the required relation lim lim Φ m(x y)φ(y)v r (y) 2 2 Φ(x) u 0 (x, p 0 ) u 0 (x, p) 1/2. To complete the proof it only remains to observe that, in view of (25), νx({p 0 0 }) = 0 and therefore u 0 (x, p) u 0 (x, p 0 ) as p p 0. The following statement is rather well-known. Lemma 17. Let U r (x) be a sequence bounded in L 2 ( ) L 1 ( ) and weakly convergent to zero, a(ξ) be a bounded function on such that a(ξ) 0 as ξ. Then a(ξ)f (U r )(ξ) 0 in L 2 ( ). Proof. First, observe that by the assumption a(ξ) 0 at infinity for any ε > 0 we can choose R > 0 such that a(ξ) < ε for ξ > R. Then a(ξ) 2 F (U r )(ξ) 2 dξ ε 2 F (U r ) 2 = ε 2 U r 2 Cε 2, (30) ξ >R where C = sup r N U r 2 is a constant independent of r.

15 DEGENERATE EQUATIONS WITH DISCONTINUOUS COEFFICIENTS 15 Furthermore, by our assumption U r 0 as r weakly in L 1. This implies that F (U r )(ξ) 0 pointwise as r. Moreover, F (U r )(ξ) U r 1 const. Hence, using the Lebesgue dominated convergence theorem, we find that a(ξ) 2 F (U r )(ξ) 2 dξ 0 (31) ξ R as r. It follows from (30), (31) that lim a(ξ) 2 F (U r )(ξ) 2 dξ Cε 2. Since ε > 0 is arbitrary, we conclude that lim a(ξ) 2 F (U r )(ξ) 2 dξ = 0, that is, a(ξ)f (U r )(ξ) 0 in L 2 ( ). The proof is complete. We now fix x, p 0, p D. Let L(p) be the smallest linear subspace containing supp µ pp0 x, and L = L(p 0 ). As follows from (29), supp µ pp0 x supp µ p0p0 x and therefore L(p) L. Suppose that f(y, λ) is a Caratheodory vector-function on R such that f(x, ) M, = max λ M f(x, λ) α M (x) L 2 loc(), (32) for all M > 0. Since the space C(R, ) is separable with respect to the standard locally convex topology generated by seminorms M,, then, by the Pettis theorem (see [5, Ch. 3]), the map x F (x) = f(x, ) C(R, ) is strongly measurable and in view of estimate (32) we see that F (x) L 2 loc (, C(R, Rn )), F (x) 2 L 1 loc (, C(R)). In particular (see [5, Ch. 3]), the set f of common Lebesgue points of the maps F (x), F (x) 2 has full measure. For x f we have lim K m (x y) F (x) F (y) M, dy = 0, lim for all M > 0. Since, evidently, K m (x y) F (x) 2 F (y) 2 M, dy = 0 F (x) F (y) 2 M, 2 F (x) F (y) M, F (x) M, + F (x) 2 F (y) 2 M,, it follows from the above limit relations that for x f K m (x y) F (x) F (y) 2 M, dy = 0, (33) lim for all M > 0. Clearly, each x f is a Lebesgue point of all functions x f(x, λ), λ R. Let = f, γx r = νx r νx. 0 Suppose that x, p 0 D, χ(λ) = θ(λ p 1 ) θ(λ p 2 ), where p 1, p 2 D. For a vector-function h(y, λ) on R, which is Borel and locally bounded with respect to the second variable, we denote I r (h)(y) = h(y, λ)dγy(λ). r In view of the strong measurability of F (x) and (32) we see that the sequence I r = I r (f χ)(y) is bounded in L 2 loc () (also see Remark 5). Moreover, this sequence weakly converges to zero as r. The latter easily follows from the fact that fχ(y) can be pointwise approximated by finite sums of functions of the kind h(y, λ) = g(y)θ(λ p), where g(y) L 2 loc () and p D. Since I r(h)(y) = g(y)ur p (y) we see that I r (y) is approximated in L 2 loc () by finite sums of the indicated functions g(y)u r p (y). By Lemma 10 the functions g(y)ur p (y) 0 as r weakly in L 2 loc () and we conclude, by the approximation arguments, that the same remains true for the original sequence I r (y).

16 16 H. HOLDEN, K. H. KARLSEN, D. MITROVIC, AND E. YU. PANOV Let X be the subspace from the definition of ultra-parabolic H-measure, X be the orthogonal complement to X. We denote by L, L the spaces obtained by orthogonal projections of L on the subspaces X, X, respectively: L = P1 (L), L = P 2 (L). Proposition 18 ([16, Prop. 4]). Assume that f(x, λ) L, and ρ(ξ) C ( ) is a function such that 0 ρ(ξ) 1 and ρ(ξ) = 0 for ξ 2 + ξ 4 1, ρ(ξ) = 1 for ξ 2 + ξ 4 2. Then for all ψ(ξ) C(S X ) lim lim ρ(ξ)ξ F (Φ m I r (f χ))(ξ) F (Φ m U p0 r )(ξ)ψ(π X (ξ))dξ = 0. Here Φ m = Φ m (x y) = K m (x y) and I r (f χ) are supposed to be functions of the variable y. Proof. Note that I r (y) f(y, λ) χ(λ) dvar γy(λ) r 2α M (y), (34) where M = sup νx r. Let us first show that for each m N ρ(ξ) ξ F (Φ m I r )(ξ) lim For that, it is sufficient to demonstrate that F (Φ m U p0 r )(ξ)ψ(π X (ξ))dξ = 0. (35) ρ(ξ) ξ ( ξ 2 + ξ 4 ) F (Φ mu p0 1/2 r )(ξ) 0 in L 2 ( ). (36) Remark that the sequence Φ m U p0 r, r N is bounded in L 2 ( ) and in L 1 ( ) (since supp Φ m is compact) and weakly converges to zero. Hence, (36) follows from Lemma 17. We only need to demonstrate that the function ρ(ξ) ξ a(ξ) = satisfies the assumptions of this lemma. First, we show that a(ξ) 1. Indeed, for ξ 2 + ξ 4 1 the value ρ(ξ) = 0 while in the case ξ 2 + ξ 4 > 1 we have ρ(ξ) ξ min( ξ, 1/ ξ ) 1. Then, observe that for ξ 2 + ξ 4 R 4 > 0 a(ξ) ξ ( ξ 2 + ξ 4 ) 1/4 R 1. Therefore, a(ξ) 0 as ξ. Thus, assumptions of Lemma 17 are satisfied and by Lemma 17 we conclude that (36), (35) hold. In view of (35), lim lim = lim lim ρ(ξ)ξ F (Φ m I r )(ξ) F (Φ m U p0 r ρ(ξ) ξ F (Φ m I r )(ξ) )(ξ)ψ(π X (ξ))dξ )(ξ)ψ(π X (ξ))dξ. F (Φ m U p0 r Let g(λ) = f(x, λ), I r = I r (gχ)(y) = g(λ)χ(λ)dγy(λ), r M = sup νy r. Then I r I r f(y, λ) f(x, λ) dvar γy(λ) r 2 F (y) F (x) M,. (37)

17 DEGENERATE EQUATIONS WITH DISCONTINUOUS COEFFICIENTS 17 This and the Plancherel identity imply that ρ(ξ) ξ F (Φ m (I r I r))(ξ) F (Φ m U p0 r )(ξ)ψ(π X (ξ))dξ ψ F (Φ m (I r I r)) 2 F (Φ m U p0 r ) 2 ψ Φ m (I r I r) 2 2 ψ ( K m (x y) F (y) F (x) 2 M, dy) 1/2. It follows from the above estimate and (33) that lim lim ρ(ξ) ξ F (Φ m I r )(ξ) F (Φ m U p0 r )(ξ)ψ(π X (ξ))dξ ρ(ξ) ξ F (Φ m I r)(ξ) F (Φ m U p0 r )(ξ)ψ(π X (ξ))dξ lim lim ρ(ξ) ξ F (Φ m (I r I r))(ξ) F (Φ m U p0 r )(ξ)ψ(π X (ξ))dξ = 0 and, in view of this relation and (37), it is sufficient to prove that lim lim ρ(ξ) ξ F (Φ m I r)(ξ) F (Φ m U p0 r )(ξ)ψ(π X (ξ))dξ = 0. (38) The vector-function g(λ) is continuous and does not depend on y. Therefore for any ε > 0 there exists a vector-valued function h(λ) of the form h(λ) = k i=1 v iθ(λ p i ), where v i L and p i D such that g χ h ε on R. Using again Plancherel s identity and the fact that I r I r (h) = (g χ h)(λ)dγy(λ) r (g χ h)(λ) dvar (γy)(λ) r 2ε, we obtain Since ρ(ξ) ξ F (Φ m I r)(ξ) F (Φ m Ur p0 )(ξ)ψ(π X (ξ))dξ ρ(ξ) ξ F (Φ m I r (h))(ξ) F (Φ m U p0 r )(ξ)ψ(π X (ξ))dξ ψ Φ m I r (g χ h) 2 2ε ψ Φ m 2 = 2ε ψ. ( ) k I r (h)(y) = v i θ(λ p i ) dγy(λ) r = i=1 it follows from (27) the limit relation lim lim ρ(ξ) ξ F (Φ m I r (h))(ξ) k =, (v i ξ)ψ(ξ). i=1 µ pip0 x k i=1 v i U pi r (y), F (Φ m Ur p0 )(ξ)ψ(π X (ξ))dξ Here we also take into account Remark 12. Since ρ(ξ)ψ(π X (ξ)) = ψ(π X (ξ)) for large ξ then, by this remark, for i = 1,..., k, ρ(ξ) ξ v i F (Φ m Ur pi )(ξ) lim F (Φ m U p0 r )(ξ)ψ(π X (ξ))dξ (39) (40)

18 18 H. HOLDEN, K. H. KARLSEN, D. MITROVIC, AND E. YU. PANOV = lim = ξ vi F (Φ m Ur pi )(ξ) µ pip0 (y, ξ), K m (x y)(v i ξ)ψ(ξ) F (Φ m Ur p0 )(ξ)ψ(π X (ξ))dξ. Now observe that supp µ pip0 L(p 0 ) = L, and for each ξ L v i ξ = 0 because ξ L while v i L. Hence lim lim x k µ pip0 x i=1 ρ(ξ) ξ F (Φ m I r (h))(ξ), (v i ξ)ψ(ξ) = 0, and it follows from (40) that F (Φ m U p0 r )(ξ)ψ(π X (ξ))dξ = 0. This relation together with (39) yields lim lim ρ(ξ) ξ F (Φ m I r)(ξ) F (Φ m U p0 r )(ξ)ψ(π X (ξ))dξ 2ε ψ, and since ε > 0 is arbitrary we claim that (38) holds. This completes the proof. Let Q(λ) be a continuous matrix-valued function, which ranges in the space Sym n of symmetric matrices of order n, and Q(λ)ξ ξ = 0 for all ξ L = P 2 (L) (recall that P 2 is the orthogonal projection onto X ). Let p 1, p 2 D, χ(λ) = θ(λ p 1 ) θ(λ p 2 ), J r (y) = J r (Q)(y) = χ(λ)q(λ)dγ r y(λ), and let ρ(ξ) be a function as in Proposition 18. Proposition 19 ([16, Prop. 5]). Under the above notation for each ψ(ξ) C(S X ) lim lim ρ(ξ)f (Φ m J r )(ξ) ξ ξ F (Φ m U p0 r )(ξ)ψ(π X (ξ))dξ = 0. (41) Proof. Since the space Y of symmetric matrices A, satisfying the property Aξ ξ = 0 for ξ L, is linear, for every ε > 0 one can find a step function H(λ) = k i=1 θ(λ p i )Q i, where p i D, Q i Y for each i = 1,..., k such that χ(λ)q(λ) H(λ) < ε for all λ R. We denote J r(y) = H(λ)dγy(λ) r and observe that J r(y) = k i=1 U pi r (y)q i, (42) J r (y) J r(y) Q(λ) H(λ) χ(λ) dvar γy(λ) r 2ε. (43) We also remark that F (Φ m J r )(ξ) ξ ξ F (Φ mj r )(ξ) ξ 2 F (Φ m J r )(ξ).

19 DEGENERATE EQUATIONS WITH DISCONTINUOUS COEFFICIENTS 19 The latter estimate and (43) imply that ρ(ξ)f (Φ m J r )(ξ) ξ ξ F (Φ m U p0 r )(ξ)ψ(π X (ξ))dξ ρ(ξ)f (Φ m J r)(ξ) ξ ξ F (Φ m U p0 r )(ξ)ψ(π X (ξ))dξ = ρ(ξ)f (Φ m (J r J r))(ξ) ξ ξ F (Φ m U p0 r )(ξ)ψ(π X (ξ))dξ ψ F (Φ m (J r J r)) 2 F (Φ m U p0 r ) 2 = ψ Φ m (J r J r) 2 Φ m U p0 r 2 ψ Φ m (J r J r) 2 = ψ ( K m (x y) J r (y) J r(y) 2 dy ) 1/2 (44) 2ε ψ. We also use that Ur p0 ρ(ξ)f (Φ m J r)(ξ) ξ ξ = 1 and therefore Φ m U p0 r 2 1. In view of (42) k ρ(ξ)f (Φ m U pi i=1 F (Φ m Ur p0 )(ξ)ψ(π X (ξ))dξ r )(ξ)q i ξ ξ and by relation (27) and Remark 12 we find lim lim ρ(ξ)f (Φ m J r)(ξ) ξ ξ k = i=1 F (Φ m Ur p0 )(ξ)ψ(π X (ξ))dξ, F (Φ m Ur p0 )(ξ)ψ(π X (ξ))dξ µ p ip 0 x ψ(ξ)q i ξ ξ = 0, (45) because supp µ pip0 x L and therefore Q i ξ ξ = 0 on supp µ p ip 0 x (recall that Q i ξ ξ = 0 for any ξ L). By (44) and (45) we obtain the relation lim lim ρ(ξ)f (Φ m J r )(ξ) ξ ξ F (Φ m U p0 r )(ξ)ψ(π X (ξ))dξ 2ε ψ and since ε > 0 is arbitrary, we conclude that (41) holds. The proof is complete. In the sequel we will need the following simple result. Lemma 20. Let { ξ k : k = 1,..., l } L be a basis in L. Then there exists a positive constant C such that for every v, Q Sym n v 1 + Q 1 C max k=1,...,l iv ξ k + Q ξ k ξ k, where v 1 = P v, Q 1 = P Q P, P, and P are orthogonal projections on the spaces L, L, respectively, and i = 1. Proof. We introduce the linear spaces S = { Q Sym n : Q = P Q P }, H = L S and remark that p(v, Q) = max iv ξ k + Q ξ k ξ k is a norm in H. Indeed, it is k=1,...,l clear that p is a seminorm. To prove that p is a norm, suppose that p(v, Q) = 0. Then v ξ k = Q ξ k ξ k = 0 and since vectors ξ k, ξ k generate spaces L, L, respectively,

20 20 H. HOLDEN, K. H. KARLSEN, D. MITROVIC, AND E. YU. PANOV we claim that v ξ = 0 for all ξ L and Qξ ξ = 0 for all ξ L. Since v L we see that v = 0. Furthermore, since Q S we find that for every ξ Qξ ξ = P Q P ξ ξ = Q P ξ P ξ = 0, and we conclude that Q = 0. It is well-known that any two norms in a finitedimensional space are equivalent. Applying this property to the norms p(v, Q) and p 1 (v, Q) = v + Q and using the relations v ξ k = v 1 ξ k, Q ξ k ξ k = Q P ξ k P ξ k = Q 1 ξk ξ k, k = 1,... l, we find that for some constant C > 0 v 1 + Q 1 C max iv 1 ξ k + Q 1 ξk ξ k = C max iv ξ k + Q ξ k ξ k, k=1,...,l k=1,...,l as was to be proved. Corollary 21. There exist functions ψ k (ξ) C(S X ), k = 1,..., l = dim L and a constant C > 0 such that, in the notation of Lemma 20, for all v, Q Sym n such that Q 0 v 1 + Q 1 C max k=1,...,l µp0p0 x, (iv ξ + Q ξ ξ)ψ k (ξ). (46) Proof. We remark that the measure µ p0p0 x 0. If µ p0p0 x = 0, then both sides of the inequality (46) equal zero, and this inequality is evidently satisfied. Thus, suppose that µ p0p0 x (S X ) > 0. Since L is a linear span of supp µ p0p0 x, we can choose functions ψ k (ξ) C(S X ), k = 1,..., l such that ψ k (ξ) 0, ψ k (ξ)dµ p0p0 x = 1 for all k = 1,..., l, and the family ξ k = ξψ k (ξ)dµ x p0p0, k = 1,..., l, is a basis in L. By Lemma 20 there exists a constant C > 0 such that for all v, Q Sym n v 1 + Q 1 C max k=1,...,l iv ξ k + Q ξ k ξ k, (47) where v 1 = P v, Q 1 = P Q P. Now, we observe that ξ k = ξψ k (ξ)dµ p0p0 x (ξ), ξk = ξψ k (ξ)dµ p0p0 x (ξ). Therefore, v ξ k = and if Q 0 then Q ξ k ξ k = Q ξψ k (ξ)dµ p0p0 x (ξ) v ξψ k (ξ)dµ p0p0 x (ξ), ξψ k (ξ)dµ p0p0 x (ξ) Q ξ ξψ k (ξ)dµ p0p0 x (ξ) by Jensen s inequality applied to the convex function ξ Q ξ ξ. In view of the above relation, (46) readily follows from (47) (we also take into account that for real a the function f(x) = ia+x increases on [0, + )). The proof is complete. 4. Localization principle and strong precompactness of bounded sequences of measure-valued functions In this section we need some results about Fourier multipliers in spaces L d, d > 1. Recall that a function a(ξ) L ( ) is a Fourier multiplier in L d if the pseudodifferential operator A with the symbol a(ξ), defined as F (Au)(ξ) = a(ξ)f (u)(ξ), u = u(x) L 2 ( ) L d ( ), can be extended as a bounded operator on L d ( ), that is, Au d C u d, u L 2 ( ) L d ( ),

21 DEGENERATE EQUATIONS WITH DISCONTINUOUS COEFFICIENTS 21 for a constant C. We denote by M d the space of Fourier multipliers in L d. We also denote n Ṙ n = (R \ {0}) n = { ξ = (ξ 1,..., ξ n ) : ξ k 0 }. The following statement readily follows from the known Marcinkiewicz multiplier theorem (see [20, Ch. 4]). Theorem 22. Suppose that a(ξ) C n (Ṙn ) is a function such that for some constant C k=1 ξ α D α a(ξ) C, ξ Ṙn, (48) for every multi-index α = (α 1,..., α n ) such that α = α α n n. Then a(ξ) M d for all d > 1. Here we use the notation ξ α = n i=1 (ξ i) αi, D α = ( ) n αi. i=1 ξ i Actually, it is sufficient to require that (48) is satisfied for multi-indices α such that α i {0, 1}, i = 1,..., n. We will use the statement of Theorem 22 for symbols of special type. Namely, assume that X is a linear subspace of, and π X : S X be the projection defined in Section 2. Corollary 23. If ψ C n (S X ), then ψ(π X (ξ)) M d for every d > 1. Proof. Using an orthogonal transform, we can assume that X = R k = {ξ : ξ = (y 1,..., y k, 0,..., 0) } while X = {ξ : ξ = (0,..., 0, z 1,..., z n k ) }. By the definition of π X we have π X (t 2 y, tz) = π X (y, z) for each t > 0 and ξ = (y, z), ξ 0. The function a(y, z) = ψ(π X (y, z)) satisfies the same property a(t 2 y, tz) = a(y, z). As is easy to see a(y, z) C n ( \ {0}) and it follows from the above homogeneity relation that D α y D β z a(y, z) = D α y D β z a(t 2 y, tz) = t 2 α + β (D α y D β z a)(t 2 y, tz). (49) Here α = (α 1,..., α k ), β = (β 1,..., β n k ) are multi-indices corresponding to variables y X, z X, respectively, and α + β n. Putting t = ( y 2 + z 4 ) 1/4 in (49), we obtain that D α y D β z a(y, z) = D α y D β z a(t 2 y, tz) = ( y 2 + z 4 ) α /2 β /4 (D α y D β z a)(y, z ), (50) where (y, z ) = π X (y, z). Since the derivatives D α y D β z a are bounded on S X it follows from (50) that for some constant C > 0 y α z β D α y D β z a(y, z) y α z β C ( y 2 + z 4 ) C y α z β C, α /2+ β /4 ( y 2 + z 4 ) α /2 ( y 2 + z 4 ) β /4 for all multi-indices (α, β) such that α + β n. By Theorem 22 we conclude that a(ξ) M d for every d > 1. The proof is complete. Now we consider the symbol a(ξ) = ρ(ξ) (1 + ξ 2 ), where ρ(ξ) C ( ) is a function with the properties indicated in Proposition 14, namely: 0 ρ(ξ) 1, ρ(ξ) = 0 for ξ 2 + ξ 4 1, ρ(ξ) = 1 for ξ 2 + ξ 4 2. Another consequence of Theorem 22 is the following result. Corollary 24. a(ξ) M d for every d > 1.

22 22 H. HOLDEN, K. H. KARLSEN, D. MITROVIC, AND E. YU. PANOV Proof. Obviously, a(ξ) C ( \ {0}). As in the proof of Corollary 23, we can suppose that X = R k = {ξ ξ = (y 1,..., y k, 0,..., 0) }. Then for ξ = (y, z) X X, ξ 0 where we denote a(y, z) = ρ(y, z)(1 + y 2 + z 2 ) 1/2 ( y 2 + z 4 ) 1/2 = ρ(y, z)a 1 (y, z)a 2 (y, z), a 1 (y, z) = (1 + y 2 + z 2 ) 1/2, a 2 (y, z) = ( y 2 + z 4 ) 1/2. In correspondence with (48) we have to show that for all α, β, α + β n y α z β D α y D β z (a 1 (y, z)a 2 (y, z)) C (51) in the domain y 2 + z 4 1 (here we take into account the properties of ρ(ξ)) for some constant C. In order to prove (51), we estimate derivatives of functions a 1, a 2. Evidently, D α y D β z a 1 (y, z) A m (1 + y 2 + z 2 ) (1 α β )/2 A m (1 + y 2 + z 2 ) 1/2 y α z β, where A m is a constant depending only on m = α + β. Furthermore, we observe that the function a 2 (y, z) satisfies the homogeneity relation a 2 (t 2 y, tz) = t 2 a 2 (y, z). It follows from this relation that D α y D β z a 2 (y, z) = t 2 D α y D β z a 2 (t 2 y, tz) = t 2 α + β +2 (D α y D β z a 2 )(t 2 y, tz). Taking in this equality t = ( y 2 + z 4 ) 1/4, we arrive at D α y D β z a 2 (y, z) = ( y 2 + z 4 ) (2 α + β +2)/4 (D α y D β z a 2 )(y, z ), (y, z ) = π X (y, z) S X. Since the derivatives Dy α Dz β a 2 are bounded on S X the latter equality yields the estimates Dy α Dz β a 2 (y, z) B m ( y 2 + z 4 ) (2 α + β +2)/4 (53) B m ( y 2 + z 4 ) 1/2 y α z β, where the constants B m depend on m = α + β. By the Leibniz formula we derive from (52) and (53) the estimates (52) D α y D β z a 1 (y, z)a 2 (y, z) C m (1 + y 2 + z 2 ) 1/2 ( y 2 + z 4 ) 1/2 y α z β, (54) where C m is a constant. As is easily verified, in the domain y 2 + z y 2 + z 2 y 2 y 2 + z y 2 + z 4 + z 2 y 2 + z min( z 2, z 2 ) 3 and by (54) we conclude that in this domain for each α, β, α + β n D α y D β z a 1 (y, z)a 2 (y, z) C y α z β, C being a constant. It is clear that this implies (51). Hence, the requirements of Theorem 22 are satisfied. Therefore, a(ξ) M d for all d > 1. The proof is complete. Now we consider the bounded sequence of measure-valued functions ν k x MV() and suppose that for some d > 1 and each a, b R, a < b the sequence of distributions div x ϕ(x, s a,b (λ))dνx(λ) k D 2 B(s a,b (λ))dν k x(λ) is precompact in W 1 d,loc (). (55) Here s a,b (u) = max(a, min(u, b)) is the cut-off function and W 1 d,loc () denotes the locally convex space of distributions u(x) such that uf(x) belongs to the Sobolev