UNIQUENESS FOR ELLIPTIC OPERATORS ON Lp (RN ) WITH UNBOUNDED COEFFICIENTS GIORGIO METAFUNE∗ , DIEGO PALLARA∗ , PATRICK J. RABIER† , AND ROLAND SCHNAUBELT‡
Abstract. Let A denote a general second order differential operator in divergence form, with real coefficients satisfying only local boundedness conditions. Then, A can be viewed as an unbounded operator Ap on Lp (RN ) with maximal domain D(Ap ). The main result of this paper is a criterion for the injectivity of Ap when p ∈ (1, ∞). This criterion is next used to establish the denseness of C0∞ (RN ) in D(Ap ) or, under more general conditions, in a smaller natural domain. When p = 2, this also yields a positive answer to a selfadjointness question raised by Kato in 1981.
1. Introduction Second order elliptic operators on RN with unbounded coefficients arise in the study of Kolmogoroff and Schr¨odinger equations. In several respects, their properties differ markedly from those known in the case of bounded coefficients. For example, test functions need not form a core of the maximal domain D(Ap ) of A in Lp (RN ) and it may happen that A + λ is not injective on D(Ap ) for any λ > 0. In this paper, we provide conditions about the coefficients of A ensuring that A is injective and that test functions are a core of D(Ap ). In particular, by extending techniques from [12], we are able to treat the limiting cases of some problems that were left open in the works [2] and [9]. Throughout this paper, A denotes the second order linear differential operator on RN N N X X (1.1) A := − ∂i (aij (x)∂j ) + bi (x)∂i + c(x), i,j=1
i=1 1,∞ Wloc
1,∞ ∞ N with real coefficients aij = aji , bi ∈ := Wloc (RN ) and c ∈ L∞ loc := Lloc (R ). More generally, for every m ∈ N ∪ {0} and q ∈ [1, ∞], we shall always use the m,q m,q convenient shortcut W(loc) to denote the Sobolev space W(loc) ( RN ) (Lq(loc) when m = 0). Consistent with this notation, || · ||m,q is the norm of W m,q . If Ω RN is an open subset, || · ||m,q,Ω refers to the norm of the space W m,q (Ω). Given p ∈ (1, ∞) and u ∈ Lp , the assumptions about the coefficients imply that Au is well defined as a distribution on RN . Accordingly, we may and shall denote by Ap the unbounded operator with domain
(1.2)
D(Ap ) := {u ∈ Lp : Au ∈ Lp },
1991 Mathematics Subject Classification. 35K65, 47F05. Key words and phrases. Elliptic operators, maximal domain, core, coarea formula. A substantial part of this work was done during two separate visits of P. J. Rabier at the University of Lecce. The hospitality and support of this institution are gratefully acknowledged. 1
2
G. METAFUNE, D. PALLARA, P. J. RABIER, AND R. SCHNAUBELT
given by (1.3)
∀u ∈ D(Ap ).
Ap u := Au,
It is our goal to establish various properties, starting with the injectivity, of the operators Ap under additional assumptions about the coefficients. Uniqueness results for Kolmogorov operators (i.e., when c = 0) can also be found in Eberle’s monograph [5]. First and foremost, we shall assume ellipticity, i.e., that for every x ∈ RN , there is a constant α(x) > 0 such that N X
(1.4)
aij (x)ξi ξj ≥ α(x)|ξ|2 ,
∀ξ := (ξ1 , · · · , ξN ) ∈ RN ,
i,j=1
where, as usual, | · | is the euclidian norm on RN . Since α(x) above can be chosen to coincide with the smallest eigenvalue of the symmetric matrix (aij (x)) and since this smallest eigenvalue depends continuously upon x, (1.4) is equivalent to the local uniform ellipticity condition N X
aij (x)ξi ξj ≥ αK |ξ|2 ,
∀x ∈ K,
∀ξ := (ξ1 , · · · , ξN ) ∈ RN ,
i,j=1
where K is an arbitrary compact subset of RN and αK > 0. It then follows from elliptic regularity that 2,p D(Ap ) ⊂ Wloc .
(1.5)
In addition, from the classical a priori estimates (see for instance Gilbarg and 2,p Trudinger [8]), for every R > 0, there is a constant CR > 0 independent of u ∈ Wloc such that (1.6)
||u||2,p,BR ≤ CR (||u||0,p,B2R + ||Au||0,p,B2R ) ,
where BR denotes the ball in RN with center 0 and radius R. Together with (1.3) and (1.5), this shows at once that the operator Ap above is closed. Equivalently, D(Ap ) is a Banach space for the graph norm (1.7)
||u||D(Ap ) := ||u||0,p + ||Au||0,p .
For future use, note that C0∞ (:= C0∞ (RN )) ⊂ D(Ap ), so that D(Ap ) is dense in Lp . The additional hypotheses that we shall make about the coefficients of A in (1.1) are not standard and involve an auxiliary C N function ρ ≥ 1 on RN and a parameter s > 0. Even though auxiliary functions have already been used in related matters (mostly the selfadjointness issue when p = 2; see for instance Evans [6], Kato [9]), our approach, based on Federer’s coarea formula, follows a much different line. The hypotheses recently introduced in Rabier [12] when p = 2 and aij ∈ L∞ also involve a parameter s, but the method fails to work when p 6= 2. Thus, while the results of this paper may be viewed as a (partial) generalization of those in [12] and in spite of notable similarities, the arguments are, of necessity, markedly different. From now on, (1.8)
a(x, ξ, η) :=
N X i,j=1
aij (x)ξi ηj ,
∀ξ, η ∈ RN
OPERATORS WITH UNBOUNDED COEFFICIENTS
3
and b(x) := (b1 (x), · · · , bN (x))
(1.9)
is the vector of the first order coefficients of A. The main result of this paper (sufficient condition for the injectivity of Ap ) is given in Theorem 3.2. The denseness of C0∞ in D(Ap ) equipped with the graph norm is subsequently obtained as a by-product under slightly more general assumptions (Theorem 4.2). A generalization of the denseness property is given in Theorem 4.4, where D(Ap ) is replaced by a possibly smaller domain. A special case (Corollary 4.5) allows for an easy comparison with recent results of the same nature in [2]. To complete the paper, a simple “concrete” application of Theorem 4.2 is given in Section 5, where we answer in particular a question left open by Kato in [9] when p = 2 and b = 0.
2. Preliminary estimates 1
If ρ ≥ 0 is a C function on RN with lim|x|→∞ ρ(x) = ∞ and if A is a differential operator (1.1) with aij = aji , bi ∈ L∞ loc , we set, for every r ≥ 1 and every λ > 1, (2.1)
Mλ (ρ, r) :=
sup
{1 + |b(x) · ∇ρ(x)| + a(x, ∇ρ(x), ∇ρ(x))},
λ−1 r≤ρ(x)≤λ2 r
where a and b are given by (1.8) and (1.9), respectively. Note that Mλ (ρ, r) < ∞ (in particular, observe that the set {λ−1 r ≤ ρ ≤ λ2 r} is compact since lim|x|→∞ ρ(x) = ∞). Lemma 2.1. Given p ∈ (1, ∞) and λ > 1 as above, there is a constant C(λ, p) > 0 such that, for every C 1 function ρ ≥ 0 on RN with lim|x|→∞ ρ(x) = ∞ and every 1,∞ elliptic operator A in (1.1) with real coefficients aij = aji , bi ∈ Wloc and c ∈ L∞ loc 2,p ∇·b such that c − p ≥ 0, the following inequality holds for every u ∈ Wloc and every r≥1: ∇·b |u|p−2 a(·, ∇u, ∇u)χ{u6=0} + c − |u|p p {r≤ρ≤λr} ! Z p p ≤ C(λ, p) |Au| + Mλ (ρ, r)|u| ,
Z (2.2)
{λ−1 r 0. By substitution into (2.3), we obtain Z Z ∇·b 2 2 (p − 1 − γ) ηλ,r |u|p−2 a(·, ∇u, ∇u)χ{u6=0} + c− ηλ,r |u|p p RN RN Z Z 2 2 ≤ (Au)ηλ,r |u|p−2 u + ηλ,r b · ∇ηλ,r + γ −1 a(·, ∇ηλ,r , ∇ηλ,r ) |u|p . p RN RN Above, choose γ > 0 such that p − 1 − γ > 0. Since c − ∇·b p ≥ 0 and ηλ,r (x) = 1 when r ≤ ρ(x) ≤ λr, this yields ! Z ∇·b p p−2 min{p − 1 − γ, 1} |u| a(·, ∇u, ∇u)χ{u6=0} + c − |u| p {r≤ρ≤λr} Z Z 2 2 p−1 −1 ≤ (Au)ηλ,r |u| + ηλ,r b · ∇ηλ,r + γ a(·, ∇ηλ,r , ∇ηλ,r ) |u|p . p RN RN 1If so, χ {u6=0} is not needed in the formula, which is obvious if p > 2. If p = 2, this is due to the well known fact (see Gilbarg and Trudinger [8, Lemma 7.7], Stampacchia [13]) that ∇u = 0 a.e. on the set {u = 0}.
OPERATORS WITH UNBOUNDED COEFFICIENTS
5
Now, in the right-hand side, use 0 ≤ ηλ,r ≤ 1, ηλ,r (x) = 0 if ρ(x) ≤ λ−1 r or 0 ρ(x) ≥ λ2 r and ∇ηλ,r = ψλ,r (ρ)∇ρ = r−1 ψλ0 (ρ/r)∇ρ to get ! Z ∇·b p−2 p min{p − 1 − γ, 1} |u| a(·, ∇u, ∇u)χ{u6=0} + c − |u| p {r≤ρ≤λr} Z ≤ |Au||u|p−1 {λ−1 r 1 such that c − ∇·b p > 0 outside a ball, regardless of the value of q in (4.8). However, since (4.8) is not retained at all in [2, Theorem 4.1], this is not a full generalization. Roughly speaking, (4.8) limits the growth of b to exponential (since z grows at most exponentially in Corollary 4.5). On the other hand, Theorem 4.4 is much more general than its special case in Corollary 4.5 and, in particular, it is valid without any boundedness condition about the coefficients aij . This is not the case in [2, Theorem 4.1], where the boundedness of aij is used in the proof. Remark 4.2. If N ≥ 2 and z is C N with ∇z 6= 0 a.e., then Corollary 4.5 is 1,∞ valid when aij = aji ∈ W 1,∞ , bi ∈ Wloc andc ∈ L∞ if (4.7) is replaced loc . Also, by the stronger condition |b · ∇z| ≤ M (1 + z) 1 + c −
applicable, so that
A0p
∇·b p
, then Theorem 4.2 is
can be replaced by Ap in Corollary 4.5.
Example 4.1. Let A := −∆ + |x|β−1 x · ∇ + c0 |x|γ where c0 > 0, β ≥ 1 and γ ≥ 0. Then, C0∞ is a core of D(Ap ) if γ > β − 1 or if γ = β − 1 and c0 > p−1 (N + β − 1). This follows from Corollary 4.5 with z(x) := |x|α for large |x| and α > p−1 (β − 1) and from the second part of Remark 4.2. If N = 1, β = 3 and γ = 2, this justifies the statement in [2, Proposition 6.3], whose argument for this case (and this case only) is incorrect. In addition, that A0p may be replaced by Ap does not follow from [2]. Example 4.2. In Example 4.1, change the drift term into its negative, i.e., let A := −∆ − |x|β−1 x · ∇ + c0 |x|γ . Then, C0∞ is a core of D(A0p ) for every β ≥ 1 and every γ ≥ 0. This follows from Corollary 4.5 with the same choice z(x) := |x|α , α > p−1 (β − 1) as before. However, the second part of Remark 4.2 to the effect that A0p may be replaced by Ap in this statement is only valid under the same restriction γ > β − 1 or γ = β − 1 and c0 > p−1 (N + β − 1) as in Example 4.1. Other natural choices of z in the above examples, such as z(x) := ea|x| (a > 0) lead to weaker results. 5. On a problem of Kato We shall illustrate Theorem 4.2 with the simple case when p = 2 and (5.1)
A := −
N X
∂i (aij (x)∂i ) + c(x),
i,j=1 1,∞ (so that b = 0) and aij = aji ∈ Wloc and c ∈ L∞ loc satisfy the conditions
(5.2)
N X i,j=1
aij (x)
xi xj ≤ k0 (1 + r)τ r2
and (5.3)
c(x) ≥ c0 rθ ,
for r := |x| > 0 large enough, where k0 , c0 > 0 and τ, θ ≥ 0 are constants. Of course, we also assume that the ellipticity condition (1.4) holds. Theorem 5.1. Under the above assumptions, C0∞ is a core of D(A2 ) if 0 ≤ τ ≤ θ + 2.
OPERATORS WITH UNBOUNDED COEFFICIENTS
15
Proof. The hypotheses (i) to (iv) of Theorem 4.2 depend only on the values of ρ and of the coefficients for large r, so that we may (and shall) choose ρ such that (5.4)
ρ(x) := Log(1 + r)
for r > 0 large enough. It is obvious that this can be done in such a way that ρ is C ∞ , ρ ≥ 1 and ∇ρ vanishes only when x = 0. Since conditions (i) and (iii) of Theorem 4.2 hold irrespective of (p and) s0 , it suffices to check the validity of (ii) and (iv). For p = 2, the latter amounts to 0
k0 (1 + r)τ −2−2s ∈ L∞ , which happens if and only if s0 ≥ is possible if and only if
τ 2
− 1. In turn, such a choice of s0 with 0 < s0 < s
n τ o s > max 0, − 1 . 2 It follows from (3.9) with b = 0, (5.2) and (5.3) that condition (ii) of Theorem 4.2 holds with p = 2 if (5.5)
c0 rθ − k0 s2 (1 + r)τ −2 ≥ 0,
(5.6)
for r > 0 large enough. Since c0 > 0, this inequality is true irrespective of s if τ < θ + 2. Thus, if τ < θ + 2, the hypotheses of Theorem 4.2 are satisfied upon choosing any s > 0 satisfying (5.5). Accordingly, C0∞ is a core of D(A2 ) in this case. To complete the proof, we now consider the case τ = θ + 2. If so, (5.5) amounts to s > θ2 while (5.6) reduces to assuming that k0−1 c0 > s2 . Clearly, such an s > 0 can be found provided that θ2 < 4k0−1 c0 and so this condition alone suffices to ascertain that C0∞ is a core of A2 . However, with a little extra work, this restriction can be removed, as we now show. Upon writing any function u on RN in the form u(x) = v(x/ε) for some ε > 0, it e2 ) when A is replaced appears that it suffices to prove the denseness of C0∞ in D(A e with coefficients by the operator A e aij (x) := ε−1 aij (εx)
and
e c(x) := c(εx),
respectively. The ellipticity condition (1.4) is not affected by this change, while the hypotheses (5.2) and (5.3) above take the form (using τ = θ + 2) N X i,j=1
e aij (x)
xi xj ≤ k0 ε−1 (1 + εr)θ+2 r2
and
e c(x) ≥ c0 εθ rθ ,
for r > 0 large enough, respectively. As a result, with the same choice (5.4) of ρ, e holds if condition (ii) of Theorem 4.2 for A c0 εθ rθ − k0 s2 ε−1 (1 + εr)θ+2 (1 + r)−2 ≥ 0 for r large enough, which is the case if εs2 < k0−1 c0 . Next, by the arguments used above with the operator A, condition (iv) of Thee is satisfied if and only if s > θ . Therefore, Theorem 4.2 yields the orem 4.2 for A 2 e2 ) if εθ2 < 4k −1 c0 . Since k0 , c0 > 0, this inequality always denseness of C0∞ in D(A 0 holds if ε is chosen small enough in the first place. This proves that C0∞ is a core of A2 when τ = θ + 2.
16
G. METAFUNE, D. PALLARA, P. J. RABIER, AND R. SCHNAUBELT
It is well known that the denseness of C0∞ in D(A2 ) implies that A2 is selfadjoint. When 0 ≤ τ < θ + 2, the selfadjointness of A2 was already obtained by Kato [9], where the limiting case τ = θ + 2 was explicitly left open when N ≥ 2. Theorem 5.1 resolves the issue in the affirmative (for locally regular coefficients) and the result is sharp since there are counterexamples to essential selfadjointness when θ = 0 and τ > 2 (see Davies [3, Example 3.5] or Chapter 2, §1 of Eberle [5] and the references therein). On the other hand, a special structure of the diffusion matrix (aij ) implies selfadjointness regardless of any growth condition (Devinatz [4]). It is of course possible to generalize Theorem 5.1 when p 6= 2 by using Theorem 4.2, but this requires making more assumptions than just (5.2) and (5.3) since PN 2 ρ are involved (see (3.8)). Thus, the discussion the terms i,j=1 ∂j aij ∂i ρ and aij ∂ij depends in part upon what is assumed about ∂j aij . We shall not explore the various options. References [1] Ambrosio, L., Fusco, N. and Pallara, D., Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000. [2] Arendt, W., Metafune, G. and Pallara, D., Schr¨ odinger operators with unbounded drift, J. Oper. Theory 55 (2006) 185-211. [3] Davies, E. B., L1 properties of second order elliptic operators, Bull. London Math. Soc. 17 (1985) 417-436. [4] Devinatz, A., Essential self-adjointness of Schr¨ odinger-type operators, J.Funct. Anal. 25 (1977) 58-69. [5] Eberle, A., Uniqueness and non-uniqueness of semigroups generated by singular diffusion operators, Springer Lecture Notes in Mathematics 1718, 1999. [6] Evans, W. D., On the essential self-adjointness of powers of Schr¨ odinger-type operators, Proc. Roy. Soc. Edinburgh Sect. A 79 (1977) 61-77. [7] Federer, H., Geometric measure theory, Springer-Verlag, New York (1969). [8] Gilbarg, D. and Trudinger, N. S., Elliptic partial differential equations of second order, 2nd ed., Grund. Math. Wiss. 224. Springer-Verlag, Berlin 1983. [9] Kato, T., Remarks on the selfadjointness and related problems for differential operators, in Spectral Theory of Differential Operators, I. W. Knowles and R. T. Lewis eds, North-Holland (1981) 253-266. [10] Metafune, G. and Spina, C., An integration by parts formula in Sobolev spaces, Medirerranean J. Math. (to appear). [11] Ohtsuka, M., Area formula, Bull. Inst. Math. Acad. Sinica 6 (1978) 599-636. [12] Rabier, P. J., Elliptic problems on RN with unbounded coefficients in classical Sobolev spaces, Math. Zeit. 249 (2005) 1-30. [13] Stampacchia, G., Equations elliptiques du second ordre a ` coefficients discontinus, Presses Univ. Montreal, 1966. [14] Whitney, H., Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc. 36 (1934) 63-89. ∗ Dipartimento di Matematica “Ennio de Giorgi”, Universita ` del Salento, C. P. 193, 73100, Lecce, Italy. E-mail address:
[email protected]/
[email protected] † Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA. E-mail address:
[email protected] ‡ Department
of Mathematics, University of Karlsruhe, 76128 Karlsruhe, Germany. E-mail address:
[email protected]
