Accretivity and form boundedness of second order differential operators
aa r X i v : . [ m a t h . A P ] J un ACCRETIVITY AND FORM BOUNDEDNESS OFSECOND ORDER DIFFERENTIAL OPERATORS
VLADIMIR G. MAZ’YA AND IGOR E. VERBITSKY
In memory of Aizik Volpert
Abstract.
Let L be the general second order differential operator withcomplex-valued distributional coefficients A = ( a jk ) nj,k =1 , ~b = ( b j ) nj =1 ,and c in an open set Ω ⊆ R n ( n ≥ L u = div ( A ∇ u ) + ~b · ∇ u + c u , or non-divergence form, L u = P nj, k =1 a jk ∂ j ∂ k u + ~b · ∇ u + c u .We give a survey of the results by the authors which characterize thefollowing two properties of L :(1) −L is accretive, i.e., Re h−L u, u i ≥ L is form bounded, i.e., |hL u, u i| ≤ C k∇ u k L (Ω) ,for all complex-valued u ∈ C ∞ (Ω). Contents
1. Introduction 22. Preliminaries 53. Form boundedness 73.1. Form boundedness in the homogeneous Sobolev space 83.2. Form boundedness in W , ( R n ) 103.3. Infinitesimal form boundedness 103.4. Nash’s inequality and p -subordination 113.5. Form boundedness in W , ( R n ) 114. Accretivity 114.1. General accretivity criterion 114.2. Real-valued coefficients 124.3. Nonnegative definite Schr¨odinger operators 134.4. The one-dimensional case 134.5. Upper and lower bounds of quadratic forms 144.6. Accretivity criterion in R n Mathematics Subject Classification.
Primary 35J15, 42B37; Secondary 31B15,35J10.
Key words and phrases.
Accretivity, form boundedness, general second order differen-tial operators.The first author is supported in part by the “RUDN University Program 5-100”. Introduction
We consider the general second order differential operator(1.1) L u = n X j, k =1 a jk ∂ j ∂ k u + n X j =1 b j ∂ j u + c u, in an open set Ω ⊆ R n , with A = ( a jk ) ∈ D ′ (Ω) n × n , ~b = ( b j ) ∈ D ′ (Ω) n ,and c ∈ D ′ (Ω), where D ′ (Ω) = C ∞ (Ω) ∗ is the space of complex-valueddistributions in Ω.We discuss the accretivity property of −L (or, equivalently, dissipativity of L ), i.e.,(1.2) Re h−L u, u i ≥ , for all complex-valued functions u ∈ C ∞ (Ω).More general differential operators(1.3) L u = n X j, k =1 a jk ∂ j ∂ k u + ~b · ∇ u + div ( ~b u ) + c u, with ~b , ~b ∈ D ′ (Ω) n and c ∈ D ′ (Ω) can be treated as well, since L isimmediately reduced to L with ~b = ~b + ~b and c = c + div ~b .Our main results on the accretivity problem for general differential oper-ators are discussed in Sec. 4 below. (See Propositions 4.1 and 4.2, as wellas Theorem V for n = 1, and Theorem VI for n ≥ L = ∆ + ~b · ∇ + c, whose principal part is the Laplacian ∆, and the coefficients ~b = ( b j ) and c are locally integrable functions in R n . Then the sesquilinear form of − ˜ L isgiven by h− ˜ L u, v i = Z R n ( ∇ u · ∇ v − ~b · ∇ u v − c u v ) dx, (1.4)where u, v ∈ C ∞ ( R n ).In this special case, let(1.5) q = Re c − div (Re ~b ) , ~d = (Im ~b ) . We denote by H = ∆ + q the corresponding Schr¨odinger operator. Thequadratic form associated with −H in the case q ∈ L ( R n ) is given by(1.6) [ h ] H := h−H h, h i = Z R n ( |∇ h | − q | h | ) dx, h ∈ C ∞ ( R n ) . Theorem I.
Let ˜ L = ∆ + ~b · ∇ + c , where Re ~b ∈ W , ( R n ) , and Im ~b, c ∈ L ( R n ) . Let q , ~d be given by (1.5) . Then the operator − ˜ L is accretive ifand only if the following two conditions hold: CCRETIVITY AND FORM BOUNDEDNESS 3 (i)
The operator −H is nonnegative definite, i.e., (1.7) [ h ] H = Z R n ( |∇ h | − q | h | ) dx ≥ , for all real (or complex-valued) h ∈ C ∞ ( R n ) . (ii) The commutator inequality (1.8) (cid:12)(cid:12)(cid:12)(cid:12)Z R n ~d · ( u ∇ v − v ∇ u ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ [ u ] H [ v ] H holds for all real-valued u, v ∈ C ∞ ( R n ) . A necessary and sufficient condition for property (1.7) was obtained in[11, Proposition 5.1] (see Sec. 4.3 below). Concerning condition (1.8), weobserve that, under the upper and lower bounds on the quadratic form(1.7) discussed in Sec. 4.5, the expressions [ u ] H and [ v ] H on the right-hand side of (2.12) can be replaced, up to a constant multiple, with thecorresponding Dirichlet norms ||∇ u || L ( R n ) and ||∇ v || L ( R n ) , respectively.Then the corresponding commutator inequality(1.9) (cid:12)(cid:12)(cid:12)(cid:12)Z R n ~d · ( u ∇ v − v ∇ u ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ||∇ u || L ( R n ) ||∇ v || L ( R n ) , for all (real-valued or complex-valued) u, v ∈ C ∞ ( R n ), can be characterizedcompletely as follows (see [26, Lemma 4.8]). Theorem II.
Let ~d ∈ L ( R n ) , n ≥ . Then inequality (1.9) holds if andonly if (1.10) ~d = ~c + Div F, where F ∈ BMO( R n ) n × n is a skew-symmetric matrix field, and ~c satisfiesthe condition (1.11) Z R n | ~c | | u | dx ≤ C ||∇ u || L ( R n ) , where the constant C does not depend on u ∈ C ∞ ( R n ) .Moreover, if (1.9) holds, then (1.10) is valid with ~c = ∇ ∆ − (div ~d ) satis-fying (1.11) , and F = ∆ − (Curl ~d ) ∈ BMO( R n ) n × n .In the case n = 2 , necessarily ~c = 0 , and ~d = ( − ∂ f, ∂ f ) with f ∈ BMO( R ) in the above statements. Here the gradient ∇ , and the matrix operators Div, Curl are understood inthe sense of distributions (see Sec. 2). Expressions ∆ − (div ~d ), ∆ − (Curl ~d ),etc., are defined in terms of the weak- ∗ BMO convergence (details can befound in [26], [27]). Theorems I & II yield an explicit criterion of accretivityfor − ˜ L (see Theorem VI below in the general case). V. G. MAZ’YA AND I. E. VERBITSKY
More general commutator inequalities related to compensated compact-ness theory [3] were studied earlier by the authors [26] in the framework ofthe form boundedness problem,(1.12) |hL u, v i| ≤ C ||∇ u || L ( R n ) ||∇ v || L ( R n ) , where the constant C does not depend on u, v ∈ C ∞ ( R n ).If (1.12) holds, then hL u, v i can be extended by continuity to u, v ∈ L , ( R n ) ( n ≥ L , ( R n ) is the completion of (complex-valued) C ∞ ( R n ) functions with respect to the norm || u || L , ( R n ) = ||∇ u || L ( R n ) .Equivalently,(1.13) L : L , ( R n ) → L − , ( R n )is a bounded operator, where L − , ( R n ) = L , ( R n ) ∗ is a dual Sobolevspace. Analogous problems have been studied in [23]–[25] for the inhomoge-neous Sobolev space W , ( R n ), fractional Sobolev spaces, infinitesimal formboundedness, and other related questions (see Sec. 3 below).In the special case of the operator ˜ L , we have the following characteriza-tion of form boundedness. Theorem III.
Let ˜ L = ∆ + ~b · ∇ + q , where ~b ∈ L ( R n ) n and q ∈ L ( R n ) , n ≥ . Then the following statements hold. (i) The sesquilinear form of ˜ L given by (1.4) is bounded if and only if ~b and q can be represented respectively in the form (1.14) ~b = ~c + Div F, q = div ~h, where F is a skew-symmetric matrix field such that (1.15) F ∈ BMO( R n ) n × n , whereas ~c and ~h satisfy the condition (1.16) Z R n ( | ~c | + | ~h | ) | u | dx ≤ C ||∇ u || L ( R n ) , where the constant C does not depend on u ∈ C ∞ ( R n ) . (ii) If the sesquilinear form of ˜ L is bounded, then ~c , F , and ~h in decom-position (1.14) can be determined explicitly by ~c = ∇ ∆ − (div ~b ) , ~h = ∇ (∆ − q ) , (1.17) F = ∆ − (Curl ~b ) , (1.18) so that conditions (1.15) , (1.16) hold.If n = 2 , then (1.16) yields that ~c = 0 and ~h = 0 , so that q = 0 and ~b = ( − ∂ f, ∂ f ) with f ∈ BMO( R ) . The form boundedness problem (1.12) for the general second order differ-ential operator L in the case Ω = R n was characterized by the authors in[26] using harmonic analysis and potential theory methods. These results CCRETIVITY AND FORM BOUNDEDNESS 5 are discussed in Sec. 3 below. We observe that no ellipticity assumptionsare imposed on the principal part A of L in this context.For the Schr¨odinger operator H = ∆ + q with q ∈ D ′ (Ω), where eitherΩ = R n , or Ω is a bounded domain that supports Hardy’s inequality (see[2]), a characterization of form boundedness was obtained earlier in [22]. Adifferent approach for H = div ( P ∇· ) + q in general open sets Ω ⊆ R n , underthe uniform ellipticity assumptions on P , was developed in [11]. (We remarkthat these assumptions on P can be relaxed in a substantial way.) There isalso a quasilinear version for operators of the p -Laplace type (see [12]).Both the accretivity and form boundedness properties have numerousapplications. They include problems in mathematical quantum mechanics([31], [32]), PDE theory ([4], [6], [13], [14], [21], [28], [8], [29]), fluid mechanicsand Navier-Stokes equations ([7], [16], [33], [35]), semigroups and Markovprocesses ([18]), homogenization theory ([37]), harmonic analysis ([3], [5]),etc.We conclude the Introduction with the observation that, for the formboundedness property, the case of complex-valued coefficients is easily re-duced to the real-valued case. In contrast, for the accretivity property,complex-valued coefficients lead to additional difficulties that appear whenthe matrix Im A is not symmetric, or the imaginary part of ~b is nontrivial.2. Preliminaries
Let Ω ⊆ R n ( n ≥
1) be an open set. The matrix row divergence operatorDiv : D ′ (Ω) n × n → D ′ (Ω) n is defined on matrix fields F = ( f jk ) nj,k =1 ∈ D ′ (Ω) n × n by Div F = ( P nk =1 ∂ k f jk ) nj =1 ∈ D ′ (Ω) n . If F is skew-symmetric,i.e., f jk = − f kj , then we obviously have div (Div F ) = 0.The matrix curl operator Curl: D ′ (Ω) n → D ′ (Ω) n × n is defined on vectorfields ~f = ( f k ) nk =1 by Curl ~f = ( ∂ j f k − ∂ k f j ) nj,k =1 . Clearly, Curl ~f is alwaysa skew-symmetric matrix field.It will be convenient to use the notion of admissible measures M , (Ω),i.e., nonnegative locally finite Borel measures µ in Ω which obey the traceinequality(2.1) (cid:16) Z Ω | u | dµ (cid:17) ≤ C ||∇ u || L (Ω) , for all u ∈ C ∞ (Ω) , where the constant C does not depend on u . The least embedding constant C in (2.1) will be denoted by || µ || M , (Ω) . For admissible measures q ( x ) dx with nonnegative density q ∈ L (Ω), we write q ∈ M , (Ω).Several characterizations of M , (Ω) are known. They can be formulatedin terms of capacities [21] or Green energies [5], [30], and, in the case Ω = R n ,in terms of local maximal estimates [15], pointwise potential inequalities [22],or dyadic Carleson measures [36] (see also [26], [27]). V. G. MAZ’YA AND I. E. VERBITSKY
Suppose that the principal part A u of the general differential operator isgiven in the divergence form,(2.2) A u = div ( A ∇ u ) , u ∈ C ∞ (Ω) . Then we consider the operator(2.3) L u = div ( A ∇ u ) + ~b · ∇ u + c u, with distributional coefficients A = ( a jk ), ~b = ( b j ), and c . The correspondingsesquilinear form hL u, v i is given by(2.4) hL u, v i = −h A ∇ u, ∇ v i + h ~b · ∇ u, v i + h c u, v i , where u, v ∈ C ∞ (Ω) are complex-valued.We observe that if L is given in the non-divergence form (1.1), then L = L −
Div A · ∇ . (See, for instance, [14], [27].) Hence, we can express hL u, v i in the form(2.4), with ~b − Div A in place of ~b , for distributional coefficients A and ~b .This means that, without loss of generality, we may treat the accretivityproperty(2.5) Re h−L u, u i ≥ , for all u ∈ C ∞ (Ω) , for the divergence form operator L given by (2.3).This problem is of substantial interest even in the real-variable case, wherethe goal is to characterize operators −L with real-valued coefficients whosequadratic form is nonnegative definite,(2.6) h−L h, h i ≥ , for all real-valued h ∈ C ∞ (Ω) . In this case the operator −L is called nonnegative definite .In the special case of Schr¨odinger operators(2.7) H u = div ( P ∇ u ) + σ u, with real-valued P ∈ D ′ (Ω) n × n and σ ∈ D ′ (Ω), a characterization of thisproperty was obtained earlier in [11, Proposition 5.1] under the assumptionthat P is uniformly elliptic, i.e.,(2.8) m || ξ || ≤ P ( x ) ξ · ξ ≤ M || ξ || , for all ξ ∈ R n , a.e. x ∈ Ω , with the ellipticity constants m > M < ∞ .An analogous characterization of (2.6) for more general operators whichinclude drift terms, L = div( P ∇· ) + ~b · ∇ + c , with real-valued coefficientsand P satisfying (2.8), is given in Proposition 4.2 below.For the general differential operator in the form (2.2), we define the sym-metric part A s , and co-symmetric (or skew-symmetric) part A c , respectively,by(2.9) A s = 12 ( A + A ⊥ ) , A c = 12 ( A − A ⊥ ) . Here A = ( a jk ) ∈ D ′ (Ω) n × n , and A ⊥ = ( a kj ) is the transposed matrix. CCRETIVITY AND FORM BOUNDEDNESS 7
For −L to be accretive, the matrix A s must have a nonnegative definitereal part: P = Re A s should satisfy(2.10) P ξ · ξ ≥ ξ ∈ R n , in D ′ (Ω) . Moreover, if the corresponding Schr¨odinger operator H is defined by (2.7)with P = Re A s , σ = Re c −
12 div (Re ~b ) , then −H must be nonnegative definite:(2.11) [ h ] H = h−H h, h i = h P ∇ h, ∇ h i − h σh, h i ≥ , for all real-valued (or complex-valued) h ∈ C ∞ (Ω).The rest of the accretivity problem for L (see Sec. 4.1) is reduced to thecommutator inequality(2.12) (cid:12)(cid:12)(cid:12) h ~d, u ∇ v − v ∇ u i (cid:12)(cid:12)(cid:12) ≤ [ u ] H [ v ] H , for all real-valued u, v ∈ C ∞ (Ω), where the real-valued vector field ~d is givenby(2.13) ~d = [Im ~b − Div(Im A c )] . As mentioned in the Introduction, under some mild restrictions on H , the“norms” [ u ] H and [ v ] H on the right-hand side of (2.12) can be replaced, upto a constant multiple, with the corresponding Dirichlet norms ||∇ · || L (Ω) .This leads to explicit criteria of accretivity, such as Theorem VI below inthe case Ω = R n . 3. Form boundedness
We start with a discussion of form boundedness for the general secondorder differential operator L in the form (2.3), where a ij , b i , and c are real- orcomplex-valued distributions, on the homogeneous Sobolev space L , ( R n ),and its inhomogeneous counterpart W , ( R n ), obtained in [26].In particular, this leads to criteria of the relative form boundedness of theoperator ~b · ∇ + q with distributional coefficients ~b and q with respect tothe Laplacian ∆ on L ( R n ). Invoking the so-called KLMN Theorem (see[4, Theorem IV.4.2]; [31, Theorem X.17]), we can then demonstrate that˜ L = ∆ + ~b · ∇ + q is well defined, under appropriate smallness assumptionson ~b and q , as an m-sectorial operator on L ( R n ). In this case, the quadraticform domain of ˜ L coincides with W , ( R n ).This yields a characterization of the relative form boundedness for themagnetic Schr¨odinger operator(3.1) M = ( i ∇ + ~a ) + q, with arbitrary vector potential ~a ∈ L ( R n ) n , and q ∈ D ′ ( R n ) on L ( R n )with respect to ∆ (see [26]). V. G. MAZ’YA AND I. E. VERBITSKY
Our approach is based on factorization of functions in Sobolev spacesand integral estimates of potentials of equilibrium measures, combined withcompensated compactness arguments, commutator estimates, and the ideaof gauge invariance. Moreover, an explicit Hodge decomposition is estab-lished for form bounded vector fields in R n . In this decomposition, theirrotational part of the vector field is subject to a stronger restriction thanits divergence-free counterpart.3.1. Form boundedness in the homogeneous Sobolev space.
As wasmentioned above, without loss of generality we may assume that the prin-cipal part of the differential operator is in the divergence form, i.e., L =div ( A ∇· ) + ~b · ∇ + q .We present necessary and sufficient conditions on A , ~b , and q , obtained in[26, Theorem I], which ensure the boundedness in the homogeneous Sobolevspace L , ( R n ) of the sesquilinear form associated with L :(3.2) |hL u, v i| ≤ C || u || L , ( R n ) || v || L , ( R n ) , where C does not depend on u, v ∈ C ∞ ( R n ), and || u || L , ( R n ) = ||∇ u || L ( R n ) . Theorem IV.
Let L = div ( A ∇· ) + ~b · ∇ + q , where A ∈ D ′ ( R n ) n × n , ~b ∈ D ′ ( R n ) n and q ∈ D ′ ( R n ) , n ≥ . Then the following statements hold. (i) The sesquilinear form of L is bounded, i.e., (3.2) holds if and only if A s ∈ L ∞ ( R n ) n × n , and ~b and q can be represented respectively in the form (3.3) ~b = ~c + Div F, q = div ~h, where F is a skew-symmetric matrix field such that (3.4) F − A c ∈ BMO( R n ) n × n , whereas ~c and ~h belong to L ( R n ) n , and obey the condition (3.5) | ~c | + | ~h | ∈ M , ( R n ) . (ii) If the sesquilinear form of L is bounded, then ~c , F , and ~h in decom-position (3.3) can be determined explicitly by ~c = ∇ (∆ − div ~b ) , ~h = ∇ (∆ − q ) , (3.6) F = ∆ − Curl [ ~b − Div ( A c )] + A c , (3.7) where (3.8) ∆ − Curl [ ~b − Div ( A c )] ∈ BMO( R n ) n × n , and (3.9) |∇ (∆ − div ~b ) | + |∇ (∆ − q ) | ∈ M , ( R n ) . We remark that condition (3.8) in statement (ii) of Theorem IV may bereplaced with(3.10) ~b − Div ( A c ) ∈ BMO − ( R n ) n , CCRETIVITY AND FORM BOUNDEDNESS 9 which ensures that decomposition (3.3) holds. Here BMO − ( R n ) stands forthe space of distributions that can be represented in the form f = div ~g where ~g ∈ BMO( R n ) n (see [16]).In the special case n = 2, it is easy to see that (3.2) holds if and only if A s ∈ L ∞ ( R ) × , ~b − Div ( A c ) ∈ BMO − ( R ) , and q = div ~b = 0.As mentioned in the Introduction, expressions ∇ (∆ − q ), ∇ (∆ − div ~b ),Div(∆ − Curl ~b ), which involve nonlocal operators, are defined in the senseof distributions. This is possible, since ∆ − q , ∆ − div ~b , and ∆ − Curl ~b can be understood in terms of the convergence in the weak- ∗ topology ofBMO( R n ) of ∆ − div ( ψ N ~b ), ∆ − Curl ( ψ N ~b ), and ∆ − ( ψ N q ), respectively,as N → + ∞ . Here ψ N ( x ) = ψ ( xN ) is a smooth cut-off function, where ψ issupported in the unit ball { x : | x | < } , and ψ ( x ) = 1 if | x | ≤ . The limitsabove do not depend on the choice of ψ .It follows from Theorem IV that L is form bounded on L , ( R n ) × L , ( R n )if and only if A s ∈ L ∞ ( R n ) n × n , and ~b · ∇ + q is form bounded, where(3.11) ~b = ~b − Div( A c ) . In particular, the principal part P u = div( A ∇ u ) is form bounded if andonly if A s ∈ L ∞ ( R n ) n × n , (3.12) Div ( A c ) ∈ BMO − ( R n ) n . (3.13)A simpler condition with A c ∈ BMO( R n ) n × n in place of (3.13) is sufficient,but generally is necessary only if n = 1 , L = ~b · ∇ + q, ~b ∈ D ′ ( R n ) n , q ∈ D ′ ( R n ) . As a corollary of Theorem IV, we deduce that, if ~b ·∇ + q is form bounded,then the Hodge decomposition(3.15) ~b = ∇ (∆ − div ~b ) + Div (∆ − Curl ~b )holds, where ∆ − (Curl ~b ) ∈ BMO( R n ) n × n , and(3.16) Z | x − y |
3; in two dimensions, it follows thatdiv ~b = q = 0.We observe that condition (3.16) is generally stronger than ∆ − div ~b ∈ BMO( R n ) and ∆ − q ∈ BMO( R n ), while the divergence-free part of ~b ischaracterized by ∆ − Curl ~b ∈ BMO( R n ) n × n , for all n ≥ q − div ~b and the divergence free partof ~b . To this effect, we use Theorem II, which characterizes vector fields ~d such that the commutator inequality (1.9) holds. Theorem II is proved in[26, Lemma 4.8] using the idea of the gauge transformation ([17, Sec. 7.19];[31, Sec. X.4]): ∇ → e − iλ ∇ e + iλ , where the gauge λ is a real-valued function in L , ( R n ).The nontrivial problem of choosing an appropriate gauge is solved in [26]as follows: λ = τ log ( N µ ) , < τ < nn − , where N µ = ( − ∆) − µ is the Newtonian potential of the equilibrium measure µ associated with an arbitrary compact set e of positive capacity.With this choice of λ , the energy space L , ( R n ) is gauge invariant, and forthe irrotational part ~c = ∇ (∆ − div ~d ) we have | ~c | ∈ M , ( R n ). In addition,we have F = ∆ − Curl ~d belongs to BMO( R n ) n × n , and ~d = ~c + Div F . Theseconditions are necessary and sufficient for (1.9).Applications of Theorem IV to the magnetic Schr¨odinger operator M defined by (3.1) are given in [26, Theorem 3.4], where it is shown that M isform bounded if and only if both q + | ~a | and ~a · ∇ are form bounded.3.2. Form boundedness in W , ( R n ) . The above results are easily ex-tended to the Sobolev space W , ( R n ) ( n ≥
1) with norm || u || W , ( R n ) = ||∇ u || L ( R n ) + || u || L ( R n ) .In particular, necessary and sufficient conditions are given in [26, Theorem5.1] for the boundedness of the general second order operator L : W , ( R n ) → W − , ( R n ) . This solves the relative form boundedness problem for L , and consequentlyfor the magnetic Schr¨odinger operator M , with respect to the Laplacianon L ( R n ) (see [31, Sec. X.2]). The proofs make use of an inhomogeneousversion of the div-curl lemma ([26, Lemma 5.2]).3.3. Infinitesimal form boundedness.
Other fundamental properties ofquadratic forms associated with differential operators can be characterizedusing our methods. In particular, for the Schr¨odinger operator H = ∆ + q with q ∈ D ′ ( R n ), criteria of relative compactness were obtained in [22],whereas the infinitesimal form boundedness expressed by the inequality(3.17) |h q u, u i| ≤ ǫ ||∇ u || L ( R n ) + C ( ǫ ) || u || L ( R n ) , u ∈ C ∞ ( R n ) , for every ǫ ∈ (0 , C ( ǫ ) is a positive constant, along with Trudinger’ssubordination where C ( ǫ ) = C ǫ − β ( β > CCRETIVITY AND FORM BOUNDEDNESS 11
Nash’s inequality and p -subordination. For q ∈ D ′ ( R n ), we con-sider the p -subordination property(3.18) |h q u, u i| ≤ C ||∇ u || pL ( R n ) || u || − p ) L ( R n ) , for all u ∈ C ∞ ( R n ), where p ∈ (0 , || u || L ( R n ) in place of || u || L ( R n ) on the right-hand side,(3.19) |h q u, u i| ≤ C ||∇ u || pL ( R n ) || u || − p ) L ( R n ) . The classical Nash’s inequality corresponds to q ≡ p = nn +2 (see [17,Theorem 8.13].It is proved in [25, Theorem 6.5] that (3.18) holds if and only if q = div ~ Γ , where ~ Γ = ∇ ∆ − q , and one of the following conditions hold: ~ Γ ∈ BMO if p = 1 / ~ Γ ∈ Lip(1 − p ) if 0 < p < / Z | x − y |
We now turn to the accretivity problem for −L , where L is a second orderlinear differential operator with complex-valued distributional coefficientsdefined by (2.3) in an open set Ω ⊆ R n ( n ≥ General accretivity criterion.
Given A = ( a jk ) ∈ D ′ (Ω) n × n , wedefine its symmetric part A s and skew-symmetric part A c respectively by(2.9). The accretivity property for −L can be characterized in terms of thefollowing real-valued expressions:(4.1) P = Re A s , ~d = [Im ~b − Div (Im A c )] , σ = Re c − div (Re ~b ) , where P = ( p jk ) ∈ D ′ (Ω) n × n , ~d = ( d j ) ∈ D ′ (Ω) n , and σ ∈ D ′ (Ω). This is aconsequence of the relation (see [27, Sec.4])(4.2) Re h−L u, u i = Re h−L u, u i , u ∈ C ∞ (Ω) , where(4.3) L = div ( P ∇· ) + 2 i ~d · ∇ + σ. Moreover, in order that −L be accretive, the matrix P must be nonneg-ative definite, i.e., P ξ · ξ ≥ D ′ (Ω) for all ξ ∈ R n . In particular, each p jj ( j = 1 , . . . , n ) is a nonnegative Radon measure.A characterization of accretive operators −L is given in the followingcriterion obtained in [27, Proposition 2.1]. Proposition 4.1.
Let L = div( A ∇· ) + ~b · ∇ + c , where A ∈ D ′ (Ω) n × n , ~b ∈ D ′ (Ω) n and c ∈ D ′ (Ω) are complex-valued. Suppose that P , ~d , and σ are defined by (4.1) .The operator −L is accretive if and only if P is a nonnegative definitematrix, and the following two conditions hold: (4.4) [ h ] H = h P ∇ h, ∇ h i − h σ h, h i ≥ , for all real-valued h ∈ C ∞ (Ω) , and (4.5) (cid:12)(cid:12)(cid:12) h ~d, u ∇ v − v ∇ u i (cid:12)(cid:12)(cid:12) ≤ [ u ] H [ v ] H , for all real-valued u, v ∈ C ∞ (Ω) . Real-valued coefficients.
It follows from Proposition 4.1 that, foroperators with real-valued coefficients, condition (4.4) alone characterizesnonnegative definite operators −L in an open set Ω ⊆ R n ( n ≥ P = A s ∈ L (Ω) n × n in the sufficiency part, and that P is uniformly ellipticin the necessity part, is given in the next proposition (see [27, Theorem 2.2]). Proposition 4.2.
Let L = div( A ∇· ) + ~b · ∇ + c , where A ∈ D ′ (Ω) n × n , ~b ∈ D ′ (Ω) n and c ∈ D ′ (Ω) are real-valued. Suppose that P = A s ∈ L (Ω) n × n is a nonnegative definite matrix a.e. (i) If there exists a measurable vector field ~g in Ω such that ( P ~g ) · ~g ∈ L (Ω) , and (4.6) σ = c − div ( ~b ) ≤ div ( P ~g ) − ( P ~g ) · ~g in D ′ (Ω) , then the operator −L is nonnegative definite. (ii) Conversely, if −L is nonnegative definite, then there exists a vectorfield ~g ∈ L (Ω) n so that ( P ~g ) · ~g ∈ L (Ω) , and (4.6) holds, provided P isuniformly elliptic. The uniform ellipticity condition on P in statement (ii) of Proposition 4.2can be relaxed. This question will be treated elsewhere.Results similar to Proposition 4.2 are well known in ordinary differentialequations [9, Sec. XI.7], in relation to disconjugate Sturm-Liouville equa-tions and Riccati equations with continuous coefficients (see also [8], [23],[27]). CCRETIVITY AND FORM BOUNDEDNESS 13
Nonnegative definite Schr¨odinger operators.
As was mentionedabove, in the special case of Schr¨odinger operators H = div ( P ∇ h ) + σ ,with real-valued σ ∈ D ′ (Ω) and uniformly elliptic P , Proposition 4.2 wasobtained originally in [11, Proposition 5.1]. Under these assumptions, −H is nonnegative definite, i.e.,[ h ] H = h−H h, h i ≥ , for all h ∈ C ∞ (Ω) , if and only if there exists a vector field ~g ∈ L (Ω) n such that(4.7) σ ≤ div ( P ~g ) − P ~g · ~g in D ′ (Ω) . A simpler linear sufficient condition for −H to be nonnegative definite isgiven by σ ≤ div ( P ~g ) , where ~g ∈ L (Ω) n satisfies the inequality Z Ω ( P ~g · ~g ) h dx ≤ Z Ω | P ∇ h | dx, for all h ∈ C ∞ (Ω) . Here
P ~g · ~g ∈ M , (Ω), and so | ~g | is admissible if P is uniformly elliptic.However, such conditions are not necessary, with any constant in place of ,even when P = I ; see [11, Proposition 7.1].We observe that in Proposition 4.1 above, the nonnegative definite qua-dratic form [ h ] H is associated with the Schr¨odinger operator −H , where H has real-valued coefficients P = Re A s and σ = Re c − div (Re ~b ). Hence,(4.7) characterizes the first condition of Proposition 4.1 given by (4.4). Thesecond one, namely, the commutator condition (4.5), will be discussed fur-ther in Sections 4.5 and 4.6.4.4. The one-dimensional case.
In this section, the differential operator L u = ( a u ′ ) ′ + bu ′ + c is defined on an open interval I ⊆ R (possibly un-bounded). In this case, one can avoid commutator estimates using methodsof ordinary differential equations ([9], [10]). In the statements below wewill make use of the standard convention = 0. The following criterion ofaccretivity for complex-valued coefficients in the one-dimensional case wasobtained in [27, Theorem 2.2]. Theorem V.
Let a, b, c ∈ D ′ ( I ) . Suppose that p = Re a ∈ L ( I ) , and Im b ∈ L ( I ) . (i) The operator −L is accretive if and only if (Im b ) p ∈ L ( I ) , where p ≥ a.e., and the following quadratic form inequality holds: (4.8) Z I p ( h ′ ) dx − h Re c −
12 (Re b ) ′ , h i − Z I (Im b ) p h dx ≥ , for all real-valued h ∈ C ∞ ( I ) . (ii) If there exists a function f ∈ L ( I ) such that f p ∈ L ( I ) , and (4.9) Re c −
12 (Re b ) ′ − (Im b ) p ≤ f ′ − f p in D ′ ( I ) , then the operator −L is accretive. Conversely, if −L is accretive, and m ≤ p ( x ) ≤ M a.e. for some con-stants M, m > , then there exists a function f ∈ L ( I ) such that (4.9) holds. We remark that in Theorem V, the terms Im a and Im c play no role, butthe behavior of Im b is essential. In higher dimensions, the situation is evenmore complicated. The term Im b may contain both the irrotational anddivergence-free components, and the latter may interact with Im A c .4.5. Upper and lower bounds of quadratic forms.
For general oper-ators with complex-valued coefficients in the case n ≥
2, we recall thatthe first condition of Proposition 4.1 is necessary for the accretivity of −L ,namely,(4.10) h σ h, h i ≤ Z Ω ( P ∇ h · ∇ h ) dx, for all real-valued h ∈ C ∞ (Ω), where σ = Re c − div(Re ~b ) ∈ D ′ (Ω), andRe A s = P ∈ D ′ (Ω) n × n is a nonnegative definite matrix.Suppose now that σ has a slightly smaller upper form bound, that is,(4.11) h σ h, h i ≤ (1 − ǫ ) Z Ω ( P ∇ h · ∇ h ) dx, h ∈ C ∞ (Ω) , for some ǫ ∈ (0 , h σ h, h i ≥ − K Z Ω ( P ∇ h · ∇ h ) dx, h ∈ C ∞ (Ω) , for some constant K ≥ σ ∈ D ′ (Ω) were invoked in [11, Theorem1.1], for uniformly elliptic P .We observe that (4.11) is satisfied for any ǫ ∈ (0 , C || h || L (Ω) , if σ is infinitesimally form bounded (see Sec. 3.3). The secondterm on the right is sometimes included in the definition of accretivity of theoperator −L . We can always incorporate it as a constant term in σ − C ( ǫ ).The same is true with regards to the lower bound where we can use σ + C ( ǫ ).Assuming that both bounds (4.11) and (4.12) hold for some ǫ ∈ (0 ,
1] and K ≥
0, we obviously have, for all h ∈ C ∞ (Ω),(4.13) ǫ Z Ω ( P ∇ h · ∇ h ) dx ≤ [ h ] H ≤ ( K + 1) Z Ω ( P ∇ h · ∇ h ) dx. If P satisfies the uniform ellipticity assumptions (2.8), then from (4.13)it follows that condition (4.5) equivalent, up to a constant multiple, to(4.14) (cid:12)(cid:12)(cid:12) h ~d, u ∇ v − v ∇ u i (cid:12)(cid:12)(cid:12) ≤ C ||∇ u || L (Ω) ||∇ v || L (Ω) where C > u, v ∈ C ∞ (Ω). For Ω = R n and ~d ∈ L ( R n ), see Theorem II above.In the case Ω = R n , inequality (4.14) was characterized completely in[26, Lemma 4.8] for complex-valued u, v . However, that characterization CCRETIVITY AND FORM BOUNDEDNESS 15 obviously works in the case of real-valued u, v as well (one only needs tochange the constant C up to a factor of √ Accretivity criterion in R n . Combining the characterization of thecommutator inequality (4.14) with Proposition 4.1 yields the following ac-cretivity criterion ([27, Theorem 2.7]), where the lower bound (4.12) in usedthe necessity part, whereas the upper bound (4.11) is invoked in the suffi-ciency part.
Theorem VI.
Let L be the second order differential operator (2.3) on R n ( n ≥ with complex-valued coefficients A ∈ D ′ ( R n ) n × n , ~b ∈ D ′ ( R n ) n and c ∈ D ′ ( R n ) . Let P , ~d and σ be defined by (4.1) , where P is uniformlyelliptic. (i) Suppose that −L is accretive, i.e., (2.5) holds, and σ satisfies (4.12) for some K ≥ . Then ~d can be represented in the form (4.15) ~d = ∇ f + Div G, where f ∈ D ′ ( R n ) is real-valued, |∇ f | ∈ M , ( R n ) , and G ∈ BMO( R n ) n × n is a real-valued skew-symmetric matrix field.Moreover, f and G above can be defined explicitly as (4.16) f = ∆ − (div ~d ) , G = ∆ − (Curl ~d ) . (ii) Conversely, suppose that σ satisfies (4.11) with some ǫ ∈ (0 , . Then −L is accretive if representation (4.15) holds, where |∇ f | ∈ M , ( R n ) , and G ∈ BMO( R n ) n × n is a real-valued skew-symmetric matrix field, providedboth k|∇ f | k M , ( R n ) and the BMO -norm of G are small enough, dependingonly on ǫ . If n = 2, then in Theorem VI, we have f = 0, and ~d = ( − ∂ g, ∂ g ) with g ∈ BMO( R ). In statement (ii), the BMO-norm of g is supposed to besmall enough (depending only on ǫ ).If n = 3, one can use the usual vector-valued curl( ~g ) ∈ D ′ ( R ) in place ofDiv G in decomposition (4.15), with ~g = ∆ − (curl ~d ) in place of G in (4.16). References [1] D. R. Adams and L. I. Hedberg,
Function spaces and potential theory , Grundlehrender math. Wissenschaften , Springer-Verlag, Berlin–Heibelberg–New York, 1996.[2] A. Ancona,
On strong barriers and an inequality of Hardy for domains in R n , J.London Math. Soc. (1986), 274–290.[3] R. Coifman, P. L. Lions, Y. Meyer, and S. Semmes, Compensated compactness andHardy spaces , J. Math. Pures Appl. (1993), 247–286.[4] D. E. Edmunds and W. D. Evans, Spectral theory and differential operators , ClarendonPress, Oxford, 1987.[5] M. Frazier, F. Nazarov, and I. Verbitsky,
Global estimates for kernels of Neumannseries and Green’s functions , J. London Math. Soc. (2014), 903–918.[6] F. Giannetti, L. Greco, and G. Moscariello, Linear elliptic equations with lower orderterms , Diff. Int. Eqs. (2013), 623–638. [7] C. Guevara and N. C. Phuc, Leray’s self-similar solutions to the Navier-Stokes equa-tions with profiles in Marcinkiewicz and Morrey spaces , SIAM J. Math. Analysis (2018), 541–556.[8] T. Hara, A refined subsolution estimate of weak subsolutions to second-order linearelliptic equations with a singular vector field , Tokyo J. Math. (2015), 75–98.[9] P. Hartman, Ordinary differential equations , second ed., Classics in Appl. Math. ,SIAM, Philadelphia, PA, 2002.[10] E. Hille, Non-oscillation theorems , Trans. Amer. Math. Soc. (1948), 234–252.[11] B. J. Jaye, V. G. Maz’ya and I. E. Verbitsky, Existence and regularity of positivesolutions of elliptic equations of Schr¨odinger type , J. d’Analyse Math. (2012),577–621.[12] B. J. Jaye, V. G. Maz’ya and I. E. Verbitsky,
Quasilinear elliptic equations andweighted Sobolev-Poincar´e inequalities with distributional weights , Adv. Math. (2013), 513–542.[13] C. E. Kenig,
Harmonic analysis techniques for second order elliptic boundary valueproblems , CBMS Regional Conference Ser. Math. , Amer. Math. Soc., Providence,RI, 1994.[14] C. E. Kenig and J. Pipher, The Dirichlet problem for elliptic equations with driftterms , Publ. Math. (2001), 199–217.[15] R. Kerman and E. Sawyer, The trace inequality and eigenvalue estimates forSchr¨odinger operators , Ann. Inst. Fourier, Grenoble (1987), 207–228.[16] H. Koch and D. Tataru, Well-posedness for the Navier-Stokes equations , Adv. Math. (2001), 22–35.[17] E. H. Lieb and M. Loss,
Analysis , second ed., Amer. Math. Soc., Providence, RI,2001.[18] V. A. Liskevich, M. A. Perelmuter, and Yu. A. Semenov,
Form-bounded perturbationsof generators of sub-Markovian semigroups , Acta Appl. Math. (1996), 353–377.[19] V. G. Maz’ya, The negative spectrum of the higher-dimensional Schr¨odinger operator ,Sov. Math. Dokl. (1962), 808–810.[20] V. G. Maz’ya, On the theory of the higher-dimensional Schr¨odinger operator (Rus-sian), Izv. Akad. Nauk SSSR Ser. Mat. (1964), 1145–1172.[21] V. Maz’ya, Sobolev spaces, with applications to elliptic partial differential equations ,second augmented ed., Grundlehren der math. Wissenschaften , Berlin–New York,Springer, 2011.[22] V. G. Maz’ya and I. E. Verbitsky,
The Schr¨odinger operator on the energy space:boundedness and compactness criteria , Acta Math. (2002), 263–302.[23] V. G. Maz’ya and I. E. Verbitsky,
Boundedness and compactness criteria for theone-dimensional Schr¨odinger operator , In: Function spaces, interpolation theory andrelated topics, Proc. Jaak Peetre conf., Lund, Sweden, August 17–22, 2000, eds. M.Cwikel, A. Kufner, and G. Sparr, De Gruyter, Berlin, 2002, 369–382.[24] V. G. Maz’ya and I. E. Verbitsky,
The form boundedness criterion for the relativisticSchr¨odinger operator , Ann. Inst. Fourier (2004), 317–339.[25] V. G. Maz’ya and I. E. Verbitsky, Infinitesimal form boundedness and Trudinger’ssubordination for the Schr¨odinger operator , Invent. Math. (2005), 81–136.[26] V. G. Maz’ya and I. E. Verbitsky,
Form boundedness of the general second orderdifferential operator , Commun. Pure Appl. Math. (2006), 1286–1329.[27] V. G. Maz’ya and I. E. Verbitsky, Accretivity of the general second order linear dif-ferential operator , Acta Math. Sinica, English Ser. (2019) 832–852.[28] A. I. Nazarov and N. N. Ural’tseva, The Harnack inequality and related propertiesof solutions of elliptic and parabolic equations with divergence-free lower-order coeffi-cientts , St. Petersburg Math. J. (2012), 93–115.[29] T. Phan, Regularity gradient estimates for weak solutions of singular quasi-linearparabolic equations , J. Diff. Eq. (2017), 8329–8361.
CCRETIVITY AND FORM BOUNDEDNESS 17 [30] S. Quinn and I. E. Verbitsky,
A sublinear version of Schur’s lemma and elliptic PDE ,Analysis & PDE (2018), 439–466.[31] M. Reed and B. Simon, Methods of modern mathematical physics II: Fourier analysis,self-adjointness , Academic Press, New York–London, 1975.[32] G. V. Rozenblum, M. A. Shubin, and M. Z. Solomyak,
Spectral Theory of DifferentialOperators , Encyclopaedia of Math. Sci. , Partial Differential Equations VII, ed.M.A. Shubin, Springer-Verlag, Berlin–Heidelberg, 1994.[33] G. Seregin, L. Silvestre, V. Sverak, and A. Zlatos, On divergence-free drifts , J. Diff.Eq. (2011), 505–540.[34] E. M. Stein,
Harmonic analysis: real–variable methods, orthogonality, and oscillatoryintegrals , Princeton Math. Ser. , Monographs in Harmonic Analysis, PrincetonUniversity Press, Princeton, NJ, 1993.[35] R. Temam, Navier-Stokes equations, theory and numerical analysis , third ed., Studiesin Math. Appl. , North-Holland, Amsterdam, 1984.[36] I. E. Verbitsky, Nonlinear potentials and trace inequalities , The Maz’ya AnniversaryCollection, eds. J. Rossmann, P. Tak´ac, and G. Wildenhain, Operator Theory: Adv.Appl. (1999), 323–343.[37] V. V. Zhikov and S. E. Pastukhova,
On operator estimates in homogenization theory ,Russian Math. Surveys (2016), 417–511. Department of Mathematics, Link¨oping University, SE-581 83, Link¨oping,Sweden and RUDN University, 6 Miklukho-Maklay St., Moscow, 117198, Rus-sia
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