L coefficient operators and non-smooth Riemannian metrics

L coefficient operators and non-smooth Riemannian metrics Lashi Bandara Centre for Mathematics and its Applications Australian National University 17 October 2012 Geometric Analysis Seminar Stanford University Lashi Bandara (ANU) L operators and non-smooth metrics 17 October 2012 1 / 23

Setup Let M be a smooth, complete Riemannian manifold with metric g, Levi-Civita connection, and volume measure dµ. Lashi Bandara (ANU) L operators and non-smooth metrics 17 October 2012 2 / 23

Setup Let M be a smooth, complete Riemannian manifold with metric g, Levi-Civita connection, and volume measure dµ. Write div = in L 2 and let S = (I, ). Lashi Bandara (ANU) L operators and non-smooth metrics 17 October 2012 2 / 23

Setup Let M be a smooth, complete Riemannian manifold with metric g, Levi-Civita connection, and volume measure dµ. Write div = in L 2 and let S = (I, ). Consider the following uniformly elliptic second order differential operator L A : D(L A ) L 2 (M) L 2 (M) defined by L A u = as ASu = a div(a 11 u) a div(a 10 u) + aa 01 u + aa 00 u. where a and A = (A ij ) are L multiplication operators. Lashi Bandara (ANU) L operators and non-smooth metrics 17 October 2012 2 / 23

Setup Let M be a smooth, complete Riemannian manifold with metric g, Levi-Civita connection, and volume measure dµ. Write div = in L 2 and let S = (I, ). Consider the following uniformly elliptic second order differential operator L A : D(L A ) L 2 (M) L 2 (M) defined by L A u = as ASu = a div(a 11 u) a div(a 10 u) + aa 01 u + aa 00 u. where a and A = (A ij ) are L multiplication operators. That is, that there exist κ 1, κ 2 > 0 such that Re av, v κ 1 v 2, v L 2 Re ASu, Su κ 2 ( u 2 + u 2 ), u H 1 Lashi Bandara (ANU) L operators and non-smooth metrics 17 October 2012 2 / 23

The problem The Kato square root problem on manifolds is to determine when the following holds: Lashi Bandara (ANU) L operators and non-smooth metrics 17 October 2012 3 / 23

The problem The Kato square root problem on manifolds is to determine when the following holds: { D( LA ) = H 1 (M) L A u u + u = u H 1, u H 1 (M) Lashi Bandara (ANU) L operators and non-smooth metrics 17 October 2012 3 / 23

Theorem (B.-Mc [BMc]) Let (M, g) be a smooth, complete Riemannian manifold Ric C and inj(m) κ > 0. Suppose there exist κ 1, κ 2 > 0 such that Re av, v κ 1 v 2 Re ASu, Su κ 2 u 2 H 1 for v L 2 (M) and u H 1 (M). Then, D( L A ) = D( ) = H 1 (M) and L A u u + u = u H 1 for all u H 1 (M). Lashi Bandara (ANU) L operators and non-smooth metrics 17 October 2012 4 / 23

Stability Theorem (B.-Mc [BMc]) Let (M, g) be a smooth, complete Riemannian manifold with Ric C and inj(m) κ > 0. Suppose that there exist κ 1, κ 2 > 0 such that Re av, v κ 1 v 2 Re ASu, Su κ 2 u 2 H 1 for v L 2 (M) and u H 1 (M). Then for every η i < κ i, whenever ã η 1, Ã η 2, the estimate L A u L A+ Ã u ( ã + Ã ) u H 1 holds for all u H 1 (M). The implicit constant depends in particular on A, a and η i. Lashi Bandara (ANU) L operators and non-smooth metrics 17 October 2012 5 / 23

History of the problem In the 1960 s, Kato considered the following abstract evolution equation du dt on a Hilbert space H. + A(t)u = f(t), t [0, T ]. Lashi Bandara (ANU) L operators and non-smooth metrics 17 October 2012 6 / 23

History of the problem In the 1960 s, Kato considered the following abstract evolution equation du dt on a Hilbert space H. + A(t)u = f(t), t [0, T ]. In 1962, Kato showed in [Kato61] that for 0 α < 1/2 and A(t) maximal accretive that D(A(t) α ) = D(A(t) α ) = D α = const, and A(t) α u A(t) α u, u D α. (K α ) Lashi Bandara (ANU) L operators and non-smooth metrics 17 October 2012 6 / 23

History of the problem In the 1960 s, Kato considered the following abstract evolution equation du dt on a Hilbert space H. + A(t)u = f(t), t [0, T ]. In 1962, Kato showed in [Kato61] that for 0 α < 1/2 and A(t) maximal accretive that D(A(t) α ) = D(A(t) α ) = D α = const, and A(t) α u A(t) α u, u D α. (K α ) Counter examples were known for α > 1/2. Lashi Bandara (ANU) L operators and non-smooth metrics 17 October 2012 6 / 23

Kato asked two questions: (K1) Does (K α ) hold for α = 1/2? Lashi Bandara (ANU) L operators and non-smooth metrics 17 October 2012 7 / 23

Kato asked two questions: (K1) Does (K α ) hold for α = 1/2? (K2) For the case ω = 0, we know (K1) is automatically true, but is d A(t)u u dt for u W? Lashi Bandara (ANU) L operators and non-smooth metrics 17 October 2012 7 / 23

Kato asked two questions: (K1) Does (K α ) hold for α = 1/2? (K2) For the case ω = 0, we know (K1) is automatically true, but is d A(t)u u dt for u W? In 1972, McIntosh provided a counter example in [Mc72] demonstrating that (K1) is false in such generality. Lashi Bandara (ANU) L operators and non-smooth metrics 17 October 2012 7 / 23

Kato asked two questions: (K1) Does (K α ) hold for α = 1/2? (K2) For the case ω = 0, we know (K1) is automatically true, but is d A(t)u u dt for u W? In 1972, McIntosh provided a counter example in [Mc72] demonstrating that (K1) is false in such generality. In 1982, McIntosh showed that (K2) also did not hold in general in [Mc82]. Lashi Bandara (ANU) L operators and non-smooth metrics 17 October 2012 7 / 23

The Kato square root problem then became the following. Lashi Bandara (ANU) L operators and non-smooth metrics 17 October 2012 8 / 23

The Kato square root problem then became the following. Suppose A L is a pointwise matrix multiplication operator satisfying the following ellipticity condition: Re A u, u κ u 2, for some κ > 0. Is it then true that D( div A ) = H 1 (R n ) div A u u (K1) This was answered in the positive in 2002 by Pascal Auscher, Steve Hofmann, Michael Lacey, Alan McIntosh and Phillipe Tchamitchian in [AHLMcT]. Lashi Bandara (ANU) L operators and non-smooth metrics 17 October 2012 8 / 23

Axelsson (Rosén)-Keith-McIntosh framework In [AKMc], the authors created a first-order framework using the language of Dirac type operators to study such problems. Lashi Bandara (ANU) L operators and non-smooth metrics 17 October 2012 9 / 23

Axelsson (Rosén)-Keith-McIntosh framework In [AKMc], the authors created a first-order framework using the language of Dirac type operators to study such problems. (H1) Let Γ be a densely-defined, closed, nilpotent operator on a Hilbert space H, Lashi Bandara (ANU) L operators and non-smooth metrics 17 October 2012 9 / 23

Axelsson (Rosén)-Keith-McIntosh framework In [AKMc], the authors created a first-order framework using the language of Dirac type operators to study such problems. (H1) Let Γ be a densely-defined, closed, nilpotent operator on a Hilbert space H, (H2) Suppose that B 1, B 2 L(H ) such that here exist κ 1, κ 2 > 0 satisfying Re B 1 u, u κ 1 u 2 and Re B 2 v, v κ 2 v 2 for u R(Γ ) and v R(Γ), Lashi Bandara (ANU) L operators and non-smooth metrics 17 October 2012 9 / 23

Axelsson (Rosén)-Keith-McIntosh framework In [AKMc], the authors created a first-order framework using the language of Dirac type operators to study such problems. (H1) Let Γ be a densely-defined, closed, nilpotent operator on a Hilbert space H, (H2) Suppose that B 1, B 2 L(H ) such that here exist κ 1, κ 2 > 0 satisfying Re B 1 u, u κ 1 u 2 and Re B 2 v, v κ 2 v 2 for u R(Γ ) and v R(Γ), (H3) The operators B 1, B 2 satisfy B 1 B 2 R(Γ) N (Γ) and B 2 B 1 R(Γ ) N (Γ ). Lashi Bandara (ANU) L operators and non-smooth metrics 17 October 2012 9 / 23

Axelsson (Rosén)-Keith-McIntosh framework In [AKMc], the authors created a first-order framework using the language of Dirac type operators to study such problems. (H1) Let Γ be a densely-defined, closed, nilpotent operator on a Hilbert space H, (H2) Suppose that B 1, B 2 L(H ) such that here exist κ 1, κ 2 > 0 satisfying Re B 1 u, u κ 1 u 2 and Re B 2 v, v κ 2 v 2 for u R(Γ ) and v R(Γ), (H3) The operators B 1, B 2 satisfy B 1 B 2 R(Γ) N (Γ) and B 2 B 1 R(Γ ) N (Γ ). Let Γ B = B 1Γ B 2, Π B = Γ + Γ B and Π = Γ + Γ. Lashi Bandara (ANU) L operators and non-smooth metrics 17 October 2012 9 / 23

Quadratic estimates and Kato type problems Proposition If (H1)-(H3) are satisfied and ˆ 0 tπ B (I + t 2 Π 2 B) 1 u 2 dt t u for u R(Π B ), then (i) D(Γ) D(Γ B ) = D(Π B) = D( Π 2 B ), and (ii) Γu + Γ B u Π B u Π 2 B u, for all u D(Π B). Lashi Bandara (ANU) L operators and non-smooth metrics 17 October 2012 10 / 23

Set H = L 2 (M) (L 2 (M) L 2 (T M)). Lashi Bandara (ANU) L operators and non-smooth metrics 17 October 2012 11 / 23

Set H = L 2 (M) (L 2 (M) L 2 (T M)). Define Γ g = [ ] 0 0, B S 0 1 = [ ] a 0, and B 0 0 2 = [ ] 0 0. 0 A Lashi Bandara (ANU) L operators and non-smooth metrics 17 October 2012 11 / 23

Set H = L 2 (M) (L 2 (M) L 2 (T M)). Define Γ g = [ ] 0 0, B S 0 1 = [ ] a 0, and B 0 0 2 = [ ] 0 0. 0 A Then, [ ] Γ 0 S g = 0 0 and Π 2 B = [ ] LA 0 0 Lashi Bandara (ANU) L operators and non-smooth metrics 17 October 2012 11 / 23

Set H = L 2 (M) (L 2 (M) L 2 (T M)). Define Γ g = [ ] 0 0, B S 0 1 = [ ] a 0, and B 0 0 2 = [ ] 0 0. 0 A Then, [ ] Γ 0 S g = 0 0 and Π 2 B = [ ] LA 0 0 and Π B,g (u, 0) = (0, u, u) and Π 2 B,g (u, 0) = ( L A u, 0). Lashi Bandara (ANU) L operators and non-smooth metrics 17 October 2012 11 / 23

Theorem 1 and 2 are a direct consequence of the following. Lashi Bandara (ANU) L operators and non-smooth metrics 17 October 2012 12 / 23

Theorem 1 and 2 are a direct consequence of the following. Proposition Let M be a smooth, complete manifold with smooth metric g. Suppose there exist η, κ > 0 such that Ric g η and inj(m, g) κ. Furthermore, suppose that there exist κ 1, κ 2 such that Re av, v κ 1 v 2, v L 2 (M, g) Re ASu, Su κ 2 u 2 H 1, u H1 (M, g). Then, for u R(Π B ). ˆ 0 tπ B (I + t 2 Π 2 B) 1 u 2 dt t u 2 Lashi Bandara (ANU) L operators and non-smooth metrics 17 October 2012 12 / 23

Rough metrics We let M be a smooth, complete manifold as before but now let g be a C 0 metric. Let µ g denote the volume measure with respect to g. Lashi Bandara (ANU) L operators and non-smooth metrics 17 October 2012 13 / 23

Rough metrics We let M be a smooth, complete manifold as before but now let g be a C 0 metric. Let µ g denote the volume measure with respect to g. Definition Let h C 0 (T (2,0) M). Then, define h op,g = esssup µg(x) a.e. sup h x (u, v), u g = v g =1 and h op,s,g = sup x M sup h x (u, v). u g = v g =1 Lashi Bandara (ANU) L operators and non-smooth metrics 17 October 2012 13 / 23

Rough metrics We let M be a smooth, complete manifold as before but now let g be a C 0 metric. Let µ g denote the volume measure with respect to g. Definition Let h C 0 (T (2,0) M). Then, define h op,g = esssup µg(x) a.e. sup h x (u, v), u g = v g =1 and h op,s,g = sup x M sup h x (u, v). u g = v g =1 We say that two C 0 metrics g and g are δ-close if g g op,g < δ for δ > 0. Lashi Bandara (ANU) L operators and non-smooth metrics 17 October 2012 13 / 23

Translating from one metric to another Suppose that we now have a smooth metric g. Lashi Bandara (ANU) L operators and non-smooth metrics 17 October 2012 14 / 23

Translating from one metric to another Suppose that we now have a smooth metric g. Write f u (v) = g(u, v) g(u, v). Lashi Bandara (ANU) L operators and non-smooth metrics 17 October 2012 14 / 23

Translating from one metric to another Suppose that we now have a smooth metric g. Write f u (v) = g(u, v) g(u, v). By Riesz-Representation theorem, we find that there exists B such that g((i + B)u, v) = g(u, v). That is, exactly, we can absorb the lack of regularity of g in terms of a symmetric, bounded operator in g. Lashi Bandara (ANU) L operators and non-smooth metrics 17 October 2012 14 / 23

Translating from one metric to another Suppose that we now have a smooth metric g. Write f u (v) = g(u, v) g(u, v). By Riesz-Representation theorem, we find that there exists B such that g((i + B)u, v) = g(u, v). That is, exactly, we can absorb the lack of regularity of g in terms of a symmetric, bounded operator in g. The operator B = g g op,g. Lashi Bandara (ANU) L operators and non-smooth metrics 17 October 2012 14 / 23

Translating from one metric to another Suppose that we now have a smooth metric g. Write f u (v) = g(u, v) g(u, v). By Riesz-Representation theorem, we find that there exists B such that g((i + B)u, v) = g(u, v). That is, exactly, we can absorb the lack of regularity of g in terms of a symmetric, bounded operator in g. The operator B = g g op,g. The change of measures is given in terms of I + B. θ = dµ g dµ g Lashi Bandara (ANU) L operators and non-smooth metrics 17 October 2012 14 / 23

Stability of function spaces Proposition Let g and g be two C 0 metrics on M. If there exists δ [0, 1) such that g g op,g < δ, then (i) the spaces L 2 (M, g) = L 2 (M, g) with (1 δ) n 4 2,g 2, g (1 + δ) n 4 2,g, (ii) the Sobolev spaces H 1 (M, g) = H 1 (M, g) with (1 δ) n 4 1 + δ H 1,g H 1, g (1 + δ) n 4 1 δ H 1,g. Lashi Bandara (ANU) L operators and non-smooth metrics 17 October 2012 15 / 23

Π B under a change of metric The operator Γ g does not change under the change of metric. Lashi Bandara (ANU) L operators and non-smooth metrics 17 October 2012 16 / 23

Π B under a change of metric The operator Γ g does not change under the change of metric. However, Proposition Γ g = ΘΓ g C where Θ is a bounded multiplication operator on L2 (M) and C is a bounded multiplication operator on L 2 (M) L 2 (T M). Thus, Π B,g = Γ g + B 1 Γ gb 2 = Γ g + B 1 ΘΓ gcb 2. The idea is to reduce the Kato problem for Π B,g to a Kato problem for Π B, g = Γ g + B 1 Γ g B 2 where B 1 = B 1 Θ and B 2 = CB 2 but now with a smooth metric g. Lashi Bandara (ANU) L operators and non-smooth metrics 17 October 2012 16 / 23

Ellipticity Let us define ellipticity of L A with respect to a metric h by Re av, v h κ 1,h v 2 2,h, v L2 (M) Re ASu, Su h κ 2,h u 2 H 1,h, u H1 (M). (E) Lashi Bandara (ANU) L operators and non-smooth metrics 17 October 2012 17 / 23

Ellipticity (cont.) Then the loss in the ellipticity constants by transferring from one operator to another is as follows. Proposition If g g op,g δ < 1, then assuming (E) with respect to g implies (E) with respect to g with an appropriate change of the coefficients a and A with ellipticity constants κ 1, g = κ 1,g (1 + δ) n 2 (1 δ) and κ 2, g = κ 2,g. (1 + δ) n 2 Lashi Bandara (ANU) L operators and non-smooth metrics 17 October 2012 18 / 23

Theorem (Reduction of the rough problem to the smooth) Let M be a smooth manifold with a C 0 metric g and suppose that the ellipticity condition (E) is satisfied for (M, g). Suppose further that there exists a smooth, complete metric g satisfying: (i) there exists η > 0 such that Ric g g η, (ii) there exists κ > 0 such that inj(m, g) κ, (iii) g g op,g < 1, Then, the quadratic estimate ˆ holds for all u R(Π B,g ). 0 tπ B,g (I + t 2 Π 2 B,g) 1 u 2 g dt t u 2 2,g Lashi Bandara (ANU) L operators and non-smooth metrics 17 October 2012 19 / 23

Compact manifolds Given a C 0 metric g, we can always find a C metric g that is as close as we would like in the op,s,g norm by pasting together Euclidean metrics via a partition of unity. Further if we assume that M is compact, then automatically Ric g C g and inj(m, g) κ g > 0. Theorem Let M be a smooth, compact Riemannian manifold and let g be a C 0 metric on M. Then, the quadratic estimate ˆ 0 tπb,g (I + tπ 2 B,g) 1 u 2 dt t u 2 is satisfied for all u R(Π B,g ). Lashi Bandara (ANU) L operators and non-smooth metrics 17 October 2012 20 / 23

The noncompact case Lashi Bandara (ANU) L operators and non-smooth metrics 17 October 2012 21 / 23

The noncompact case Although we can obtain arbitrarily close smooth metrics via a partition of unity in the noncompact setting, the desired geometric properties are not automatic. Intuition is that the smoothing properties of geometric flows could be utilised to find good close smooth metrics. The non-uniqueness of existence of solutions to geometric flows are not an issue since we only require one good metric near the initial one. Lashi Bandara (ANU) L operators and non-smooth metrics 17 October 2012 21 / 23

The noncompact case Although we can obtain arbitrarily close smooth metrics via a partition of unity in the noncompact setting, the desired geometric properties are not automatic. Intuition is that the smoothing properties of geometric flows could be utilised to find good close smooth metrics. The non-uniqueness of existence of solutions to geometric flows are not an issue since we only require one good metric near the initial one. A good place to start may be the mean curvature flow since such flows have been studied with rough initial data. Lashi Bandara (ANU) L operators and non-smooth metrics 17 October 2012 21 / 23

The noncompact case Although we can obtain arbitrarily close smooth metrics via a partition of unity in the noncompact setting, the desired geometric properties are not automatic. Intuition is that the smoothing properties of geometric flows could be utilised to find good close smooth metrics. The non-uniqueness of existence of solutions to geometric flows are not an issue since we only require one good metric near the initial one. A good place to start may be the mean curvature flow since such flows have been studied with rough initial data. The first task is to understand the backward behaviour of this flow and the relationship it has to op,g. Lashi Bandara (ANU) L operators and non-smooth metrics 17 October 2012 21 / 23

References I [AHLMcT] Pascal Auscher, Steve Hofmann, Michael Lacey, Alan McIntosh, and Ph. Tchamitchian, The solution of the Kato square root problem for second order elliptic operators on R n, Ann. of Math. (2) 156 (2002), no. 2, 633 654. [AKMc] Andreas Axelsson, Stephen Keith, and Alan McIntosh, Quadratic estimates and functional calculi of perturbed Dirac operators, Invent. Math. 163 (2006), no. 3, 455 497. [BMc] L. Bandara and A. McIntosh, The Kato square root problem on vector bundles with generalised bounded geometry, ArXiv e-prints (2012). [Kato61] Tosio Kato, Fractional powers of dissipative operators, J. Math. Soc. Japan 13 (1961), 246 274. MR 0138005 (25 #1453) [Mc72] Alan McIntosh, On the comparability of A 1/2 and A 1/2, Proc. Amer. Math. Soc. 32 (1972), 430 434. MR 0290169 (44 #7354) Lashi Bandara (ANU) L operators and non-smooth metrics 17 October 2012 22 / 23

References II [Mc82], On representing closed accretive sesquilinear forms as (A 1/2 u, A 1/2 v), Nonlinear partial differential equations and their applications. Collège de France Seminar, Vol. III (Paris, 1980/1981), Res. Notes in Math., vol. 70, Pitman, Boston, Mass., 1982, pp. 252 267. MR 670278 (84k:47030) Lashi Bandara (ANU) L operators and non-smooth metrics 17 October 2012 23 / 23