CUBO A Mathematical Journal Vol.21, No¯ 01, (01–19). April 2019 http: // dx. doi. org/ 10. 4067/ S0719-06462019000100001 On algebraic and uniqueness properties of harmonic quaternion fields on 3d manifolds M.I.Belishev and A.F.Vakulenko Saint-Petersburg Department of the Steklov Mathematical Institute, St-Petersburg State University, Supported by the RFBR grant 18-01-00269. belishev@pdmi.ras.ru, vak@pdmi.ras.ru ABSTRACT Let Ω be a smooth compact oriented 3-dimensional Riemannian manifold with boun- dary. A quaternion field is a pair q = {α, u} of a function α and a vector field u on Ω. A field q is harmonic if α, u are continuous in Ω and ∇α = rot u, div u = 0 holds into Ω. The space Q(Ω) of harmonic fields is a subspace of the Banach algebra C (Ω) of continuous quaternion fields with the point-wise multiplication qq′ = {αα′ − u · u′, αu′ + α′u + u ∧ u′}. We prove a Stone-Weierstrass type theorem: the subalgebra ∨Q(Ω) generated by harmonic fields is dense in C (Ω). Some results on 2-jets of harmonic functions and the uniqueness sets of harmonic fields are provided. Comprehensive study of harmonic fields is motivated by possible applications to inverse problems of mathematical physics. RESUMEN Sea Ω una variedad Riemanniana 3-dimensional suave con borde, orientada y compacta. Un campo cuaterniónico es un par q = {α, u} dado por una función α y un campo de vectores u en Ω. Un campo q es armónico si α, u son continuos en Ω y ∇α = rot u, div u = 0 vale en todo Ω. El espacio Q(Ω) de campos armónicos es un subespacio del álgebra de Banach C (Ω) de campos cuaterniónicos continuos con la multiplicación punto a punto qq′ = {αα′ − u · u′, αu′ + α′u + u ∧ u′}. Probamos un teorema de tipo Stone-Weierstrass: la subálgebra ∨Q(Ω) generada por campos armónicos es densa en C (Ω). Se entregan también algunos resultados acerca de 2-jets de funciones armónicas y los conjuntos de unicidad campos armónicos. http://dx.doi.org/10.4067/S0719-06462019000100001 2 M.I.Belishev and A.F.Vakulenko CUBO 21, 1 (2019) Keywords and Phrases: 3d quaternion harmonic fields, real uniform Banach algebras, Stone- Weierstrass type theorem on density, uniqueness theorems. 2010 AMS Mathematics Subject Classification: 30F15, 35Qxx, 46Jxx. CUBO 21, 1 (2019) On algebraic and uniqueness properties of harmonic . . . 3 1 Introduction Motivation There is an approach to inverse problems of mathematical physics (the so-called Boundary Control method), which was originally based on the relations between inverse problems and the boundary control theory [4, 7, 9]. The BC-method recovers Riemannian manifolds via spectral and/or dy- namical boundary data. Later on, its version that makes use of connections with Banach algebras, was proposed in [2, 5, 6]. The problem of recovering the manifold via its DN-map (the so-called Impedance Tomography Problem) in dimensions > 3 isn’t yet properly solved. However, beginning from the papers [3, 10] it becomes clear that harmonic quaternion fields may play the key role in the 3d ITP. It is the reason, which has stimulated the study of their properties [8, 11]. Here we consider certain of algebraic and uniqueness properties of the harmonic quaternion fields with hope for their future application to ITP [8]. In the mean time, our results may be of certain independent interest for functional analysis: namely, the real uniform Banach algebras theory [1, 13, 15]. Main result • Let Ω be a smooth compact oriented 3-dimensional Riemannian manifold with boundary, TΩx the tangent space at x ∈ Ω, u · v and u ∧ v the inner and vector products in TΩx. Elements of the space Hx := R ⊕ TΩx (the pairs q = {α, u}) endowed with a multiplication qq′ = {αα′ − u · u′, αu′ +α′u+u∧u′} are said to be the geometric quaternions. As an algebra, Hx is isometrically isomorphic to the quaternion algebra H. • A quaternion field is a pair q = {α, u} of a function α and vector field u on Ω; in other words, q is an Hx-valued function on the manifold. The space C(Ω; H) of continuous quaternion fields endowed with the point-wise linear operations and multiplication, and the relevant sup-norm, is a real uniform Banach algebra [1, 13, 15]. A field q = {α, u} ∈ C(Ω; H) is harmonic if α, u are continuous in Ω and ∇α = rot u, div u = 0 holds into Ω. The space Q(Ω) of harmonic fields is a subspace of C(Ω, H) (but not a subalgebra!). • Let A be an algebra. For a set A ⊂ A by ∨A we denote the minimal subalgebra that contains A. The main result of the paper is a Stone-Weierstrass type Theorem 1 which claims that ∨Q(Ω) is dense in C(Ω; H). 4 M.I.Belishev and A.F.Vakulenko CUBO 21, 1 (2019) More results and comments • In the course of proving Theorem 1 we show that Q(Ω) (and, hence, ∨Q(Ω)) separates points of Ω. It is quite evident for Ω ⊂ R3 [11] but far from being evident for a 3d-manifold of arbitrary topology. The separation property is derived from the so-called H-controllability of Ω from the boundary, which is much stronger than separability. The H-controllability is proved by the use of the results [18] on existence of the global Green function and the Landis type uniqueness theorems for the second order elliptic equations [16]. The key step in proving Theorem 1 is to show that ∨Q(Ω) contains the algebra of scalar fields { {α, 0} | α ∈ CR(Ω) } . The latter resembles the trick applied in [14]. • In sec 4 we prove that the 2-jets of harmonic functions are point-wise controllable from the boundary. The proof also makes use of the elliptic uniqueness theorems. Then this result is applied to show that harmonic functions determine the Riemannian structure of 3d manifold. As we hope, it is a step towards the main prospective goal: application to the 3d impedance tomography problem on Riemannian manifolds. • One more result, which is of certain independent interest, is the following uniqueness property of harmonic quaternion fields (sec 5). If q ∈ Q(Ω) vanishes on a piece of a smooth surface then it vanishes in Ω identically. • Everywhere in the paper we deal with real functions, fields, spaces, etc. Everywhere smooth means C∞-smooth. Acknowledgements We’d like to thank Dr C.Shonkwiler for helpful remarks and useful references. 2 Quaternion fields Quaternions • Let E be an oriented 3d euclidean space, u · v and u ∧ v the scalar (inner) and vector products, |u| = √ u · u. Elements p = {α, u} of the space H := R⊕E endowed with the norm |p| = √ α2 + |u|2 and a (noncommutative) multiplication pp′ := {αα′ − u · u′, αu′ + α′u + u ∧ u′} , (2.1) are said to be geometric quaternions. The norm obeys |p2| = |p|2, CUBO 21, 1 (2019) On algebraic and uniqueness properties of harmonic . . . 5 • Let H be the algebra of (standard) quaternions. Recall that it is the real algebra generated by 1, i, j, k with the unit 1 and multiplication defined by the table i 2 = j2 = k2 = −1, ij = k, jk = i, ki = j . • For an orthogonal normalized basis ε = {e1, e2, e3} in E, the correspondence e1 7→ i, e2 7→ j, e3 7→ k determines an isometric isomorphism µε : H → H, {α, ae1 + be2 + ce3} µε7→ α1 + ai + bj + ck , (2.2) (we write H ∼= H). Any isometric isomorphism µ : H → H is of the form (2.2) by proper choice of the basis ε. Vector analysis In the sequel, the following assumptions are accepted. Convention 1. Ω is a smooth compact oriented Riemannian 3d-manifold with the smooth bound- ary ∂Ω. It is endowed with the metric tensor g ∈ C2; dµ is the Riemannian volume 3-form; ⋆ is the Hodge operator. On such a manifold, the intrinsic operations of vector analysis are well defined on smooth functions and vector fields (sections of the tangent bundle TΩ). Following [21], Chapter 10, we recall their definitions. • For a vector field u, one defines the conjugate 1-form u♭ by u♭(v) = g(u, v), ∀v. For a 1-form f, the conjugate field f♭ is defined by g(f♭, u) = f(u), ∀u. • A scalar product: {fields} × {fields} ·→ {functions} is defined point-wise by u · v = g(u, v). A vector product: {fields} × {fields} ∧→ {fields} is defined point-wise by g(u ∧ v, w) = dµ (u, v, w), ∀w. • A gradient: {functions} ∇→ {fields} and a divergence: {fields} div→ {functions} are defined by ∇α = (dα)♭ and div u = ⋆ d⋆ u♭ respectively, where d is the exterior derivative. • A rotor: {fields} rot→ {fields} is defined by rot u = (⋆ d u♭)♭. Recall the basic identities: div rot = 0 and rot ∇ = 0. The equalities ∇α = rot u and dα = ⋆ d u♭ are equivalent. • The Laplacian {functions} ∆→ {functions} is ∆ = div ∇. The vector Laplacian {fields} ~∆ → {fields} is ~∆ = ∇ div − rot rot . 6 M.I.Belishev and A.F.Vakulenko CUBO 21, 1 (2019) Remark 1. Under the above accepted assumptions on the smoothness of Ω and g, the (harmonic) functions and fields, which obey ∆α = 0 and ~∆u = 0 in the relevant weak sense, do belong to the class C2 loc : see, e.g, [12], Part II, Chapter 1. Fields Let Ω̇ := Ω\∂Ω be the set of the inner points, C(Ω) and ~C(Ω) the spaces of continuous functions and vector fields. Let Hx := R ⊕ TΩx, x ∈ Ω be the point-wise geometric quaternion algebras. • A quaternion field is a pair p = {α, u} with the components α ∈ C(Ω) and u ∈ ~C(Ω), the values p(x) = {α(x), u(x)} ∈ Hx being regarded as geometric quaternions. By C(Ω; H) we denote the space of continuous quaternion fields. One can regard them as sections of the bundle C(Ω; H) = ∪x∈ΩHx. • Elements of the subspace Q(Ω) := { p ∈ C(Ω; H) ∣ ∣ ∇α = rot u, div u = 0 in Ω̇ } are called harmonic fields. To be rigorous, here the conditions on the components of p are under- stood in the relevant sense of distributions but imply ∆α = 0 and ~∆u = 0, so that α and u are automatically smooth enough by Remark 1. 3 Density theorem Algebra C(Ω; H) The space C(Ω; H) with the point-wise multiplication (2.1) and the norm ‖p‖ = sup x∈Ω |p(x)| = sup x∈Ω √ |α(x)|2 + |u(x)|2 TΩx satisfying ‖qp‖ 6 ‖q‖‖p‖, ‖p2‖ = ‖p‖2 is a real uniform noncommutative Banach algebra. • The fields {α, 0} constitute a subalgebra C(Ω; R) of C(Ω; H), which is isometrically isomorphic to the real continuous function algebra on Ω: C(Ω; R) ∼= C R (Ω) . (3.1) We say {α, 0} to be the scalar fields and often identify them with functions α via the map α 7→ {α, 0}, which embeds CR(Ω) in C(Ω; H). • The harmonic subspace Q(Ω) ⊂ C(Ω; H) is not an algebra since, in general, p, q ∈ Q(Ω) does not imply pq ∈ Q(Ω). It is easy to see that Q(Ω) ∩ C(Ω; R) = {{c, 0} | c is a constant function} , CUBO 21, 1 (2019) On algebraic and uniqueness properties of harmonic . . . 7 whereas {1, 0} is the unit of C(Ω; H). Main result For an algebra A and a set S ⊂ A by ∨S we denote a minimal (sub)algebra in A , which contains S. Our main results is the following. Theorem 1. The algebra ∨Q(Ω) is dense in C(Ω; H). The proof occupies the rest of sec 3. Green function • A well-known in geometry fact is that the assumptions of Convention 1, in particular, provide the existence of a compact 3-dimensional C∞- manifold Ω′ ⋑ Ω endowed with the tensor g′ ∈ C2 such that g′|Ω = g. This enables one to apply the results by M.Mitrea and M.Taylor [18] (existence of the fundamental solution, Green function, Poisson formula, etc) which are valid for much weaker smoothness restrictions on g and ∂Ω. Also, one can apply the results on the uniqueness of continuation of solutions to the elliptic PDE [12, 16]. • The following results are mostly taken from [18]. Also we use some well-known facts of the elliptic 2-nd order equations theory [17, 12, 16]. By Wlp(Ω) we denote the Sobolev space of functions which possess the (generalized) derivatives of the order l = 1, 2, . . . belonging to Lp(Ω) (p > 1). Recall that Ω̇ = Ω \ ∂Ω. Also we put D := {(x, y) ∈ Ω × Ω | x = y}. The distance in Ω is denoted by rxy. Let D(Ω̇) be a space of the smooth compactly supported into Ω functions (test functions) endowed with the standard topology, D ′(Ω̇) the corresponding distributions. For an h ∈ L2(Ω), the Dirichlet problem ∆v = h in Ω̇ v = 0 on ∂Ω has a unique solution vh ∈ W22(Ω) vanishing at the boundary. The solution is represented in the form vh(x) = ∫ Ω G(x, y) h(y) dµ(y), x ∈ Ω (3.2) via the Green function G, which possesses the following properties. 1. G ∈ C2 loc ([Ω × Ω] \ D); G(x, y) = G(y, x), (x, y) 6∈ D; G(x, ·)|∂Ω = 0, x ∈ Ω̇ . (3.3) 8 M.I.Belishev and A.F.Vakulenko CUBO 21, 1 (2019) For the closed sets K, K′ ⊂ Ω provided K ∩ K′ = ∅ the map y 7→ G(·, y) is continuous from K to C2(K′). 2. The estimates G(x, y) 6 c rxy , |∇yG(x, y)| 6 c r2xy hold and imply G(x, ·) ∈ W1p(Ω) for x ∈ Ω, 1 6 p < 32 . 3. As a distribution of the class D ′(Ω̇) on the test functions (of the variable y) of the class D(Ω̇), the Green function satisfies ∆yG(x, ·) = δx, (3.4) where δx is the Dirac measure supported at x. Note that in (3.4), and below in (3.8), (3.9), the variable x ∈ Ω̇ plays the role of parameter. 4. For f ∈ C∞(∂Ω), the inhomogeneous boundary value problem ∆w = 0 in Ω̇ (3.5) w = f on ∂Ω (3.6) has a unique classical solution w = wf(x), which is represented in the form wf(x) = ∫ ∂Ω ∂νyG(x, y) f(y) dσ(y), x ∈ Ω̇ , (3.7) where νy is the outward unit normal at the boundary, dσ is the boundary surface element. This is a Poisson formula derived from (3.2) by integration by parts. Function f in (3.6) is said to be a boundary control. • Fix a point x ∈ Ω̇ and a vector e ∈ TΩx, |e| = 1. Let γe be the geodesic that emanates from x in direction e. Define a functional ∂xeδx ∈ D ′(Ω̇) by 〈∂xeδx, ϕ〉 := lim γe∋ x′→x ϕ(x′) − ϕ(x) rxx′ = 〈 lim γe∋ x′→x δx′ − δx rxx′ , ϕ 〉 = e · ∇ϕ(x) . The relevant limit passage in (3.4) determines a derivative ∂xeG(x, ·) ∈ D ′(Ω̇) which satisfies ∆y[∂ x eG(x, ·)] = ∂xeδx . (3.8) In the mean time, by the properties 1 and 2, ∂xeG(·, y) is a (classical) function belonging to Lp(Ω) for 1 6 p < 3 2 . Moreover it is harmonic (and hence C2-smooth) in Ω \ {x} and satisfies ∂xeG(x, ·)|∂Ω = 0 , x ∈ Ω̇. (3.9) • The relevant limit passage in the Poisson formula (3.7) implies e · ∇wf(x) = ∫ ∂Ω ∂νy [∂ x eG(x, y)] f(y) dσ(y), x ∈ Ω̇ . (3.10) CUBO 21, 1 (2019) On algebraic and uniqueness properties of harmonic . . . 9 H-controllability • The following result plays the key role in the proof of Theorem 1. Recall that Hx = R⊕TΩx ∼= H, and Ω obeys Convention 1. For a set of points A = {a1, . . . , aN} ⊂ Ω define a 4N-dimensional space HA := ⊕ ∑N i=1 Hai and a map MA : C ∞(∂Ω) → HA: f 7→ ⊕ N∑ i=1 {wf(ai), ∇wf(ai)} (each summand {wf(ai), ∇wf(ai)} belongs to the corresponding Hai). We say Ω to be H- controllable from boundary if this map is surjective for any finite set A. Lemma 1. The manifold Ω is H-controllable from boundary. Proof. The opposite means that HA⊖Ran MA 6= {0}, i.e. there is a nonzero element ⊕ ∑N i=1 {αi, βiei} ∈ HA (αi, βi ∈ R, |ei| = 1) such that N∑ i=1 αiw f(ai) + βi ei · ∇wf(ai) = 0 (3.11) holds for all f ∈ C∞(∂Ω). Show that such an assumption leads to contradiction. 1. Let A ⊂ Ω̇, i.e., all ai are the interior points. A function Φ(y) := N∑ i=1 αiG(ai, y) + βi∂ x ei G(ai, y) (3.12) satisfies ∆Φ = 0 in Ω \ A (3.13) Φ|∂Ω = 0 (3.14) by (3.3), (3.4), (3.8), and (3.9). The relations (3.7), (3.10) and (3.11) easily follow to ∫ ∂Ω ∂νΦ(y) f(y) dσ(y) = 0 that implies ∂νΦ|∂Ω = 0 (3.15) by arbitrariness of f. 10 M.I.Belishev and A.F.Vakulenko CUBO 21, 1 (2019) 2. So, Φ is harmonic in Ω \ A and has the zero Cauchy data at the boundary: see (3.14) and (3.15). By the well-known uniqueness property of solutions to elliptic PDE (see, e.g., [16], sec. 4.3, Remark 4.17), we get Φ = 0 in Ω \ A, i.e., almost everywhere in Ω. Since G(ai, ·) ∈ W1p(Ω) and ∂eiG(ai, ·) ∈ Lp(Ω), we have Φ ∈ Lp(Ω) for some p > 1. Therefore, Φ is a summable function equal zero a.e. in Ω. Thus, Φ = 0 as a distribution of the class D ′(Ω̇). In the mean time, by (3.4) and (3.8) one has ∆Φ = N∑ i=1 αiδai + βi∂ x ei δai 6= 0 , i.e., Φ is a nonzero element of D ′(Ω̇). We arrive at the contradiction that proves the Lemma for A ∈ Ω̇. 3. Let A contain the points of ∂Ω. The smoothness assumptions on Ω enable one to provide Ω′, g′ obeying Convention 1 and such that Ω ⋐ Ω′ and g′|Ω = g holds. Then one has A ⊂ Ω̇′ that reduces this case to the previous one. Note that relations between controllability and uniqueness theorems (like the one used in the proof) are widely exploited in control theory for PDE (see, e.g., [9]). • Recall that wf is a harmonic function that solves (3.5), (3.6). As immediate consequence of Lemma 1 we have Corollary 1. The algebra ∨ { |∇wf|2 | f ∈ C∞(Ω) } is dense in CR(Ω). Indeed, by Lemma 1, for any a, b ∈ Ω there is a smooth f such that |∇wf(a)|2 6= |∇wf(b)|2, i.e., the functions |∇wf(·)|2 separate points of Ω. In the mean time, by the same Lemma, there is no x0 ∈ Ω, at which all these functions vanish simultaneously. Hence, by the classical Stone- Weierstrass Theorem (see, e.g., [19]), the above mentioned density does hold. Note that {0, ∇wf} ∈ Q(Ω) and {0, ∇wf}2 = −{|∇wf(·)|2, 0} ∈ ∨Q(Ω). Hence, the algebra ∨ { {|∇wf|2, 0} | f ∈ C∞(Ω) } is a subalgebra in ∨Q(Ω). By (3.1), Corollary 1 implies that this algebra is dense in C(Ω; R). As a result, denoting C := ∨Q(Ω) we arrive at the important relation C ⊃ C(Ω; R) . (3.16) CUBO 21, 1 (2019) On algebraic and uniqueness properties of harmonic . . . 11 Strong separation We say that a family F ⊂ C(Ω; H) strongly separates points (of Ω) if for any a, b ∈ Ω and ha ∈ Ha, hb ∈ Hb there is a p ∈ F such that p(a) = ha and p(b) = hb holds [13]. Lemma 2. The space Q(Ω) strongly separates points. Proof. • Let ~L2(Ω) be the space of square-integrable vector fields and H := {v ∈ ~L2(Ω) | div v = 0, rot v = 0} its harmonic subspace. The well-known Hodge-Morrey-Friedrichs decomposition claims that H = G ⊕ N = R ⊕ D , (3.17) where G := {v ∈ H | v = ∇α}, N := {v ∈ H | v · ν = 0} , R := {v ∈ H | v = rot u}, D := {v ∈ H | v ∧ ν = 0} . (see, e.g., [21], Corollary 3.5.2). The subspaces N and D determined by the boundary conditions are called the Neumann and Dirichlet spaces respectively. Their finite dimensions are equal to the Betti numbers: dim N = β1, dim D = β2 [21]. Note that N ∩ D = {0} [3, 21]. Also note that dim G = dim R = ∞. • As a consequence of (3.17), a field v ∈ H is represented in the form v = ∇α = rot u if and only if v ∈ G ∩ R or, equivalently, v⊥[N +̇D]. If w = wf(x) solves (3.5), (3.6) then for any d ∈ D one has (∇wf, d) = ∫ Ω ∇wf · d dµ = ∫ ∂Ω f d·ν dσ . In the mean time, since ∇wf ∈ G , the representation ∇wf = rot u holds if and only if ∇wf⊥D, which is equivalent to ∫ ∂Ω f d·ν dσ = 0 , d ∈ D. (3.18) In particular, taking f = 1 one has wf = 1 in Ω and gets ∫ ∂Ω d·ν dσ = 0 , d ∈ D. (3.19) • Now, fix two distinct points a, b ∈ Ω and elements ha = {ca, ka} ∈ Ha, hb = {cb, kb} ∈ Hb. To prove the Lemma we need to show that there is a smooth f, which provides wf(a) = ca, w f (b) = cb; ∇wf = rot u; u(a) = ha, u(b) = hb . (3.20) 12 M.I.Belishev and A.F.Vakulenko CUBO 21, 1 (2019) Step 1. At first assume a, b ∈ Ω̇. Let Px(y) := ∂νyG(x, y) be the Poisson kernel. By (3.7) for f = 1 we have ∫ ∂Ω Px(y) dσ(y) = 1 , x ∈ Ω . (3.21) In accordance with (3.7) and (3.18), to satisfy the relations wf(a) = ca, w f(b) = cb; ∇wf = rot u in (3.20) we need to find f provided ∫ ∂Ω Pa(y) f(y) dσ(y) = ca , ∫ ∂Ω Pb(y) f(y) dσ(y) = cb ; ∫ ∂Ω f(y) d(y)·ν dσ(y) = 0 , d ∈ D, or, equivalently, (Pa, f) = ca, (Pb, f) = cb , f⊥ ν·D (3.22) (the inner products in L2(∂Ω)), where ν · D := {ν · d | d ∈ D}. Comparing (3.19) with (3.21), we conclude that neither Pa nor Pb belong to ν · D. In the mean time, Pa 6= Pb as elements of L2(∂Ω). Indeed, otherwise we’d have wf(a) = wf(b) for any f that is impossible by Lemma 2. Hence, span{Pa, Pb} ∩ ν · D may consist of {c(Pa − Pb) | c ∈ R} only. As a result, to proof the solvability of the linear system (3.22) (with respect to f) in the case of ca 6= cb we must show that Pa − Pb 6∈ ν · D. Step 2. Assume the opposite: there is a d ∈ D such that Pa − Pb = d · ν, and show that this assumption leads to a contradiction. Compare the fields ∇[G(a, ·) − G(b, ·)] and d. Since G(a, ·) = G(b, ·) = 0 on ∂Ω both of them are normal on the boundary. Hence, by the assumption, they are equal on ∂Ω. In the mean time, the field ∇[G(a, ·) − G(b, ·)] is harmonic in Ω̇ \ [{a} ∪ {b}], whereas d is harmonic in the whole Ω̇. The coincidence at the boundary implies the coincidence in the domain of harmonicity. Hence, ∇[G(a, ·) − G(b, ·)] can be extended by continuity to the whole Ω and ∇[G(a, ·) − G(b, ·)] = d everywhere. However, the latter is impossible since div ∇[G(a, ·) − G(b, ·)] = ∆[G(a, ·) − G(b, ·)] = δa − δb , whereas div d = 0 everywhere in Ω̇. This contradiction shows that Pa − Pb 6∈ ν · D. Step 3. The case of a and/or b belonging to the boundary is reduced to the previous one by the collar theorem arguments, which were applied at the end of the proof of Lemma 1. Corollary 2. The algebra ∨Q(Ω) ⊂ C(Ω; H) strongly separates points of Ω. This property plays important role in proving density theorems [13]. CUBO 21, 1 (2019) On algebraic and uniqueness properties of harmonic . . . 13 Completing the proof of Theorem 1 Recall that C = ∨Q(Ω) and prove that C = C(Ω; H). The fact, which will play the key role, is the embedding C ⊃ C(Ω; R) ∼= CR(Ω): see (3.16). • Fix an x ∈ Ω and choose the smooth boundary controls fx1, fx2, fx3 such that ∇wf x 1 (x), ∇wfx2 (x), ∇wfx3 (x) constitute a basis of TΩx. It is possible owing to Lemma 1. By continuity, there is a ball Br(x)[x] ⊂ Ω centered at x, of (small enough) radius r(x), such that ∇wf x 1 (y), ∇wfx2 (y), ∇wfx3 (y) is a basis of TΩy for each y ∈ Br(x)[x]. Let such a choice be done for each x ∈ Ω. • The balls provide an open cover Ω = ∪x∈ΩBr(x)[x]. By compactness there is a finite subcover Ω = ∪Nn=1Brn[xn], where rn := r(xn). Let η1, . . . , ηN be a partition of unit subordinated to the subcover, so that η1, . . . , ηN ∈ C∞(Ω), supp ηn ⊂ Brn[xn], N∑ n=1 ηn ≡ 1 in Ω holds. • Take p = {α, u} ∈ C(Ω; H) and represent p = N∑ n=1 ηnp = { N∑ n=1 ηnα, N∑ n=1 ηnu} = N∑ n=1 {ηnα, 0} + N∑ n=1 {0, ηnu} with {ηnα, 0} ∈ C(Ω; R) ⊂ C . In the mean time, one has ηnu = 3∑ k=1 κ n k ∇wf xn k with the certain κnk ∈ CR(Ω) supported in Brn[xn]. Note that {κnk , 0} ∈ C(Ω; R) ⊂ C . Summarizing, we arrive at the representation p = N∑ n=1 {ηnα, 0} + N∑ n=1 3∑ k=1 {κnk , 0}{0, ∇wf xn k } , where all cofactors and summands do belong to C . Thus p ∈ C and, hence, C(Ω; H) = C . Theorem 1 is proved. Remark 2. Analyzing the proof, it is easy to recognize that the family W := { {0, ∇wf} | f is smooth } , which is smaller than Q(Ω), also generates the whole of the continuous field algebra: ∨W = C(Ω; H). 14 M.I.Belishev and A.F.Vakulenko CUBO 21, 1 (2019) 4 Controllability of 2-jets Fix an a ∈ Ω̇; let x1, x2, x3 be the local coordinates in a neighborhood ω ∋ a. With a smooth function φ one associates the row of its 0,1,2-order derivatives ja[φ] := {φ(a); φx1(a), φx2(a), φx3(a); φx1x1(a), φx1x2(a), φx1x3(a), φx2x2(a), φx2x3(a), φx3x3(a)} ∈ R10, which provides a coordinate representation of its second jet at the point a [20]. For short, we say ja[φ] to be a 2-jet of φ at a and consider R 10 with the (standard) inner product 〈j, j′〉 as a space of 2-jets. Recall that in coordinates the Laplacian acts by ∆φ = g− 1 2 [g 1 2 gikφxk]xi , where {gik} is the inverse to the metric tensor matrix {gik} and g = det{gik} (summation over repeating indexes is in the use). We say the row λa := = {0; g− 1 2 [g 1 2 gi1]xi, g − 1 2 [g 1 2 gi2]xi, g − 1 2 [g 1 2 gi3]xi; g 11, 2g12, 2g13, g22, 2g23, g33} ∣ ∣ x=a to be the Laplace jet and represent (∆φ)(a) = 〈λa, ja[φ]〉. The harmonicity ∆w = 0 is equivalent to the orthogonality 〈ja[w], λa〉 = 0, a ∈ ω. Therefore one has ja[w] ∈ R10 ⊖ span λa. Let us show that the 2-jets of harmonic functions exhaust the subspace R10 ⊖ span λa. This result may be interpreted as a point-wise boundary controllability of 2-jets by harmonic functions. Recall that wf is a solution to (3.5), (3.6). Lemma 3. For any a ∈ Ω and s ∈ R10 ⊖ span λa there is a smooth f such that ja[wf] = s. Proof. Taking into account the structure of the Laplace jet, we may deal with s = {0; s1, s2, s3; s11, . . . , s33}, and let it be such that 0 6= s ∈ R10 ⊖ span λa but 〈s, ja[wf]〉 = 0 for any smooth f. Show that such an assumption leads to contradiction. • For a differential operator L with smooth coefficients in Ω, by L∗ we denote its adjoint by Lagrange that is defined by (Lη, ζ)L2(Ω) = (η, L ∗ζ)L2(Ω), η, ζ ∈ D(Ω̇) . For a distribution h ∈ D ′(Ω̇) one defines Lh by (Lh, η) := (h, L∗η)L2(Ω), η ∈ D(Ω̇). CUBO 21, 1 (2019) On algebraic and uniqueness properties of harmonic . . . 15 Let S be a differential operator, which acts by (Sv)(x) = = [s1vx1 + s2vx2 + s3vx3 + s11vx1x1 + s12vx1x2 + · · · + s33vx3x3] (x) = = 〈s, jx[v]〉, x ∈ ω in a coordinate neighborhood ω of a ∈ Ω̇, where the (constant) coefficients are the components of the above chosen jet s. • Let δa ∈ D ′(Ω̇) be the Dirac measure supported at the point a ∈ Ω̇. Consider the problem ∆H = S∗δa (4.1) H ∣ ∣ ∂Ω = 0 . (4.2) The equation is understood as a relation in D ′(Ω̇); its r.h.s. is a distribution acting by (S∗δa, η)L2(Ω) = (Sη)(a). The boundary condition does make sense since H is harmonic outside supp S∗δa = {a}. Also, the normal derivative ∂νH is a smooth function on ∂Ω. Formally by Green, for a function v ∈ C2(Ω) one has 〈s, ja[v]〉 = (Sv)(a) = ∫ Ω δa Sv dµ = ∫ Ω S∗δa v dµ (4.1) = ∫ Ω ∆H v dµ = (4.2) = ∫ Ω H ∆v dµ + ∫ ∂Ω ∂νH v dσ . To justify the final equality 〈s, ja[v]〉 = ∫ Ω H ∆v dµ + ∫ ∂Ω ∂νH v dσ (4.3) one can use the standard regularization technique, approximating δa by δ ε a ∈ D(Ω̇) supported near a. • By the choice of s, for v = wf the equality (4.3) provides ∫ ∂Ω ∂νH w f dσ = ∫ ∂Ω ∂νH f dσ = 0 . By arbitrariness of f we get ∂νH = 0 on ∂Ω. So, H is harmonic in Ω\{a} and has the zero Cauchy data on the boundary. By the uniqueness theorem, H vanishes everywhere outside a. Hence, the distribution H is supported at a. The well-known fact of the distribution theory is that such an H is a linear combination of δa and its derivatives. In the mean time, comparing the orders of singularities in the left and right hand sides of (4.1), one easily concludes that H = cδa 16 M.I.Belishev and A.F.Vakulenko CUBO 21, 1 (2019) with c = const 6= 0. Indeed, otherwise ∆H contains the derivatives of δa of the order > 3 that makes the equality (4.1) impossible. For an η ∈ D(Ω̇) one has 〈s, ja[η]〉 = (δa, Sη) = (S∗δa, η) (4.1) = (∆cδa, η) = (cδa, ∆η) = 〈cλa, ja[η]〉 . Comparing the beginning with the end and referring to the evident {ja[η] | η ∈ D(Ω̇)} = R10, we arrive at s = cλa that contradicts to the starting assumption s⊥λa. • The case a ∈ ∂Ω is reduced to the previous one by means of the trick already used at the end of the proof of Lemma 1: embedding Ω ⋐ Ω′. As is easy to recognize, Lemma 3 implies the assertion of Lemma 1 for the case of the single point a. However, Lemma 3 may be generalized on the finite set a1, . . . , aN so that the relevant boundary controllability of 2-jets of harmonic functions holds up to the natural defect in ⊕ ∑ i R 10 ai . Determination of metric from harmonic functions The metric on Ω determines the family of harmonic functions. The converse is also true in the following sense. • Let c > 0 be a smooth function on Ω and cg a conformal deformation of the metric g. By ∆cg and ∆g we denote the corresponding Laplacians. A simple calculation leads to the relation ∆cgy = c −1∆gy − 2 −1 ∇c−1 · ∇y , (4.4) which is specific for the 3d case. Taking y = wf, we see that the metrics cg and g have the same reserve of harmonic functions wf if and only if ∇c−1 · ∇wf = 0 holds for any smooth f. In the mean time, by Lemma 1 the gradients ∇wf = 0 constitute the local bases in Ω. Hence, the latter equality implies ∇c−1 = 0, i.e., c = const. • Fix a point a in a coordinate neighborhood ω ∋ a. By λga we denote the Laplace jet of the given metric g. By Lemma 3, the space of jets is R 10 a = {ja[φ] | φ is smooth} = {ja[w f] | f is smooth} ⊕ span λga . (4.5) Therefore, writing (∆wf)(a) = 0 in the form 〈λga, ja[wf]〉 = 0, f is smooth and varying f = f1, f2, . . . , we get a linear homogeneous algebraic system with respect to the components of the jet λ g a, which determines them up to a factor, which may depend on a. Along CUBO 21, 1 (2019) On algebraic and uniqueness properties of harmonic . . . 17 with the components, we determine the tensor g up to a factor, possibly depending on a. However, by the above mentioned geometric reasons, this factor is a constant. Thus, the family {wf | f is smooth} determines the metric g up to a constant positive factor. If g is known at least at a single point x0 ∈ Ω, then it is uniquely determined everywhere. Notice in addition that in two-dimensional case relation (4.4) is of the form ∆cgy = c −1∆gy, so that the metrics cg and g determine the same reserve of harmonic functions. It is the reason, because of which in 2d impedance tomography problem the metric is recovered up to conformal equivalence [2]. • Here we describe a trick, which is used in dynamical/spectral inverse problems and 2d impedance tomography problem, for recovering the metric via boundary data[9]. The hope is that it may be useful in future investigation of 3d ITP. Assume that a topological space Ω̃ is homeomorphic to Ω via a homeomorphism β : Ω → Ω̃. Also assume that the family of functions {w̃f = wf ◦ β−1 | f is smooth} is given. The following procedure enables one to determine the metric g̃ = β∗g in Ω̃. 1. Fix a point a ∈ Ω̃ and choose its neighborhood ω̃ with the coordinates x1, x2, x3. By the way, Lemma 1 enables one to use the images w̃f as local coordinates. 2. Find span λ g̃ a by (4.5) (replacing functions w f on ω with w̃f on ω̃). As was shown above, the family of these subspaces given for a ∈ ω̃ determines the metric up to a constant factor. So, cg̃ is recovered. Assuming g̃ to be known at least at a single point a0 ∈ ω̃, one recovers g̃ uniquely. 3. Covering Ω̃ by the coordinate neighborhoods and repeating the previous steps, we determine g̃ in Ω̃. 5 Uniqueness properties of harmonic fields Roughly speaking, the following result means that the set of zeros of a harmonic quaternion field may be at most of dimension 1. Lemma 4. Let Σ ∈ Ω be a C2-smooth surface (2-dim submanifold). If p ∈ Q(Ω) obeys p|Σ = 0 then p = 0 in the whole Ω. Proof. Since the claimed result is of local character, we assume Σ to be a both-side surface endowed with a smooth field of the unit normals ν. Also, Σ possesses the (induced) Riemannian metric and 18 M.I.Belishev and A.F.Vakulenko CUBO 21, 1 (2019) is provided with the corresponding operations on vector fields. In particular, a divergence, which is denoted by divΣ, is well defined. • For a point x ∈ Σ and vector v ∈ TΩx we represent v = vθ + vν : vν = v · ν ν, vθ = v − vν and, by default, identify vθ with the proper vector of TΣx. By the latter, for a smooth vector field v given in a neighborhood of Σ, the value [divΣ vθ](x) is of clear meaning. Also, recall the well-known vectot analysis relation ν · rot v = divΣ ν ∧ vθ on Σ (5.1) (see, e.g. [21]). • Begin with the case Σ ⊂ Ω̇. Let p = {α, u} ∈ Q(Ω), so that ∇α = rot u, div u = 0 in Ω̇ (5.2) holds. Let p|Σ = 0. Since α|Σ = 0, we have (∇α)θ|Σ = 0 that implies (rot u)θ ∣ ∣ Σ = 0 by (5.2). In the mean time, u|Σ = 0 is equivalent to uθ = uν = 0 on Σ; hence (rot u)ν|Σ = divΣ ν ∧ uθ = 0 by virtue of (5.1). Thus we get (rot u)θ|Σ = (rot u)ν|Σ = 0, i.e. rot u|Σ = 0. The latter equality and (5.2) lead to (∇α)|Σ = 0 (along with α|Σ = 0). So, α is a harmonic function with the zero Cauchy data on Σ. Therefore α = 0 in Ω by the elliptic uniqueness theorems [16]. As a result, rot u = ∇α = 0 everywhere in Ω. Since div u = 0, the vector field u is harmonic in Ω and vanishes on Σ. Therefore, locally near the points x ∈ Σ one represents u = ∇ϕ with a harmonic function ϕ provided ∇ϕ|Σ = 0. Such a function is a constant; hence u = 0 near Σ. By its harmonicity, u vanishes globally in Ω. So, we have p = 0 in Ω. • The case Σ ⊂ ∂Ω is reduced to the previous one by means of the trick already used at the end of the proof of Lemma 1: embedding Ω ⋐ Ω′. CUBO 21, 1 (2019) On algebraic and uniqueness properties of harmonic . . . 19 References [1] M.Abel and K.Jarosz. Noncommutative uniform algebras. Studia Mathematica, 162 (3) (2004), 213–218. [2] M.I.Belishev. The Calderon problem for two-dimensional manifolds by the BC-method. 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