General solution of the Inhomogeneous Div-Curl system and Consequences

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1 General solution o the Inhomogeneous Div-Curl system and Consequences Briceyda Berenice Delgado López Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional March CINVESTAV INRIA Sophia Antipolis 1/37

2 Index 1 Div-Curl system Monogenic unctions 2 associated to the equation DW = 0 associated to the equation DW = D W CINVESTAV INRIA Sophia Antipolis 2/37

3 Monogenic unctions We will give a complete solution to the reconstruction o a vector ield rom its divergence and curl, i.e., the system div W = g 0, curl W = g, (1) or appropriate assumptions on the scalar ield g 0 and the vector ield g and their domain o deinition in three-space. CINVESTAV INRIA Sophia Antipolis 3/37

4 Monogenic unctions Let H the non-commutative algebra o quaternions over the real ield R. Let x = x i=1 e ix i H, where x i R. The subspace VecH := span R {e 1, e 2, e 3 } o H is identiied with the Euclidean space R 3 as ollows x 1 e 1 + x 2 e 2 + x 3 e 3 (x 1, x 2, x 3 ) R 3. Let Ω R 3 be an open subset with smooth boundary. Deine the Cauchy-Riemann type dierential operator as D = e 1 + e 2 + e 3. x 1 x 2 x 3 It acts over dierentiable unctions w : Ω H o the orm w(x) = w 0 (x) + 3 i=1 e iw i (x), where w i : Ω R, i = 0, 1, 2, 3. CINVESTAV INRIA Sophia Antipolis 4/37

5 Monogenic unctions The ollowing result tell us that is impossible to deine an H holomorphic unctions via the existence o the limit o the dierence quotient, like in the complex case: Theorem Let w C 1 (Ω) be a unction deined in a domain Ω H. I or all points in Ω the limit ĺım h 0 h 1 [w(x + h) w(x)], exists, then in Ω the unction w has the orm w(x) = a + xb a, b H. CINVESTAV INRIA Sophia Antipolis 5/37

6 Monogenic unctions Deinition A C 1 unction w : Ω H is called let-monogenic (resp. right-monogenic) in Ω i Dw = 0 en Ω (wd = 0 in Ω). We will say simply monogenic to reer to let-monogenic unctions. Even more, since Dw = div w + curl w + w 0, then { div w = 0, Dw = 0 curl w = w 0. CINVESTAV INRIA Sophia Antipolis 6/37

7 Monogenic unctions Examples: E(x) := 1 x 4π x 3, x 0, is an example o a let- and right-monogenic unction, called Cauchy kernel. w(x) = x 3 + x 1 e 2 x 3 e 3, is let-monogenic but no right-monogenic, since Dw = 0 y wd = 2e 3. CINVESTAV INRIA Sophia Antipolis 7/37

8 Monogenic unctions Let us denote M(Ω) = Sol (Ω, R 3 ) Irr (Ω, R 3 ), where Sol (Ω, R 3 ) = { w : div w = 0 in Ω} C 1 (Ω, R 3 ), Irr (Ω, R 3 ) = { w : curl w = 0 in Ω} C 1 (Ω, R 3 ), the Solenoidal e Irrotational vector ields, respectively. The elements w M(Ω) are locally the gradient o real valued harmonic unctions. We write Har (Ω, A) = {w : Ω A, w = 0}, where A = R, R 3 or H, or the corresponding sets o harmonic unctions deined in A. Since = D 2, then let and right monogenic unctions are harmonic. CINVESTAV INRIA Sophia Antipolis 8/37

9 Monogenic unctions The monogenic completion operator S Ω : Har (Ω, R) Har (Ω, R 3 ) is given by ( 1 S Ω [w 0 ](x) = Vec = ) tdw 0 (tx)x dt tdw 0 (tx) x dt, x Ω. or harmonic unctions w 0 deined in star-shaped open sets Ω with respect to the origin. CINVESTAV INRIA Sophia Antipolis 9/37

10 Monogenic unctions Let Ω R 3 be an open set with smooth boundary and w C 0 (Ω, H). The operator T Ω deined by y x T Ω [w](x) := w(y)dy, x Ω, Ω 4π y x 3 is called the Teodorescu transorm, which acts like a right inverse o D. More even, i w L 1 (Ω, H), then DT Ω [w] = w, weakly. Let w C 1 (Ω, H). The Cauchy-Bitsadze operator is deined as F Ω [w](x) := E(y x)dy w(y), x R 3 \ Ω, And DF Ω [w] = 0. Ω CINVESTAV INRIA Sophia Antipolis 10/37

11 Monogenic unctions The operator T Ω can be descompossed in the ollowing way T 0,Ω [ w](x) = E(y x) w(y)dy, Ω T1,Ω [w 0 ](x) = w 0 (y)e(y x)dy, Ω T2,Ω [ w](x) = E(y x) w(y)dy, where denotes the scalar (or inner) product o vectors and denotes the cross product. That is, Ω T Ω [w 0 + w] = T 0,Ω [ w] + T 1,Ω [w 0 ] + T 2,Ω [ w]. CINVESTAV INRIA Sophia Antipolis 11/37

12 Monogenic unctions The ollowing act is essential in the construction o the right inverse o curl: T 0,Ω [ w] Har (Ω, R) w Sol (Ω, R 3 ). Theorem Let Ω R 3 be a star-shaped open set. The operator T2,Ω S Ω T 0,Ω (2) is a right inverse or the curl on the class o unctions Sol (Ω, R 3 ). Where S Ω is the monogenic completion operator. Furthermore, ( T 2,Ω S Ω T 0,Ω ): Sol (Ω, R 3 ) Sol (Ω, R 3 ). CINVESTAV INRIA Sophia Antipolis 12/37

13 Monogenic unctions Then we can solve the homogeneous div-curl system under the assumptions g Sol (Ω, R 3 ): And the solution is given by where h Har (Ω, R) is arbitrary. div W = 0, curl W = g, (3) W = T 2,Ω [ g] S Ω T 0,Ω [ g] + h, CINVESTAV INRIA Sophia Antipolis 13/37

14 Monogenic unctions Applying a correction, and with g 0 C 0 (Ω, R), and g Sol (Ω, R 3 ). Then a general solution to the inhomogeneous Div-Curl system is given by div W = g 0, curl W = g, (4) W = T 1,Ω [g 0 ] + T 2,Ω [ g] S Ω T 0,Ω [ g] + h, (5) where h Har (Ω, R) is arbitrary. More even, W is a weak solution o the div-curl system (4) when g 0 L 2 (Ω, R), g L 2 (Ω, R 3 ) and div g = 0 weakly. CINVESTAV INRIA Sophia Antipolis 14/37

15 Monogenic unctions Let W : Ω H, : Ω R a non-vanishing unction. Deine the main Vekua equation by DW = D The operator D D C H corresponding to this equation appears in dierent actorizations, or example when u is scalar ( ) σ u = σ 1/2 σ1/2 σ 1/2 u σ 1/2 W. ( = σ 1/2 (D + M D ) D D ) C H σ 1/2 u, where σ = 2, C H is the quaternion conjugate operator and M D the right multiplication operator. is CINVESTAV INRIA Sophia Antipolis 15/37

16 Monogenic unctions We said that W = W 0 + W = W i=1 e iw i satisies the main Vekua equation i and only i the scalar part W 0 and the vector part W satisy div ( W ) = 0, curl ( W ) = 2 ( W0 ). In other words, D( W ) = 2 ( W0 ). And we have a div-curl system, we will give an explicit solution to solve them. CINVESTAV INRIA Sophia Antipolis 16/37

17 Monogenic unctions Let W such that DW = D W. Then 2 W 0 = 0. W 0 is a solution o the stationary Schrödinger equation W 0 + r 0 W 0 = 0, with r 0 = rot ( 2 rot ( W ) ) = 0.. u = W 0 + W is a solution o the R-linear Beltrami equation Du = Du. In the book Applied pseudoanalytic unction theory rom the author Vladislav V. Kravchenko, 2009, is given a proo o the above results. CINVESTAV INRIA Sophia Antipolis 17/37

18 Monogenic unctions In the complex case, Ω C, W = W 0 + iw 1 : Ω C : Ω R, the main Vekua equation is given by W = W, where = 1 2 ( x + i y ). And the imaginary part W 1 also satisies a conductivity and a Schrödinger equation: 2 (W 1 ) = 0, W 1 + r 1 W 1 = 0, where r 1 = ( ) 1. And the corresponding conjugate Beltrami equation is W = W. CINVESTAV INRIA Sophia Antipolis 18/37

19 Monogenic unctions I the scalar part W 0 : Ω R is known, how to costruct 2 hyperconjugate pairs? Theorem Let 2 be a conductivity o class C 2 in an open star-shaped set Ω R 3. Suppose that W 0 C 2 (Ω, R) satisies the conductivity equation 2 (W 0 / ) = 0 in Ω. Then exists a unction W such that W 0 + W such that DW = D W. The unction W is unique up to the gradient o a real harmonic unction. The assumptions o the above Theorem can be relaxed to say that, W 0 H 1 (Ω, R) and satisy the conductivity equation weakly. CINVESTAV INRIA Sophia Antipolis 19/37

20 Monogenic unctions The ollowing system o equations corresponds to the static Maxwell system, in a medium when just the permeability 2 is variable: div ( 2 H) = 0, div E = 0, curl H = g, curl E = 2 H. (6) Here E and H represent electric and magnetic ields, respectively. We will apply our results to this system and to the double curl equation curl ( 2 curl E) = g, (7) which is immediate rom the last two equations o (6). CINVESTAV INRIA Sophia Antipolis 20/37

21 Monogenic unctions Theorem Let the domain Ω R 3 be a star-shaped open set, and assume that 2 is a continuous proper conductivity in Ω. Let g L 2 (Ω, R 3 ) satisy div g = 0. Then there exists a generalized solution ( E, H) to the system (6) and its general orm is given by E = T 2,Ω [ 2 ( B + h]) S Ω [T 0,Ω [ 2 ( B + h]] + h 1, H = B + h, (8) where h 1 is an arbitrary real valued harmonic unction, h satisy the inhomogeneous conductivity equation div ( 2 h) = 2 B and B = T 2,Ω [ g] S Ω [T 0,Ω [ g]]. CINVESTAV INRIA Sophia Antipolis 21/37

Such methods have a variety o dierent applications or instance in

22 Monogenic unctions In impedance tomography one aims to determine the internal structure o a body rom electrical measurements on its surace. Such methods have a variety o dierent applications or instance in engineering and medical diagnostics. CINVESTAV INRIA Sophia Antipolis 22/37

23 Monogenic unctions Sobolev spaces In particular, we are interested in the Sobolev spaces H 1 (Ω) and H 1/2 ( Ω), where H 1 (Ω) = { u L 2 (Ω) : u L 2 (Ω) }, u 2 H 1 = u 2 L 2 + u 2 L 2. While H 1/2 ( Ω) are the boundary value unctions in H 1 (Ω): H 1/2 ( Ω) = { u L 2 ( Ω) u H 1 (Ω) con u Ω = u }, u H 1/2 = in { u H 1 u Ω = u}. CINVESTAV INRIA Sophia Antipolis 23/37

24 Monogenic unctions In 1980 A. Calderón showed that the impedance tomography problem admits a clear ans precise mathematical ormulation: Let Ω R n be a bounded domain with connected complement and σ : Ω (0, ) is measurable, with σ and 1/σ bounded. Given the boundary values φ H 1/2 ( Ω) exists a unique solution u H 1 (Ω) to σ u = 0 in Ω, u Ω = φ. This so-called conductivity equation describes the behavior o the electric potential in a conductive body. CINVESTAV INRIA Sophia Antipolis 24/37

25 associated to the equation DW = 0 associated to the equation DW = D W Theorem (Plemelj-Sokhotski ormula) Let be Hölder continuous on a suiciently smooth surace Ω. Then at any regular point t Ω we have n.t.- ĺım x t F Ω [w](x) = 1 2 [±w(t) + S Ω[w](t)], where x Ω ±, with Ω + = Ω and Ω = R 3 \ Ω. The notation n.t.-ĺım x t means that the limit should be taken non-tangential. Holomorphic unctions in the Plane and n-dimensional Space rom the authors Klaus Gürlebeck, Klaus Habetha and Wolgang Sprößig, CINVESTAV INRIA Sophia Antipolis 25/37

26 associated to the equation DW = 0 associated to the equation DW = D W Deine the ollowing principal value integral obtained rom the Cauchy-Bitsadze integral S Ω [w](x) := 2PV E(y x)dy w(y), x Ω, Ω From now on, Ω = B 3 = { x R 3 : x < 1 } and Ω = S 2 = { x R 3 : x = 1 }. Since the unitary normal vector to Ω is η(y) = y, then the Cauchy kernel multiplied by the normal vector is reduced to y x 4π y x 3 y = 1 4π ( 1 2 y x + x y y x 3 ). CINVESTAV INRIA Sophia Antipolis 26/37

27 associated to the equation DW = 0 associated to the equation DW = D W The ollowing operators appear in the decomposition o the singular integral S Ω M[w](x) := 1 w(y) 4π Ω y x ds y, M 1 [w](x) := PV 1 x 2 y 3 x 3 y 2 2π Ω y x 3 w(y)ds y, M 2 [w](x) := PV 1 x 3 y 1 x 1 y 3 2π Ω y x 3 w(y)ds y, M 3 [w](x) := PV 1 x 1 y 2 x 2 y 1 2π y x 3 w(y)ds y. Ω 3 S Ω = M + e i M i. i=1 CINVESTAV INRIA Sophia Antipolis 27/37

28 associated to the equation DW = 0 associated to the equation DW = D W According to the article [3], we deine the H o u as H : L 2 ( Ω, R) L 2 ( Ω, R 3 ) ( 3 ) H(u) : = e i M i (I + M) 1 u. i=1 The authors o the article noticed that i we deine the ollowing strategic R-valued unction h := 2(I + M) 1 u = 2 (I + Sc (S Ω )) 1 u. [3] Hilbert Transorms on the Sphere with the Cliord Algebra Setting, 2009, rom the authors Tao Qian and Yan Yang. CINVESTAV INRIA Sophia Antipolis 28/37

29 associated to the equation DW = 0 associated to the equation DW = D W Thus H(u) := 1 2 Vec (S Ωh). And the monogenic extension in Ω is given by the Cauchy operator F Ω (h). On the other hand, i we take the non-tangential limit then n.t. ĺım x t F Ω (h)(x) = 1 2 (h(t) + S Ωh(t)) = 1 2 h(t) Sc(S Ωh(t)) Vec(S Ωh(t)) = 1 (I + M)h(t) + H(u)(t) 2 = (u + H(u)) (t). CINVESTAV INRIA Sophia Antipolis 29/37

30 associated to the equation DW = 0 associated to the equation DW = D W Now, we are interested in to deine the associated to the main Vekua equation DW = D W : From now on the conductivity σ = 2 H 1 (Ω, R). Suppose that ϕ H 1/2 ( Ω, R) is known, and σ, 1 σ : Ω (0, ) are measurables and bounded. Then there exists an unique extension W 0 H 1 (Ω) such that ( ) 2 W0 = 0 in Ω, W 0 Ω = ϕ in Ω. CINVESTAV INRIA Sophia Antipolis 30/37

31 associated to the equation DW = 0 associated to the equation DW = D W Using the decomposition o the Teodorescu operator T Ω : L 2 (Ω) H 1 (Ω), we have that T Ω [ 2 (W 0 / ) ] = T 0,Ω [ 2 (W 0 / ) ] + T 2,Ω [ 2 (W 0 / ) ]. And the corresponding boundary value unction o the scalar and the vectorial part o T Ω are denoted as [ u 0 (t) = ĺım T 0,Ω 2 (W 0 / ) ], x t [ u (t) = ĺım T2,Ω 2 (W 0 / ) ] x t or t Ω. Notice that g H 1/2 ( Ω), since the trace operator γ : H 1 (Ω) H 1/2 ( Ω). CINVESTAV INRIA Sophia Antipolis 31/37

32 Deinition Div-Curl system associated to the equation DW = 0 associated to the equation DW = D W Finally, deine the H associated to the main Vekua equation as H : H 1/2 ( Ω, R) L 2 ( Ω, R 3 ) ϕ u H(u 0 ), where u 0 and u are built as above using the extension or ϕ as solution o the conductivity equation. Analogously, like in the monogenic case Then h : = 2(I + M) 1 u 0. H (ϕ) = u 1 2 Vec(S Ωh ). CINVESTAV INRIA Sophia Antipolis 32/37

33 associated to the equation DW = 0 associated to the equation DW = D W In particular, when 1 we have that H = H. The associated to the main Vekua equation H : H 1/2 ( Ω, R) L 2 ( Ω, R 3 ) is a bounded operator. More even H (u) Sol ( Ω, R 3 ), or all u H 1/2 ( Ω, R). CINVESTAV INRIA Sophia Antipolis 33/37

34 associated to the equation DW = 0 associated to the equation DW = D W Theorem Let Ω = B 3. Let H 1 (Ω, R) a proper conductivity. Suppose that ϕ H 1/2 ( Ω, R). Then there exists an extension W = W i=1 e iw i in Ω such that DW = D W 0 Ω = ϕ. W in Ω, CINVESTAV INRIA Sophia Antipolis 34/37

35 associated to the equation DW = 0 associated to the equation DW = D W Sketch o the proo: The extension o the vector part W is obtained directly rom the H and the integral operators T Ω and F Ω : Thus W : = T Ω [ 2 (W 0 / )] + F Ω [H (ϕ)]. =T 0,Ω [ 2 (W 0 / ) ] + T 2,Ω [ 2 (W 0 / ) ] + F Ω [ u H(u 0 )]. W := W 0 + W satisy the main Vekua equation. W is purely vectorial. CINVESTAV INRIA Sophia Antipolis 35/37

36 associated to the equation DW = 0 associated to the equation DW = D W Although the above extension is not unique, it is the only one that satisies n.t. ĺım x t W (x) = H (ϕ),, x Ω. CINVESTAV INRIA Sophia Antipolis 36/37

37 associated to the equation DW = 0 associated to the equation DW = D W MERCI! CINVESTAV INRIA Sophia Antipolis 37/37

A Simple Explanation of the Sobolev Gradient Method

A Simple Explanation of the Sobolev Gradient Method A Simple Explanation o the Sobolev Gradient Method R. J. Renka July 3, 2006 Abstract We have observed that the term Sobolev gradient is used more oten than it is understood. Also, the term is oten used