Results on the elastic boundary map, the augmented system, planar transport, exact solutions and the null conic, with their assumptions

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• Positive elastic energy — Lemma 1.1. For real mu > 0 and 3 lambda + 2 mu > 0 and every real symmetric 3x3 matrix A, lambda (tr A)^2 + 2 mu |A|^2 >= min(2 mu, 3 lambda + 2 mu) |A|^2. No hypothesis.
• Korn's identity with zero boundary values — Lemma 2.1. For phi in H^1_0(Omega; R^3), the integral of |grad phi|^2 equals twice the integral of |e(phi)|^2 minus the integral of (div phi)^2. No hypothesis.
• The Dirichlet problem for the Lamé system — Proposition 2.3. For v in H^1(Omega; R^3) there is exactly one weak solution u of L u = 0 with u - v in H^1_0, depending only on the class of v modulo H^1_0, with ||u||_{H^1} <= C ||v||_{H^1}. No hypothesis.
• The displacement-to-traction map — Definition 3.1. The bounded symmetric map Lambda from the trace space H^1/H^1_0 to its dual given by the energy form of the solutions of the Dirichlet problem. No hypothesis.
• Global uniqueness for isotropic elasticity (OpenAI) — Theorem 4.1. On a bounded connected domain in R^3 with smooth boundary, smooth moduli with mu > 0 and 3 lambda + 2 mu > 0 are determined by the displacement-to-traction map; proved in the seventh lesson. Uses Hypothesis (BD) (boundary determination of all derivatives of the moduli, Tan-Liu 2023, Theorem 1.2); Hypothesis (ER) (interior elliptic regularity for systems, with uniform constants for smooth families, Dyatlov's notes, Section 15.1); Hypothesis (CE) (the Carleman estimate with a convexified linear weight, Salo-Tzou 2009, Lemma 2.1 with s = -1); Hypothesis (HF) (invertible solutions of d/dz-bar V = A V over a whole disk, Forstneric 2009, Theorem 1.1).
• Gluing functions with vanishing jets — Lemma 2.2. If H is smooth on R^3 with all derivatives zero on the boundary of Omega, then H on the closure of Omega extended by zero is smooth. No hypothesis.
• A common exterior extension of the moduli — Lemma 3.1. Two pairs with the same boundary map have smooth extensions to R^3 satisfying the positivity conditions, equal outside Omega and equal to one constant pair outside a ball. Uses Hypothesis (BD) (boundary determination of all derivatives of the moduli, Tan-Liu 2023, Theorem 1.2).
• Transfer of solutions — Proposition 4.1. Every smooth solution of L_1 u_1 = 0 on a ball containing the closure of B_r has a smooth solution u_2 of L_2 u_2 = 0 on the ball that equals u_1 outside the closure of Omega. Uses Hypothesis (BD) (boundary determination of all derivatives of the moduli, Tan-Liu 2023, Theorem 1.2); Hypothesis (ER) (interior elliptic regularity for systems, with uniform constants for smooth families, Dyatlov's notes, Section 15.1).
• The augmented system — Lemma 1.1. L u = R(u, div u) and div R(u,d) - S(u,d) = P(div u - d) for all smooth pairs (u, d); so L u = 0 gives R(u, div u) = 0 and S(u, div u) = 0. No hypothesis.
• The normalized augmented operator — Lemma 1.2. The operator (m^-1 R, l^-1 (S - 2 m^-1 grad m . R)) is the Laplacian times the identity plus lower-order terms; for two pairs equal outside Omega the difference has order at most one with coefficients supported in the closure of Omega. No hypothesis.
• A supported Carleman estimate — Proposition 3.1. For U in C^infinity_c(B; C^4) and small h = 1/tau, ||E_tau^-1 U||_{H^1_h} <= C h ||E_tau^-1 cal-L U||_{H^-1_h}, with E_tau = exp(tau theta . x) and theta null. Uses Hypothesis (CE) (the Carleman estimate with a convexified linear weight, Salo-Tzou 2009, Lemma 2.1 with s = -1).
• Solvability with an exponential phase — Corollary 4.1. The conjugated Lamé operator A_theta has, for every f in L^2(B), a distributional solution z in L^2(B) with ||z|| <= C ||f||. Uses Hypothesis (CE) (the Carleman estimate with a convexified linear weight, Salo-Tzou 2009, Lemma 2.1 with s = -1).
• Interior estimates at the scale 1/tau — Lemma 5.1. Solutions of A_theta z = f satisfy the sum over j <= 2 of h^j ||grad^j z||_{L^2(B')} <= C(||z||_{L^2(B)} + h^2 ||f||_{L^2(B)}). Uses Hypothesis (ER) (interior elliptic regularity for systems, with uniform constants for smooth families, Dyatlov's notes, Section 15.1).
• The Cauchy transform — Lemma 1.1. For bounded f supported in a disk of radius delta, |T f| <= 2 delta sup |f|, T f is continuous and d/dz-bar T f = f in the sense of distributions. No hypothesis.
• Regularity for d/dz-bar — Lemma 1.2. Locally square-integrable solutions of d/dz-bar W = b W + g with smooth b, g are smooth. No hypothesis.
• Local frames — Lemma 2.1. Every point has a disk on which d/dz-bar V = A V has an invertible solution V with |V - Id| <= 1/2. No hypothesis.
• Frames depending on a parameter — Lemma 2.2. For smooth A on a disk times an interval, there are locally in the parameter invertible solutions of d/dz-bar V = A V on any smaller disk, smooth in all variables. Uses Hypothesis (HF) (invertible solutions of d/dz-bar V = A V over a whole disk, Forstneric 2009, Theorem 1.1).
• Inhomogeneous planar equations — Lemma 2.3. d/dz-bar v = A v + F has smooth solutions on smaller disks, vanishing for parameters where F vanishes outside a compact set. Uses Hypothesis (HF) (invertible solutions of d/dz-bar V = A V over a whole disk, Forstneric 2009, Theorem 1.1).
• Supported planar transport — Lemma 3.1. Compactly supported L^2 solutions of the homogeneous planar equation vanish; supported solutions satisfy Sobolev estimates; if supported solutions of the inhomogeneous equation exist for a dense set of parameters, they exist for all, are unique and depend smoothly on the parameters. No hypothesis.
• Conjugation by the exponential phase — Lemma 1.1. E_tau^-1 R(E_tau a, E_tau b) = R(a,b) + tau R^theta(a,b), and similarly for S and P, with explicit first-order operators R^theta, S^theta, P^theta. No hypothesis.
• Compatibility of the leading operators — Lemma 1.2. theta . R^theta(a,b) = P^theta(theta . a), and div R^theta + theta . R - S^theta = P(theta . a) + P^theta(div a - b). No hypothesis.
• Exact solutions with prescribed leading terms (OpenAI) — Proposition 3.1. Every local frame of the constrained transport equation on a cylinder is realized, after a cutoff, as the leading displacement and divergence of exact solutions u_tau of L u = 0 on B_5r, with E_tau^-1 (u_tau, div u_tau) converging in L^2(B_4r) and weighted first derivatives O(tau). Uses Hypothesis (ER) (interior elliptic regularity for systems, with uniform constants for smooth families, Dyatlov's notes, Section 15.1); Hypothesis (CE) (the Carleman estimate with a convexified linear weight, Salo-Tzou 2009, Lemma 2.1 with s = -1); Hypothesis (HF) (invertible solutions of d/dz-bar V = A V over a whole disk, Forstneric 2009, Theorem 1.1).
• The transport normal form — Lemma 1.1. In the variables p = sqrt(m) a, s = -(k / 2 sqrt(m)) b - g . p / 2, the constrained transport equation becomes D p = theta s, D s = Q_theta . p + T_theta s with T_theta = D log(k/l) and an explicit Q_theta linear in theta. No hypothesis.
• A parametrization of the null conic — Lemma 2.1. [z0 : z1] -> [(z0^2 - z1^2, i(z0^2 + z1^2), 2 z0 z1)] is a bijection of the complex projective line onto the conic of null directions, with explicit inverse charts. No hypothesis.
• Matching of the transports (OpenAI) — Proposition 4.1. There is a unique smooth field G(x, [theta]) of endomorphisms of E_theta with D_theta G = M_2 G - G M_1 and G = Id for |x| > 2r; it is holomorphic in the direction. Uses Hypothesis (BD) (boundary determination of all derivatives of the moduli, Tan-Liu 2023, Theorem 1.2); Hypothesis (ER) (interior elliptic regularity for systems, with uniform constants for smooth families, Dyatlov's notes, Section 15.1); Hypothesis (CE) (the Carleman estimate with a convexified linear weight, Salo-Tzou 2009, Lemma 2.1 with s = -1); Hypothesis (HF) (invertible solutions of d/dz-bar V = A V over a whole disk, Forstneric 2009, Theorem 1.1).
• Holomorphic functions and sections over the null conic — Lemma 1.1. Holomorphic functions on the conic are constant, null vectors span C^3, and every holomorphic section of the bundle H of planes theta-perp is zero. No hypothesis.
• A frame of H and the quotient by the null line — Lemma 1.2. h1 = (z0, i z0, z1), h2 = (-z1, i z1, z0) frame H, theta = z0 h1 + z1 h2, the form pi = -z1 xi1 + z0 xi2 identifies H/N with the trivial bundle, and holomorphic endomorphism fields of H vanishing on N are zero. No hypothesis.
• Rigidity of the comparison (OpenAI) — Proposition 2.1. The comparison field is the identity, so the transport matrices of the two media agree on E_theta for every null theta. Uses Hypothesis (BD) (boundary determination of all derivatives of the moduli, Tan-Liu 2023, Theorem 1.2); Hypothesis (ER) (interior elliptic regularity for systems, with uniform constants for smooth families, Dyatlov's notes, Section 15.1); Hypothesis (CE) (the Carleman estimate with a convexified linear weight, Salo-Tzou 2009, Lemma 2.1 with s = -1); Hypothesis (HF) (invertible solutions of d/dz-bar V = A V over a whole disk, Forstneric 2009, Theorem 1.1).
• Finite-type uniqueness — Lemma 3.1. If d_i d_j eta = sum_a B_ij^a d_a eta + delta_ij q on a connected open set in R^n, n >= 2, and eta vanishes on a nonempty open subset, then eta = q = 0. No hypothesis.

Reading order

Seven lessons. The first defines the boundary map and states the theorem; the second reduces the comparison to solutions that differ only inside the domain; the third and fourth prove the analytic tools (a Carleman estimate for the augmented system, and planar transport equations), and can be read on their own; the fifth constructs exact solutions; the sixth compares the two media; the seventh proves the theorem. Four results from the literature are used as named hypotheses and are stated where they are used.