The Korn–Lichtenstein Theorem and the Elliptic Canonical Form

Derives Proposition A13.4. Expand, using \(\zeta_{x} = \xi_{x}+\ii\eta_{x}\) and \(\zeta_{y} = \xi_{y}+\ii\eta_{y}\):

The real parts sum to \(\bigl(a\xi_{x}^{2}+2b\xi_{x}\xi_{y}+c\xi_{y}^{2}\bigr) -\bigl(a\eta_{x}^{2}+2b\eta_{x}\eta_{y}+c\eta_{y}^{2}\bigr) = \tilde a - \tilde c\) by Equation (14.12), and the imaginary parts sum to twice the polarisation \(a\xi_{x}\eta_{x}+b(\xi_{x}\eta_{y}+\eta_{x}\xi_{y})+c\xi_{y}\eta_{y} = \tilde b\).

Derives Proposition A13.5. The quadratic \(a t^{2}+2bt+c\) has the two roots \(t_{\pm} = \bigl(-b\pm\ii\delta\bigr)/a\), so \(a\zeta_{x}^{2}+2b\zeta_{x}\zeta_{y}+c\zeta_{y}^{2} = a\bigl(\zeta_{x}-t_{+}\zeta_{y}\bigr) \bigl(\zeta_{x}-t_{-}\zeta_{y}\bigr)\) and the equation of Proposition A13.4 holds if and only if one of the two factors vanishes. Take the factor \(\zeta_{x}-t_{-}\zeta_{y} = 0\), i.e.\ \(a\zeta_{x} + \bigl(b+\ii\delta\bigr)\zeta_{y} = 0\) (the other branch is the complex conjugate statement and produces \(\bar\zeta\)). Separating real and imaginary parts of \(a(\xi_{x}+\ii\eta_{x}) + (b+\ii\delta)(\xi_{y}+\ii\eta_{y}) = 0\),

The first is the second half of Equation (A13.5). Substituting it into the second and using \(\delta^{2} = ac-b^{2}\), so that \(b^{2}+\delta^{2} = ac\),

and dividing by \(a \neq 0\) gives the first half. The steps are reversible, which is the “only if”.

For Equation (A13.6), substitute Equation (A13.5):

which is the displayed quadratic form. That form is \(\vect{v}\transpose A\,\vect{v}/\delta\) with \(\vect{v} = (\xi_{x},\xi_{y})\) and \(A\) the positive definite matrix of Equation (14.11) (Remark A13.3), so it is positive whenever \(\vect{v} \neq 0\) and zero otherwise. Finally Proposition A13.4 gives \(\tilde a = \tilde c\) and \(\tilde b = 0\), and Equation (A13.6) identifies \(\tilde a = \delta J_{0}\), which is Equation (A13.3).

Derives Proposition A13.7. Substitute \(\zeta_{x} = \pp_{z}\zeta+\pp_{\bar z}\zeta\) and \(\zeta_{y} = \ii\bigl(\pp_{z}\zeta-\pp_{\bar z}\zeta\bigr)\) from Equation (A13.1) into \(a\zeta_{x}+(b+\ii\delta)\zeta_{y}=0\) and use \((b+\ii\delta)\ii = -\delta+\ii b\):

The coefficient of \(\pp_{\bar z}\zeta\) has modulus squared \((a+\delta)^{2}+b^{2} > 0\) and never vanishes, so the equation may be solved for \(\pp_{\bar z}\zeta\), giving Equation (A13.9). Its modulus is Equation (A13.10), and \((a-\delta)^{2} < (a+\delta)^{2}\) because \(a\) and \(\delta\) are both positive; equality of the numerator with \(0\) requires \(b = 0\) and \(a = \delta = \sqrt{ac}\), i.e. \(a = c\) and \(b=0\).

For Equation (A13.11), write \(A_{1} = \zeta_{x}\) and \(B_{1} = \ii\zeta_{y}\), so that \(\pp_{z}\zeta = \tfrac12(A_{1}-B_{1})\) and \(\pp_{\bar z}\zeta = \tfrac12(A_{1}+B_{1})\). Then

and \(\bar\zeta_{x}\zeta_{y} = (\xi_{x}-\ii\eta_{x})(\xi_{y}+\ii\eta_{y})\) has imaginary part \(\xi_{x}\eta_{y}-\eta_{x}\xi_{y} = J_{0}\). Inserting Equation (A13.9) gives the second equality.

Derives Lemma A13.8. \(A(P_{0})\) is symmetric and positive definite (Remark A13.3), so it factors as \(A(P_{0}) = S\,S\transpose\) with \(S\) real and invertible — take \(S = O\Lambda^{1/2}\) with \(A(P_{0}) = O\Lambda O\transpose\) its orthogonal diagonalization, the eigenvalues in \(\Lambda\) being positive. Change variables by \(\hat{\vect{x}} = S^{-1}\vect{x}\), whose Jacobian is the constant matrix \(J = S^{-1}\); by Equation (14.6) the new principal matrix is

Congruence with a constant matrix preserves Hölder continuity and, by Theorem 14.8, ellipticity. At \(P_{0}\) one then has \(\hat a = \hat c = 1\), \(\hat b = 0\), \(\hat\delta = 1\), and Equation (A13.9) gives \(\mu(P_{0}) = -(1-1+0)/(1+1-0) = 0\). Since a composition of two changes of coordinates with non-vanishing Jacobians is again one, proving Theorem A13.1 for the normalized problem proves it in general.

Derives Lemma A13.9. \(\mu\) is a rational function of \(a,b,\delta\) with non-vanishing denominator, hence of class \(C^{\alpha}\) on a neighbourhood of \(0\); shrink that neighbourhood to a disc \(D_{\rho}\) and take \(r < \rho/2\) so that \(rz \in D_{\rho}\) for \(z \in D_{2}\). Write \(\mu^{(r)}(z) = \mu(rz)\). Since \(\mu(0)=0\) by Lemma A13.8,

and directly from Equation (A13.2), \([\mu^{(r)}]_{\alpha,D_{2}} = r^{\alpha}[\mu]_{\alpha}\). For the product with the cutoff, use \(\abs{\chi f(z)-\chi f(w)} \le \abs{\chi(z)}\abs{f(z)-f(w)} + \abs{f(w)}\abs{\chi(z)-\chi(w)}\), so that

which with Equation (A13.15) gives \([\nu_{r}]_{\alpha} \le r^{\alpha}[\mu]_{\alpha} + (2r)^{\alpha}[\mu]_{\alpha}[\chi]_{\alpha}\). Adding \(\norm{\nu_{r}}_{\infty} \le (2r)^{\alpha}[\mu]_{\alpha}\) gives Equation (A13.14) with \(K = 1+2^{\alpha}\bigl(1+[\chi]_{\alpha}\bigr)\). The bound \(\norm{\nu_{r}}_{\infty} \le \sup\abs{\mu} < 1\) is Equation (A13.10) and \(0\le\chi\le1\). Under the stronger hypothesis \(a,b,c \in C^{1,\alpha}\) one has \(\mu \in C^{1,\alpha}\) and \(\pp\mu^{(r)} = r\,(\pp\mu)(rz)\), so every seminorm entering the \(C^{1,\alpha}\) norm of \(\mu^{(r)}\) carries at least one factor \(r\), and the same product estimate applies.

Derives Lemma A13.12. Let \(G(z) = (2\pi)^{-1}\ln\abs{z}\), which is harmonic on \(\C \setminus \set{0}\) because \(\nabla^{2}\ln r = r^{-1}\pp_{r}(r\,\pp_{r}\ln r) = r^{-1}\pp_{r}(1)=0\) in plane polar coordinates. Let \(\varphi\) be a test function. Since \(\ln\abs{z}\) is integrable near the origin, \(\int G\nabla^{2}\varphi\,\dd A = \lim_{\varepsilon\to0^{+}}\int_{r>\varepsilon}G\nabla^{2}\varphi\,\dd A\). On \(r>\varepsilon\) apply Green's second identity Equation (14.41) with \(p=1\), \(u=\varphi\), \(v=G\); the term \(\varphi\nabla^{2}G\) vanishes, and the only boundary is the circle \(r=\varepsilon\), whose outward normal for the region \(r>\varepsilon\) points at the origin, so \(\pp_{\nu} = -\pp_{r}\) there. Hence

using \(\pp_{r}G = (2\pi r)^{-1}\). The circle has length \(2\pi\varepsilon\), so the first term tends to \(-\varphi(0)\) by continuity of \(\varphi\) and the second is bounded by \(\varepsilon\abs{\ln\varepsilon}\sup\abs{\nabla\varphi} \to 0\). Thus \(\int G\nabla^{2}\varphi\,\dd A = \varphi(0)\), which is the first half of Equation (A13.18). For the second, note \(\nabla^{2} = 4\pp_{z}\pp_{\bar z}\) and \(\pp_{z}\ln\abs{z} = \pp_{z}\tfrac12\ln\bigl(z\bar z\bigr) = 1/(2z)\), so \(\delta = 4\pp_{\bar z}\pp_{z}G = 4\pp_{\bar z}\bigl(1/(4\pi z)\bigr) = \pp_{\bar z}\bigl(1/(\pi z)\bigr)\). Finally Equation (A13.17) is the convolution of \(f\) with \(1/(\pi z)\), and differentiating a convolution moves the derivative onto either factor.

Derives Proposition A13.16. Each \(h_{n}\) lies in \(X\): it is a product one of whose factors is \(\nu\), which vanishes off \(\overline{D_{2}}\), and it is Hölder because \(Th_{n-1}\) is, by Equation (A13.20). By the product estimate Equation (A13.16) applied to \(\nu\,Tg\) and then Equation (A13.20),

for every \(g \in X\). Hence \([h_{n}]_{\alpha} \le \varepsilon_{\nu}^{\,n}[\nu]_{\alpha}\) by induction, and the series Equation (A13.22) is dominated term by term by a geometric series of ratio \(\varepsilon_{\nu}<1\). Since \([\,\cdot\,]_{\alpha}\) is a complete norm on \(X\) (Remark A13.15), the series converges there, with the bound stated in Equation (A13.23); the same domination and Lemma 13.6 give uniform convergence, so the sum may be rearranged and \(T\) applied term by term, \(T\) being bounded. Then

which is Equation (A13.23). Uniqueness: if \(h,h'\in X\) both satisfy it, then \([h-h']_{\alpha} = [\nu T(h-h')]_{\alpha} \le \varepsilon_{\nu}[h-h']_{\alpha}\) by Equation (A13.24), and \(\varepsilon_{\nu}<1\) forces \([h-h']_{\alpha}=0\), hence \(h=h'\) since both vanish off a compact set.

Derives Proposition A13.17. By part (ii) of Theorem A13.13 applied to \(h \in X\),

both continuous and Hölder by part (iii), so \(\zeta \in C^{1,\alpha}\). The Beltrami equation \(\pp_{\bar z}\zeta = \nu\,\pp_{z}\zeta\) reads, after Equation (A13.27), precisely \(h = \nu(1+Th)\), which is Equation (A13.23). For the Jacobian, Equation (A13.20) and Equation (A13.23) give

by Equation (A13.25), so \(\abs{\pp_{z}\zeta} = \abs{1+Th} \ge 1 - \tfrac12 = \tfrac12\) everywhere. Equation (A13.11) then yields the second half of Equation (A13.26), and \(\norm{\nu}_{\infty}<1\) makes it strictly positive.

Proof of Theorem A13.1. Derives Theorem A13.1. By Remark A13.3 assume \(a>0\), and by Lemma A13.8 assume further that \(A(P_{0}) = \identity\) and \(\mu(P_{0})=0\), \(P_{0}\) being the origin of \(\C\); both reductions are linear changes of the independent variables with constant non-vanishing Jacobian, and composing them with the map constructed below gives the map of the theorem.

Let \(\nu_{r}\) be the cutoff coefficient of Lemma A13.9. By Equation (A13.14) both \(\varepsilon_{\nu_{r}}\) of Equation (A13.21) and the left-hand side of Equation (A13.25) are bounded by \(C_{\alpha}K r^{\alpha}[\mu]_{\alpha}\) times a constant, so choosing \(r\) small enough makes \(\varepsilon_{\nu_{r}} \le \tfrac12\) and Equation (A13.25) hold. Fix such an \(r\). Propositions A13.16 and A13.17 then produce \(\hat\zeta \in C^{1,\alpha}(\C)\) with \(\pp_{\bar z}\hat\zeta = \nu_{r}\pp_{z}\hat\zeta\) and Jacobian bounded below by a positive constant.

On \(\overline{D_{1}}\) one has \(\nu_{r}(z) = \mu(rz)\), so \(\zeta(w) = \hat\zeta(w/r)\) satisfies Equation (A13.9) on \(D_{r}\) by the scaling identity of Remark A13.10, with Jacobian multiplied by \(r^{-2}>0\) and hence still bounded away from \(0\). Set \(V = D_{r}\) and \((\xi,\eta) = (\Re\zeta,\Im\zeta)\), which is \(C^{1,\alpha}\). By Proposition A13.7 the pair satisfies the branch \(a\zeta_{x}+(b+\ii\delta)\zeta_{y}=0\) of the complex characteristic equation, so by Propositions A13.4 and A13.5 the transformed coefficients obey \(\tilde b = 0\) and \(\tilde a = \tilde c = \delta J_{0} > 0\), which is Equation (A13.3). The transformed principal part is therefore \(\tilde a\,u_{\xi\xi}+\tilde c\,u_{\eta\eta} = \delta J_{0}\bigl(u_{\xi\xi}+u_{\eta\eta}\bigr)\), as claimed.

For the last sentence of the theorem, suppose \(a,b,c\in C^{1,\alpha}\). Then \(\mu \in C^{1,\alpha}\), and by the last clause of Lemma A13.9 the quantity \(\norm{\nu_{r}}_{C^{1,\alpha}}\) is \(O(r)\). Running Proposition A13.16 in the space of \(C^{1,\alpha}\) functions vanishing off \(\overline{D_{2}}\) — legitimate because part (iv) of Theorem A13.13 bounds \(T\) there, and because the product estimate Equation (A13.16) has an evident \(C^{1,\alpha}\) counterpart obtained by applying it to \(f\) and to each \(\pp f\) — gives \(h \in C^{1,\alpha}\), hence \(\zeta \in C^{2,\alpha}\) by Equation (A13.27). The change of coordinates then has continuous second derivatives, the extra term of Equation (14.7) is an honest continuous first-order coefficient, and the transformed equation is the third canonical form of Equation (14.10) in the classical sense.