The Cauchy–Kovalevskaya Theorem

Contents

Formal series, the recursion, and the reduction
Majorants
The majorant problem has a closed-form solution
Proof of the theorem

This appendix proves Theorem 14.22 of Partial Differential Equations: a Cauchy problem in normal form, with real-analytic coefficients and real-analytic data prescribed on a non-characteristic surface, has a real-analytic solution near the point in question, and it is the only analytic one. The method is Kovalevskaya's [Kowalevsky:1875] and is the model for every convergence proof of this type: write the solution as a formal power series, observe that the equation and the data determine its coefficients by a recursion whose coefficients are non-negative, replace the data by larger data for which the same recursion can be summed in closed form, and conclude that the original series converges because it is dominated term by term by one that does.

The argument is arranged in four steps. The formal series and its recursion come first (Formal series, the recursion, and the reduction), together with the reduction of a normal problem of order $k$ to a first-order quasilinear system with vanishing data — carried out at the level of formal series, which is all that is needed and avoids the compatibility questions a reduction of the differential problem would raise. The notion of a majorant and the standard majorizing function then follow (Majorants), the explicit solution of the majorant problem (The majorant problem has a closed-form solution), and the assembly (Proof of the theorem). A closing remark exhibits Lewy's equation, which shows that the analyticity of the data cannot be weakened even to smoothness [Lewy:1957].

Notation. $t \in \R$ and $x = (x_{1},\dots,x_{n}) \in \R^{n}$; $\beta \in \N^{n}$ and $\gamma \in \N^{m}$ are multi-indices in the sense of Equation (14.2), with $\N$ containing $0$; $x^{\beta} = x_{1}^{\beta_{1}}\cdots x_{n}^{\beta_{n}}$; $e_{k}$ is the $k$-th coordinate multi-index; and a multi-index in the $1+n$ variables $(t,x)$ is written $\alpha = (\alpha_{0},\alpha')$ with $\pp^{\alpha} = \pp_{t}^{\alpha_{0}}\pp_{x}^{\alpha'}$. A function is real analytic near a point if on some polydisc about it the function is the sum of an absolutely convergent power series; that is the only property of analyticity used below.

Formal series, the recursion, and the reduction

Definition A14.1 (Normal Cauchy problem of order $k$).

A normal Cauchy problem of order $k$ for the scalar unknown $w$ is

\begin{equation}\tag{A14.1} \pp_{t}^{k}w = G\bigl(t,x,\set{\pp^{\gamma}w}_{\abs{\gamma}\le k,\ \gamma_{0}<k} \bigr)\ec\qquad \pp_{t}^{p}w(0,x) = \varphi_{p}(x)\quad(p = 0,\dots,k-1)\ec \end{equation}

with $G$ and the $\varphi_{p}$ real analytic near the relevant origins. Rests on Definition 14.1 and Proposition 14.11.

Definition A14.2 (First-order quasilinear system with zero data).

For the unknown $u = (u_{1},\dots,u_{m})$,

\begin{equation}\tag{A14.2} \pp_{t}u_{i} = \sum_{j=1}^{m}\sum_{k=1}^{n} a_{ij}^{k}(x,u)\,\pp_{k}u_{j} + b_{i}(x,u)\ec \qquad u_{i}(0,x) = 0\ec \end{equation}

with $a_{ij}^{k}$ and $b_{i}$ real analytic on a neighbourhood of $(x,u) = (0,0)$ in $\R^{n+m}$. Rests on Definitions 14.2 and A14.1.

Lemma A14.3 (The recursion).

Write the coefficients of Equation (A14.2) as convergent series $a_{ij}^{k} = \sum_{\beta,\gamma}a_{ij,\beta\gamma}^{k}\,x^{\beta} u^{\gamma}$ and $b_{i} = \sum_{\beta,\gamma}b_{i,\beta\gamma}\,x^{\beta}u^{\gamma}$, and seek a formal solution $u_{i} = \sum_{\beta,l}u_{i,\beta,l}\,x^{\beta}t^{l}$. Then the coefficients $u_{i,\beta,l}$ are uniquely determined; they vanish for $l=0$; and for every $l \ge 1$

\begin{equation}\tag{A14.3} u_{i,\beta,l} = R_{i,\beta,l}\bigl(\set{a_{ij,\beta'\gamma}^{k}}, \set{b_{i,\beta'\gamma}}\bigr)\ec \end{equation}

where $R_{i,\beta,l}$ is a polynomial with non-negative rational coefficients, the same polynomial for every problem of the shape Equation (A14.2) with the same $m$ and $n$. Rests on Definition A14.2 and Equation (14.2).

Proof.

Derives Lemma A14.3. The data give $u_{i,\beta,0} = 0$ for every $i,\beta$. Substitute the series into Equation (A14.2). On the left,

\begin{equation}\tag{A14.4} \pp_{t}u_{i} = \sum_{\beta,l}(l+1)\,u_{i,\beta,l+1}\,x^{\beta}t^{l}\ep \end{equation}

On the right, $\pp_{k}u_{j} = \sum_{\beta,l}(\beta_{k}+1)u_{j,\beta+e_{k},l}\,x^{\beta}t^{l}$, whose coefficients are non-negative integers times coefficients of $u$; and $a_{ij}^{k}(x,u)$, being a power series in $x$ and in $u$ composed with the series for $u$, has coefficients that are polynomials with non-negative integer coefficients in the $a_{ij,\beta'\gamma}^{k}$ and the $u_{j',\beta',l'}$, since composition and multiplication of formal series involve only sums of products. The same holds for $b_{i}$. Multiplying the two series and collecting the coefficient of $x^{\beta}t^{l}$, the right-hand side is such a polynomial in which every $u$-coefficient carries a time index $l' \le l$: a factor $t^{l'}$ from one series can only be accompanied by factors $t^{l''}$ with $l'+l''\le l$, and no negative powers occur.

Comparing with Equation (A14.4),

\begin{equation}\tag{A14.5} u_{i,\beta,l+1} = \frac{1}{l+1}\,Q_{i,\beta,l}\bigl(\set{a},\set{b}, \set{u_{j,\beta',l'}}_{l'\le l}\bigr)\ec \end{equation}

with $Q$ a polynomial with non-negative integer coefficients. Induction on $l$ therefore determines every coefficient uniquely from the vanishing ones at $l=0$, and substituting the previously obtained expressions into Equation (A14.5) expresses $u_{i,\beta,l}$ as a polynomial in the coefficients of $a$ and $b$ alone. Non-negativity survives every substitution — the only new factors introduced are the positive rationals $1/(l+1)$ — which is Equation (A14.3). That $R$ does not depend on the particular $a,b$ is clear from the construction: only $m$, $n$ and the indices entered.

∎

Lemma A14.4 (Formal reduction of a normal problem).

Every normal Cauchy problem Equation (A14.1) has a unique formal power series solution $\hat w$ about the origin. Moreover there is a system Equation (A14.2), with $m$ and with coefficients $a_{ij}^{k},b_{i}$ built analytically from $G$ and the $\varphi_{p}$, whose unique formal solution $\hat u$ has one distinguished component equal to

\begin{equation}\tag{A14.6} \hat u_{(0,0)} = \hat w - \sum_{p=0}^{k-1}\varphi_{p}(x)\,\frac{t^{p}}{p!}\ep \end{equation}

Rests on Lemma A14.3 and Definition A14.1.

Proof.

Derives Lemma A14.4. Homogenization. Put $v = w - \sum_{p<k}\varphi_{p}(x)t^{p}/p!$. Since $\pp_{t}^{p}\bigl(\varphi_{q}t^{q}/q!\bigr)(0,x) = \delta_{pq}\varphi_{q}(x)$ for $p,q<k$, the new unknown satisfies $\pp_{t}^{p}v(0,x) = 0$ for $p<k$, and Equation (A14.1) becomes an equation of the same shape with a new right-hand side that is $G$ evaluated on the shifted arguments — again analytic near the origin, because it is the composition of $G$ with polynomials in $t$ whose coefficients are derivatives of the analytic $\varphi_{p}$.

The system. Take as unknowns the quantities $u_{\alpha}$ indexed by the multi-indices $\alpha$ in $(t,x)$ with $\abs{\alpha}\le k-1$, intended to be $\pp^{\alpha}v$, together with one extra unknown $u_{\ast}$ intended to be $t$. Their equations are

$\pp_{t}u_{\ast} = 1$;
$\pp_{t}u_{\alpha} = u_{\alpha+e_{t}}$ whenever $\abs{\alpha}\le k-2$, so that $\abs{\alpha+e_{t}}\le k-1$;
for $\abs{\alpha} = k-1$ with $\alpha_{0} < k-1$ — so that $\alpha$ carries at least one $x$-derivative, say $\alpha \ge e_{x_{j}}$ — $\pp_{t}u_{\alpha} = \pp_{x_{j}}u_{\alpha+e_{t}-e_{x_{j}}}$, the index on the right having length $k-1$;
for $\alpha = (k-1)e_{t}$, $\pp_{t}u_{\alpha} = G\bigl(u_{\ast},x,\{\cdot\}\bigr)$, where each argument $\pp^{\gamma}v$ with $\abs{\gamma}\le k-1$ is the unknown $u_{\gamma}$ and each argument with $\abs{\gamma}=k$, $\gamma_{0}<k$, carries an $x$-derivative and is written $\pp_{x_{j}}u_{\gamma-e_{x_{j}}}$.

Every right-hand side is analytic in $(x,u)$ near the origin — the $t$-slot of $G$ has been replaced by the unknown $u_{\ast}$ — and depends linearly on the first $x$-derivatives of the unknowns, which is the shape of Equation (A14.2). All data vanish: $u_{\ast}(0,x)=0$, and $\pp^{\alpha}v(0,x) = \pp_{x}^{\alpha'}\bigl[\pp_{t}^{\alpha_{0}}v(0,x)\bigr] = 0$ because $\alpha_{0}\le k-1$.

The two formal solutions agree. The recursion of Equation (A14.1) determines a unique formal $\hat v$: rewriting the equation as $\pp_{t}^{k}v = \widetilde G$ and matching coefficients of $x^{\beta}t^{l}$ expresses $l$-th and higher $t$-coefficients in terms of lower ones exactly as in Lemma A14.3, the data supplying the $k$ lowest. Now form the formal derivatives $\pp^{\alpha}\hat v$ and $\hat u_{\ast} = t$. Formal differentiation is a ring homomorphism compatible with formal composition, so applying $\pp^{\alpha}$ to the formal identity $\pp_{t}^{k}\hat v = \widetilde G(\cdots)$ and to the trivial identities $\pp_{t}\pp^{\alpha}\hat v = \pp^{\alpha+e_{t}}\hat v$ shows that the family $\bigl(\pp^{\alpha}\hat v,\ t\bigr)$ satisfies every equation (i)–(iv) formally, with the correct vanishing data. By the uniqueness half of Lemma A14.3 it is $\hat u$, and in particular $\hat u_{(0,0)} = \hat v$, which is Equation (A14.6).

∎

Remark A14.5.

The reduction has been made at the level of formal series on purpose. A reduction of the differential problems would require showing that a solution of the system has components that really are the derivatives of its first one, which is an extra argument; here nothing of the kind is needed, because all the reduction is asked to do is to transport convergence from the system to the original problem, and convergence is a property of a formal series.

Majorants

Definition A14.6 (Majorant).

For formal power series $f = \sum_{\sigma}f_{\sigma}w^{\sigma}$ and $F = \sum_{\sigma}F_{\sigma}w^{\sigma}$ in the same variables, write $f \ll F$ — “$F$ majorizes $f$” — if $\abs{f_{\sigma}} \le F_{\sigma}$ for every multi-index $\sigma$. In particular every coefficient of $F$ is then non-negative. Rests on Equation (14.2).

Lemma A14.7 (The standard majorant).

Let $g = \sum_{\sigma}g_{\sigma}w^{\sigma}$ be real analytic near the origin of $\R^{N}$, and let $r>0$ be such that $C = \sum_{\sigma}\abs{g_{\sigma}}\,r^{\abs{\sigma}} < \infty$. Then $\abs{g_{\sigma}} \le C\,r^{-\abs{\sigma}}$ and

\begin{equation}\tag{A14.7} g \ll \frac{C\,r}{r - \bigl(w_{1}+\cdots+w_{N}\bigr)}\ep \end{equation}

Rests on Definition A14.6.

Proof.

Derives Lemma A14.7. The bound $\abs{g_{\sigma}}r^{\abs{\sigma}} \le C$ is immediate, every term of the defining sum being non-negative. For the majorant, expand the geometric series and then the powers by the multinomial theorem:

\begin{equation}\tag{A14.8} \frac{Cr}{r-\bigl(w_{1}+\cdots+w_{N}\bigr)} = C\sum_{q\ge0}\frac{\bigl(w_{1}+\cdots+w_{N}\bigr)^{q}}{r^{q}} = C\sum_{\sigma}\frac{\abs{\sigma}!}{\sigma!\,r^{\abs{\sigma}}}\, w^{\sigma}\ec \end{equation}

where $\sigma! = \sigma_{1}!\cdots\sigma_{N}!$. The multinomial coefficient $\abs{\sigma}!/\sigma!$ is a positive integer, hence at least $1$, so the coefficient of $w^{\sigma}$ in Equation (A14.8) is at least $C r^{-\abs{\sigma}} \ge \abs{g_{\sigma}}$.

∎

Lemma A14.8 (Domination).

Let Equation (A14.2) have coefficients $a,b$ and let $A_{ij}^{k}, B_{i}$ be power series with $a_{ij}^{k} \ll A_{ij}^{k}$ and $b_{i} \ll B_{i}$. Let $U_{i,\beta,l}$ be the coefficients of the formal solution of the majorant problem — the system Equation (A14.2) with $A,B$ in place of $a,b$ and the same vanishing data. Then

\begin{equation}\tag{A14.9} \abs{u_{i,\beta,l}} \le U_{i,\beta,l} \qquad\text{for every } i,\beta,l\ep \end{equation}

Rests on Lemma A14.3 and Definition A14.6.

Proof.

Derives Lemma A14.8. By Lemma A14.3 both sets of coefficients are given by the same polynomials $R_{i,\beta,l}$, evaluated at the coefficients of $a,b$ and of $A,B$ respectively. A polynomial with non-negative coefficients satisfies $\abs{R(\vect{s})} \le R(\abs{\vect{s}})$ by the triangle inequality applied monomial by monomial, and is non-decreasing in each argument on the non-negative orthant; hence

\begin{equation*} \abs{u_{i,\beta,l}} = \abs{R_{i,\beta,l}(a,b)} \le R_{i,\beta,l}\bigl(\abs{a},\abs{b}\bigr) \le R_{i,\beta,l}(A,B) = U_{i,\beta,l}\ec \end{equation*}

the middle inequality using $\abs{a} \le A$ and $\abs{b} \le B$ coefficientwise.

∎

The majorant problem has a closed-form solution

Proposition A14.9 (Explicit solution of the majorant problem).

Fix $C,r>0$ and take, in the notation of Lemma A14.7 with the $N = n+m$ variables $(x,u)$,

\begin{equation}\tag{A14.10} A_{ij}^{k}(x,u) = B_{i}(x,u) = \frac{Cr}{r - s - \bigl(u_{1}+\cdots+u_{m}\bigr)}\ec \qquad s = x_{1}+\cdots+x_{n}\ep \end{equation}

Put $K = m(n+1)$. Then the functions $U_{i} = V$, $i=1,\dots,m$, with

\begin{equation}\tag{A14.11} V(t,x) = \frac{1}{K}\left[(r-s) - \sqrt{(r-s)^{2} - 2KCr\,t}\;\right]\ec \end{equation}

solve the majorant problem, are real analytic on the polydisc

\begin{equation}\tag{A14.12} \abs{x_{k}} < \frac{r}{4n} \quad (k = 1,\dots,n)\ec\qquad \abs{t} < \frac{9\,r}{32\,K\,C}\ec \end{equation}

and all their Taylor coefficients at the origin are non-negative. Rests on Lemma A14.7 and Definition A14.2.

Proof.

Derives Proposition A14.9. Write $\sigma = r-s$ and $R = \sqrt{\sigma^{2}-2KCrt}$, so $V = (\sigma - R)/K$. Since $\pp_{t}R = -KCr/R$ and $\pp_{\sigma}R = \sigma/R$,

\begin{equation}\tag{A14.13} \pp_{t}V = \frac{Cr}{R}\ec\qquad \pp_{s}V = -\pp_{\sigma}V = -\frac{1}{K}\left(1-\frac{\sigma}{R}\right)\ep \end{equation}

With $U_{i}=V$ for every $i$ one has $U_{1}+\cdots+U_{m} = mV$ and $\sum_{j,k}\pp_{k}U_{j} = mn\,\pp_{s}V$, so the majorant problem reads

\begin{equation}\tag{A14.14} \pp_{t}V = \frac{Cr}{\sigma - mV}\bigl(mn\,\pp_{s}V + 1\bigr)\ep \end{equation}

Substituting Equation (A14.13), the requirement Equation (A14.14) is

\begin{equation*} \frac{Cr}{R} = \frac{Cr\left[1 - \dfrac{mn}{K} + \dfrac{mn\,\sigma}{K\,R}\right]} {\sigma\left(1-\dfrac{m}{K}\right) + \dfrac{m}{K}R}\ec \end{equation*}

that is, after cross-multiplying by $R$ and by the denominator,

\begin{equation}\tag{A14.15} \sigma\left(1-\frac{m}{K}\right) + \frac{m}{K}R = R\left(1 - \frac{mn}{K}\right) + \frac{mn}{K}\,\sigma\ep \end{equation}

Because $\sigma$ and $R$ are functionally independent, Equation (A14.15) holds identically if and only if the coefficients match separately: $1 - m/K = mn/K$ and $m/K = 1 - mn/K$. Both give $K = m + mn = m(n+1)$, which is the choice made. At $t = 0$, $R = \abs{\sigma} = \sigma$ for $s<r$, so $V(0,x) = 0$: the data hold.

Analyticity and positivity. Factor

\begin{equation}\tag{A14.16} V = \frac{\sigma}{K}\left[1-\sqrt{1-\frac{2KCr\,t}{\sigma^{2}}}\;\right] = \frac{1}{K}\sum_{p\ge1}c_{p}\,\bigl(2KCr\bigr)^{p}\, t^{p}\,\sigma^{1-2p}\ec \end{equation}

where $1-\sqrt{1-y} = \sum_{p\ge1}c_{p}y^{p}$ is the binomial series, whose coefficients $c_{p} = \dfrac{(2p-2)!}{2^{2p-1}\,p!\,(p-1)!}$ are strictly positive, and which converges absolutely for $\abs{y}<1$. Each factor $\sigma^{1-2p} = r^{1-2p}\bigl(1-s/r\bigr)^{-(2p-1)}$ expands, for $\abs{s}<r$, as $r^{1-2p}\sum_{d\ge0}\binom{2p-2+d}{d}(s/r)^{d}$, again with positive coefficients, and $s^{d} = (x_{1}+\cdots+x_{n})^{d}$ expands with non-negative integer coefficients in the $x^{\beta}$. Hence every Taylor coefficient of $V$ is non-negative, and the multiple series converges absolutely wherever $\abs{s} < r$ and $2KCr\abs{t} < \bigl(r-\abs{s}\bigr)^{2}$. On the polydisc Equation (A14.12) one has $\abs{s} \le \sum_{k}\abs{x_{k}} < r/4$, so $\bigl(r-\abs{s}\bigr)^{2} > 9r^{2}/16$ and the second condition follows from $\abs{t} < 9r/(32KC)$.

∎

Remark A14.10.

The choice Equation (A14.10) is the whole trick, and its two features are independent. Making all the coefficient functions the same single function of $s$ and of $u_{1}+\cdots+u_{m}$ collapses the $m$-component system to one scalar equation in two variables; and making that function the geometric kernel $Cr/(r-s-\sum u_{i})$ makes the resulting scalar equation quasilinear of first order with the closed-form solution Equation (A14.11). Notice that Equation (A14.14) is exactly an equation of the type treated by Proposition 14.15: written as $\bigl(\sigma - mV\bigr)\pp_{t}V - Crmn\,\pp_{s}V = Cr$ it is quasilinear of first order in the two variables $(t,s)$, with characteristic system $\dot t = \sigma - mV$, $\dot s = -Crmn$, $\dot V = Cr$. It is because that system can be integrated — or, as done above, because the resulting closed form can be guessed and then verified — that the majorant problem is solvable at all; the square root in Equation (A14.11) is the mark of the quadratic relation between $V$ and $t$ that the last two characteristic equations impose. That the exponent works out to $K = m(n+1)$ and not to something depending on the data is what makes the radius of convergence in Equation (A14.12) explicit.

Proof of the theorem

Theorem A14.11 (Cauchy–Kovalevskaya, first-order form).

The problem Equation (A14.2) has a real-analytic solution on a polydisc about the origin of $\R^{1+n}$, and it is the only analytic solution there. Rests on Lemma A14.8, Proposition A14.9 and Lemma A14.7.

Proof.

Derives Theorem A14.11. By hypothesis the $a_{ij}^{k}$ and $b_{i}$ are analytic near $(x,u) = (0,0)$; choose $r>0$ so small that all of them are represented by absolutely convergent series on the polydisc of radius $r$ in the $n+m$ variables, and let $C$ be the largest of the finitely many sums $\sum_{\beta\gamma}\abs{a_{ij,\beta\gamma}^{k}}r^{\abs{\beta}+\abs{\gamma}}$ and $\sum_{\beta\gamma}\abs{b_{i,\beta\gamma}}r^{\abs{\beta}+\abs{\gamma}}$. By Lemma A14.7 every coefficient is then majorized by the function Equation (A14.10).

Let $U_{i,\beta,l}$ be the formal solution of the majorant problem. By Proposition A14.9 the function $V$ of Equation (A14.11) solves that problem and is analytic on the polydisc Equation (A14.12); its Taylor coefficients therefore satisfy the recursion of Lemma A14.3 for the majorant data, and by the uniqueness half of that lemma they are the $U_{i,\beta,l}$. Combining with Lemma A14.8,

\begin{equation}\tag{A14.17} \sum_{\beta,l}\abs{u_{i,\beta,l}}\,\abs{x}^{\beta}\abs{t}^{l} \le \sum_{\beta,l}U_{i,\beta,l}\,\abs{x}^{\beta}\abs{t}^{l} = V\bigl(\abs{t},\abs{x}\bigr) < \infty \end{equation}

on Equation (A14.12), where $\abs{x}^{\beta}$ means $\abs{x_{1}}^{\beta_{1}}\cdots$; the middle equality holds because all the $U$ are non-negative, so the series is its own absolute value. Hence the formal series for each $u_{i}$ converges absolutely on that polydisc and defines a function there.

The sum solves the problem. On any closed subpolydisc of ratio $\theta<1$ the differentiated series are dominated term by term by $\theta^{-1}\sup_{q\ge0}\bigl[(q+1)\theta^{q}\bigr]$ times the original one, hence converge uniformly by Lemma 13.6, so term-by-term differentiation in $t$ and in each $x_{k}$ is legitimate, one variable at a time, by Theorem 11.51. The coefficients satisfy Equation (A14.5) by construction, so the analytic function $\pp_{t}u_{i} - \sum_{j,k}a_{ij}^{k}(x,u)\pp_{k}u_{j} - b_{i}(x,u)$ has every Taylor coefficient equal to zero and therefore vanishes identically; and $u_{i}(0,x)=0$ because $u_{i,\beta,0}=0$.

Uniqueness. An analytic solution on any polydisc about the origin has a Taylor series which, by the computation of Lemma A14.3, must satisfy the same recursion with the same initial values; the coefficients are therefore those just constructed, and two analytic functions with the same Taylor series at a point agree near it.

∎

Proof of Theorem 14.22. Derives Theorem 14.22. Let the problem be in the normal form Equation (A14.1), which is what normal Cauchy problem means, with the initial surface already the slice $\set{t=0}$. By Lemma A14.4 its unique formal solution $\hat w$ is recovered by Equation (A14.6) from the distinguished component of the formal solution $\hat u$ of an associated system Equation (A14.2). By Theorem A14.11 that formal solution converges absolutely on a polydisc about the origin; hence so does $\hat u_{(0,0)}$, and hence so does $\hat w$, the finitely many extra terms $\varphi_{p}(x)t^{p}/p!$ being analytic. Its sum $w$ is analytic, and its Taylor coefficients satisfy the recursion of the normal problem, so — by the argument already used in Theorem A14.11 — the analytic function $\pp_{t}^{k}w - G(\cdots)$ has all Taylor coefficients zero and vanishes, while $\pp_{t}^{p}w(0,x)=\varphi_{p}(x)$ for $p<k$ by construction. Any other analytic solution has the same Taylor coefficients, by the same recursion, and therefore coincides with $w$ near the origin.

∎

Remark A14.12 (What is quoted here).

The proof above is complete for a problem presented in the normal form Equation (A14.1), which is the form Theorem 14.22 names. Bringing a general non-characteristic Cauchy problem to that form uses two facts that this treatise does not prove:

Flattening the initial surface. If the analytic hypersurface $S$ is $\set{\phi = 0}$ with $\nabla\phi\neq0$, the map $x \longmapsto \bigl(\phi(x),\psi_{2}(x),\dots\bigr)$ is an analytic change of coordinates near the point, by the analytic inverse function theorem.
Solving for the highest normal derivative. For a fully nonlinear equation $F\bigl(x,\set{\pp^{\alpha}u}\bigr)=0$, non-characteristic means $\pp F/\pp\bigl(\pp_{t}^{k}u\bigr) \neq 0$ at the point (Proposition 14.11 is the linear instance of the same computation), and the analytic implicit function theorem then supplies an analytic $G$ with $\pp_{t}^{k}u = G(\cdots)$.

Both are the analytic versions of theorems whose smooth versions belong to Real Analysis; both are themselves usually proved by the majorant method of this section, so nothing circular is being borrowed, but neither is derived here and neither is stated in the treatise. For a linear or quasilinear equation — which is every equation this book actually solves — step (ii) is not needed at all: being non-characteristic then means that the coefficient of $\pp_{t}^{k}u$ is nonzero, by Equation (14.8) and the computation in the proof of Proposition 14.11, and one divides by it. Nothing else in this section is quoted: the recursion, the reduction, the majorant, the closed-form solution and the convergence estimate are all carried out above.

Remark A14.13 (Lewy's equation: analyticity cannot be dropped).

Every hypothesis of Theorem 14.22 was used, and the analyticity of the data — not merely of the coefficients — is the one whose necessity is least obvious and most complete. Consider on $\R^{3}$, with coordinates $(x_{1},x_{2},x_{3})$, the first-order operator

\begin{equation}\tag{A14.18} L u = \pp_{1}u + \ii\,\pp_{2}u - 2\ii\bigl(x_{1}+\ii x_{2}\bigr)\,\pp_{3}u\ep \end{equation}

Its coefficients are polynomials, hence analytic, and it is elementary to check that it is non-characteristic in the $x_{3}$-direction at points where $x_{1}+\ii x_{2} \neq 0$. Lewy proved that there exist $f \in C^{\infty}(\R^{3})$ — real-valued, and depending on $x_{3}$ alone — for which $L u = f$ has no solution of class $C^{1}$ on any neighbourhood of any point whatever [Lewy:1957]. The failure is therefore not of uniqueness or of regularity but of existence, it is local and universal, and it is produced by an operator whose coefficients satisfy every hypothesis of Cauchy–Kovalevskaya. What is lost when $f$ ceases to be analytic is exactly what the proof above consumed: the formal series still exists in the analytic case because the recursion Equation (A14.5) always runs, and it is the majorant that turns it into a function. With a merely smooth right-hand side there is no series to majorize, and Equation (A14.18) shows that nothing takes its place.

The reader should also keep in view the second limitation, which the chapter states in Remark 14.23 and which is independent of this one: even where the theorem applies it establishes existence and not well-posedness. Analytic data cannot be varied independently in disjoint regions, so continuous dependence in the sense of Definition 14.24 is not even being asked for, and the Cauchy problem Cauchy–Kovalevskaya solves for the Laplace equation is the ill-posed one of Remark 14.26.

Remark A14.14.

The Cauchy–Kovalevskaya Theorem discharges the proof obligation of Theorem 14.22 in Partial Differential Equations. The result is used in the treatise wherever an analytic construction is needed and nothing weaker will do: it is the existence statement behind the classical, real-analytic branch of Proposition 14.13 — the complex first integral constructed there by complexifying Equation (14.13) is supplied by this theorem, and The Korn–Lichtenstein Theorem and the Elliptic Canonical Form is what replaces it when the coefficients are only $C^{1}$ — and it is the reason a formal power-series ansatz may be trusted at all when the data of a problem are analytic. Its limitations are the substance of Remark A14.13 and Remark 14.23, and they are the reason the rest of Partial Differential Equations proceeds by energy estimates, transforms and Green's functions rather than by series.