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The Cauchy problem for a partial differential equation here prescribes both the value $u{|}_{\Gamma }=g$ and the normal derivative ${\partial }_{n}u{|}_{\Gamma }=h$, together with $\Delta u=0$. Only one of these traces would be boundary data for a usual elliptic boundary problem, rather than full Cauchy data.

Every real hypersurface is a non-characteristic hypersurface for the Laplace equation. In local real analytic coordinates flattening the real analytic hypersurface $\Gamma$, the coefficient of the second transverse derivative is nonzero: its principal coefficient is the squared length of the conormal. The equation can therefore be solved for that second derivative. The normal derivative data determine the transverse first derivative, because the coefficient relating them is nonzero and the tangential first derivatives are already determined by $g$.

The coefficients, flattened Cauchy data, and coordinate change are all real analytic. The Cauchy-Kovalevskaya theorem applies, giving a unique local real analytic solution around each point of $\Gamma$. This is a local existence assertion, not a claim of stable dependence in arbitrary Sobolev space norms.

The Cauchy-Kovalevskaya theorem cannot be applied to merely $C^2$, non-$C^3$ Cauchy data. It requires real analytic data.

There is also no $C^2$ solution of the Laplace equation on a neighborhood of a point where one of these prescribed traces fails to be $C^3$. By interior elliptic regularity, any such solution would be smooth and real analytic. On the real analytic hypersurface retained from the preceding part, both its restriction and its normal derivative would then be real analytic, hence $C^3$. This contradicts the prescribed trace. There is no solution on a neighborhood of all of $\Gamma$ with the stated non-$C^3$ data. This does not exclude solutions near other points where the data happen to be real analytic.

The first-order Cauchy-Kovalevskaya theorem says that an analytic PDE system solved for the derivative normal to an analytic noncharacteristic initial hypersurface, with analytic initial data, has a unique local analytic solution near each point of that hypersurface. The general theorem likewise applies to higher-order equations solved for the highest normal derivative with analytic data for the lower normal derivatives.

Here the characteristic vector is $(x,y,a)$ and the normal to $z=0$ is $(0,0,1)$. Its normal component is $a$, so the theorem applies precisely when $a\ne 0$. The method of characteristics gives $\dot{x}=x$, $\dot{y}=y$, $\dot{z}=a$, $\dot{u}=u$. Starting from $(\xi ,\eta ,0)$ at parameter $s=0$ gives $x=\xi e^s$, $y=\eta e^s$, $z=as$, $u=e^{s}f(\xi ,\eta )$. Eliminating $s=z/a$ yieldsThis is locally defined wherever the arguments remain in the analytic domain of $f$, and verifies both PDE and initial data.

As $a\to 0$, the normal component vanishes and the characteristic curves become tangent to the initial plane. The crossing parameter $s=z/a$ and slopes $dx/dz=x/a$, $dy/dz=y/a$ become singular. At $a=0$, arbitrary analytic data are not admissible: already on the plane they must obey $x{f}_x+y{f}_y=f$. Even compatible data do not determine derivatives normal to the plane. This is failure of the noncharacteristic condition, not a failure of the analytic theorem.

Non-characteristic. The first-order principal symbol is ${\nu }_t-i{\nu }_x$, which equals one on $(0,1)$.

Set $z=x+it$. The equation is equivalent to ${\partial }_{\bar{z}}\varphi =({\varphi }_x+i{\varphi }_t)/2=0$. Because $\varphi$ is continuously differentiable, the Cauchy-Riemann equations imply that it is a holomorphic function of $z$. Its convergent complex Taylor series restricts to a real power series along $t=0$, so $g$ is a real analytic function.

For smooth-data instability of the Cauchy-Riemann Cauchy problem, chooseThese are entire functions and satisfy the equation. Their initial derivatives obey ${\sup }_x|{\partial }_x^j{g}_n|\le n^j{e}^{-\sqrt{n}}$, so every finite sum tends to zero. ButThus analytic solutions can exist while continuous dependence on initial data fails for this smooth-data topology.

The Cauchy estimate follows directly from the differentiated Cauchy integral formula:The circle has length $2\pi r$, giving the bound for every integer $j\ge 0$.

Put $w=\varphi -\psi$ and integrate on the straight segment $z=\tau t/|t|$. For fixed $|x|<sR$, a disc centred at $x$ with radius approaching $R(\sigma (\tau )-s)$ stays inside $|\xi |\le \sigma (\tau )R$. Applying the first-derivative Cauchy estimate there givesIntegrating its modulus and taking the supremum over $|x|<sR$ is exactly the asserted integral estimate. The source cancels from $T\varphi -T\psi$.

To obtain a contraction mapping, use the weighted holomorphic norm on a shrinking time domain. Its finite-norm space consists of holomorphic functions on ${\mathcal{C}}_{\alpha }$ with zero initial value; the apparent quotient at $t=0$ is interpreted by a limit. Completeness follows because convergence in this norm implies locally uniform convergence of holomorphic functions, including near $t=0$, and the limiting pointwise bounds give convergence in the norm.

Here is the explicit contraction estimate on a shrinking holomorphic domain. Write $N=\Vert w{\Vert }_{\alpha }$, $r=|t|$, $A=\alpha (1-s)$, and chooseThen $\sigma -s=(A-\tau )/(2\alpha )$ and $\alpha (1-\sigma )-\tau =(A-\tau )/2$. The whole auxiliary polydisc lies inside ${\mathcal{C}}_{\alpha }$, and its supremum of $|w|$ is at most $2N\tau /(A-\tau )$. ConsequentlyThus $\Vert T\varphi -T\psi {\Vert }_{\alpha }\le (4\alpha /R)\Vert \varphi -\psi {\Vert }_{\alpha }$. Choose $0<\alpha <\min (\eta ,R/4)$. The source is bounded on the closed polydisc, say by $M_f$, and $\Vert T0{\Vert }_{\alpha }\le \alpha M_f$. Therefore $T$ maps the Banach space into itself and is a strict contraction mapping. The Banach fixed-point theorem gives a holomorphic fixed point with $\varphi (x,0)=0$. Differentiating the integral identity yields ${\varphi }_t-i{\varphi }_x=f$, establishing this case of the Cauchy-Kovalevskaya theorem.

Finally, near each real initial point, a real analytic function $g$ has a holomorphic extension $G$. Apply the same construction to $\varphi =G(x)+v$ with $v(x,0)=0$ and source $i{G}^{\prime }(x)$, restricting the discs if necessary. This gives a local real analytic function of $(x,t)$ with the prescribed initial value. Equivalently the solution is $\boxed{\varphi (x,t)=G(x+it)}$ wherever the extension is defined.

The general-order Cauchy-Kovalevskaya theorem applies to analytic data on a non-characteristic hypersurface. In local coordinates $(t,z)$, a real analytic system of order $m$ in an unknown vector $U$ is in normal form whenwith $F$ real analytic near the initial jet. Analytic data ${\partial }_t^{j}U(0,z)=h_j(z)$ for $0\le j<m$ determine a unique real analytic solution near each initial point. More general systems, including different orders for the components, have the same conclusion after solving for the highest normal derivatives: the corresponding coefficient matrix, or highest-derivative Jacobian for a nonlinear system, must be invertible. This is the system's non-characteristic hypersurface condition. The uniqueness asserted by the Cauchy-Kovalevskaya theorem is initially uniqueness in the real analytic class.

A real analytic coordinate change flattens a real analytic initial hypersurface. Non-characteristicity permits solving for its highest normal derivatives by the real analytic implicit function theorem. Introduce all derivatives through order $m-1$ as additional unknowns. Their normal derivatives are either another jet variable, a tangential first derivative of a jet variable, or the right-hand side of the original equation. This gives a first-order real analytic system. The derivative-compatibility identities have zero initial data and are preserved by the system; equivalently, its real analytic coefficient recurrence reproduces the derivatives of $U$. This explains both reductions, rather than assuming arbitrary real analytic hypersurfaces are already flat.

For the first example, the principal symbol of the Laplace equation at the conormal $d{x}_1$ is $1$, so the plane is a non-characteristic hypersurface. The theorem therefore applies to real analytic $u{|}_{x_1=0}$ and ${\partial }_{1}u{|}_{x_1=0}$:This does not give Hadamard well-posedness in smooth or Sobolev space norms. For $n\ge 2$, the harmonic functionshave zero value data and normal derivative data $e^{-\sqrt{N}}\cos (N{x}_2)$ tending to zero with every fixed tangential derivative. At any fixed $x_1=\delta >0$, however, $u_N(\delta ,0)$ diverges. Thus the elliptic Cauchy problem for a partial differential equation is unstable despite real analytic solvability. The exceptional case $n=1$ is the elementary affine ordinary differential equation and has no such tangential high-frequency instability.

For the wave equation, the principal symbol is $-{\tau }^2+|\xi {|}^2$, so its value on the conormal $dt$ is $-1$. Consequently the Cauchy-Kovalevskaya theorem applies to real analytic displacement and velocity data on this non-characteristic hypersurface:In addition, this Cauchy problem for a partial differential equation has Hadamard well-posedness for nonanalytic data in the usual energy spaces. For example, data in $H^1({\mathbb{R}}^n)\times L^2({\mathbb{R}}^n)$ give a unique solution in $C_t{H}^1\cap C_t^1{L}^2$ on bounded time intervals, with continuous dependence; conservation of $\int (|u_t{|}^2+|\nabla u{|}^2)$ is its basic estimate. Thus real analytic existence and energy-space stability are different assertions, and in this example both are available.

In the flattened coordinates, ${\partial }_x={\partial }_{\tilde{x}}$ and ${\partial }_y=2\tilde{y}{\partial }_{\tilde{x}}+{\partial }_{\tilde{y}}$. The transformed formal adjoint consequently has coefficients obtained from finitely many real analytic coefficient pullbacks and polynomial factors. They have uniform analyticity bounds on a fixed small ball. By the preceding non-characteristic argument, its coefficient of ${\partial }_{\tilde{x}}^m$ has modulus bounded away from zero there, uniformly for small $\epsilon$. Division by it preserves uniform real analytic bounds.

The transformed forcing ${\tilde{g}}_{\epsilon }$ has analogous bounds, with constants that may depend on $g$. Apply the Cauchy-Kovalevskaya theorem in normal form with zero normal data through order $m-1$ at $\tilde{x}=0$. The supplied majorant series radius result yields a common solution radius $r_g>0$ for all sufficiently small parameters. Tangential differentiation of zero normal data shows that the entire boundary jet through order $m-1$ is zero; changing coordinates preserves this vanishing.

For $(x,y)$ in the closed cap, $|\tilde{x}|\le \epsilon$ and $|\tilde{y}|\le \sqrt{\epsilon }$. Choose $\epsilon$ so small that $\sqrt{{\epsilon }^2+\epsilon }<r_g$, with a strict margin. The whole closed cap then lies in the common real analytic solution neighborhood. Pulling the solution back provesIn fact $v$ is real analytic on a neighborhood of its closure, which is useful for the boundary integration below.

Fix the coefficient normal form and a positive analyticity exponent ${\zeta }_0$ once and for all. If $g$ is a polynomial, its pullback ${\tilde{g}}_{\epsilon }=g(\tilde{x}-{\tilde{y}}^2+\epsilon ,\tilde{y})$ is again a polynomial, with degree bounded independently of $\epsilon$. On a fixed closed coordinate ball, its finitely many nonzero derivatives are uniformly bounded as the parameter ranges over a fixed compact small interval. Hence there is a finite $A_g\ge 1$ such thatAll higher derivatives vanish, so arbitrarily high polynomial degree causes no failure of this bound; it only changes $A_g$.

The normalized forcing belongs to one common majorant class. The Cauchy-Kovalevskaya theorem therefore gives a radius $r_{\ast }>0$ independent of $g$. The problem is linear with zero data, so multiplying the normalized solution by $A_g$ solves the original forcing without changing its radius. Choose one $\epsilon$ with the cap inside this radius. ThusThis is the uniform Cauchy radius for polynomial forcing. It asserts a common domain, not a uniform solution-amplitude bound over all polynomials.

One prescribes analytic Cauchy data: the value of $u$ and one first derivative transverse to $\Sigma$, for examplewhere $\varphi$ and $\psi$ are real analytic on $\Sigma$. Being a non-characteristic hypersurface allows the equation to solve for the second derivative in the transverse direction. The Cauchy-Kovalevskaya theorem then gives one and only one local real analytic solution near $p$.

Expanding the equation givesso its principal symbol isIf a characteristic curve is locally a graph $y=y(x)$, its conormal is proportional to $(-y^{\prime },1)$. The characteristic equation is thereforeOn each region separated by $y=\pm 1$, separation of variables givesThus all the characteristic curves areThe inner curves approach the horizontal characteristics $y=\pm 1$ as $x\to \pm \infty$; the outer hyperbolic-cotangent branches have vertical asymptotes and also approach $y=\pm 1$. This describes the requested sketch.

The initial line $x=0$ has conormal $(1,0)$, and $p(y,1,0)=1$, so it is a non-characteristic hypersurface at every point. Since the coefficients and prescribed data are real analytic, the Cauchy-Kovalevskaya theorem gives a unique real analytic solution in a neighborhood of each point of $\{x=0\}$, hence in a neighborhood of that line.

The initial line $x=0$ has conormal $dx$, and the principal symbol on this conormal is $1$. It is therefore a non-characteristic hypersurface at every $(0,t_0)$. The equation's coefficients and the prescribed Cauchy data $u(0,t)=\cos t$ and $u_x(0,t)=0$ are real analytic functions. The Cauchy-Kovalevskaya theorem consequently gives a unique analytic solution in a neighbourhood of $(0,t_0)$ for every

WriteAs a differential operator of order two, its principal symbol isIf $\nu =({\nu }_t,{\nu }_x,{\nu }_y)$ is a unit normal to the real-analytic hypersurface $\Sigma$, the non-characteristic hypersurface condition is thereforeUnder this condition the equation can be solved for the second derivative normal to $\Sigma$. The Cauchy-Kovalevskaya theorem then gives a unique real-analytic solution in a neighbourhood of $\Sigma$ for the prescribed analytic Cauchy data $u{|}_{\Sigma }$ and ${\partial }_{\nu }u{|}_{\Sigma }$.

Consider an analytic quasilinear partial differential equationLet the analytic initial hypersurface be $\Gamma =\{\varphi =0\}$ and prescribe $u=u_0$ and one transverse derivative ${\partial }_{N}u=u_1$ on $\Gamma$. The tangential derivatives of $u_0$ together with $u_1$ determine the full first jet $Du$ on $\Gamma$. The hypersurface is non-characteristic at $x_0$ with respect to these data whenThis is precisely the principal symbol of a partial differential equation evaluated on the conormal $d\varphi (x_0)$.

The Cauchy-Kovalevskaya theorem then gives a unique real-analytic solution near $x_0$. In coordinates flattening $\Gamma$ to $x_n=0$, non-characteristicity lets the equation solve analytically for ${\partial }_n^{2}u$, after which the analytic equation and the two initial jets determine every higher Taylor coefficient.

The initial line is $t=t_0$, whose conormal is $dt$. Evaluating the principal symbol on it givesIt is therefore a non-characteristic hypersurface at $(t_0,x_0)$ exactly when $t_0{x}_0\ne 0$. At such a point the equation can be writtenwhose right-hand side is real analytic locally, and the prescribed Cauchy data are also real analytic. The Cauchy-Kovalevskaya theorem consequently gives a unique local analytic solution exactly at the points

The scalar Cauchy-Kovalevskaya theorem says that a partial differential equation solved for its highest derivative normal to a real-analytic non-characteristic hypersurface, with real-analytic coefficients and Cauchy data, has a unique local real-analytic solution. In coordinates, an equationhas such a solution near the origin when $F$ and the prescribed values of ${\partial }_t^j\varphi (0,x)$ for $0\le j<m$ are real analytic.

Choose a real-analytic primitive $F_0$ of $f$ near zero and apply the theorem to the scalar Laplace equationThe line $t=0$ is non-characteristic because the coefficient of ${\varphi }_{tt}$ is one. DefineThen $u(0,x)=f(x)$ and $v(0,x)=g(x)$, while equality of mixed derivatives and the Laplace equation giveThus $(u,v)$ is the required local real-analytic solution.

The Cauchy-Kovalevskaya theorem says that an order-$k$ scalar quasilinear partial differential equation with real-analytic coefficients has a unique local real-analytic solution near each point of a real-analytic non-characteristic hypersurface, provided the prescribed Cauchy dataare real analytic there. The uniqueness is among local real-analytic solutions agreeing with all of those data.

For ${\psi }_0=(r^2+z^2)/2$,The prescribed function is $g=\sqrt{D}$, so part c gives $u_z=-z$. Hence the non-characteristic condition reduces toThe Cauchy-Kovalevskaya theorem therefore guarantees a unique local real-analytic solution at exactly those pointsfor which

For a linear real analytic non-characteristic Cauchy problem for a partial differential equation with zero data, the local real analytic radius may be chosen uniformly over polynomial forcing terms. Fix an analyticity exponent $R^{-1}$. Each polynomial, including a polynomial obtained by a polynomial coordinate change, satisfies the corresponding factorial derivative bound with some finite prefactor, uniformly over a compact parameter range. Divide the forcing by that prefactor to put all such data in one fixed majorant class. The Cauchy-Kovalevskaya theorem supplies a common radius, and linearity rescales the solution back without shrinking it. This argument does not bound the amplitudes of the resulting solutions uniformly.

Let $R,\alpha >0$ and ${\mathcal{C}}_{\alpha }=\{(x,t)\in {\mathbb{C}}^2:|x|<R,|t|<\alpha (1-|x|/R)\}$. The displayed norm makes the holomorphic functions with finite norm a Banach space. Finiteness forces $u(x,0)=0$: for fixed $x$, the weight grows like $1/|t|$ near zero. A Cauchy sequence in this norm converges uniformly on every compact subset of ${\mathcal{C}}_{\alpha }$, including at $t=0$. Its limit is holomorphic by locally uniform convergence of holomorphic functions, and the pointwise norm bounds show convergence in the original norm. The shrinking domain compensates for loss of an $x$-derivative in a Cauchy-Kovalevskaya theorem argument.