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For an order-$k$ scalar quasilinear partial differential equation with real-analytic coefficients, real-analytic Cauchy data on a real-analytic non-characteristic hypersurface determine a unique real-analytic solution in a neighbourhood of each point of that hypersurface.

A real analytic system solved for the highest normal derivatives, with real analytic Cauchy data, has a unique local real analytic solution. For common order $m$, its normal form is ${\partial }_t^{m}U=F(t,x,\{{\partial }_t^j{\partial }_x^{\alpha }U:j<m,j+|\alpha |\le m\})$. An invertible coefficient/Jacobian matrix for the highest normal derivatives is the system's non-characteristic hypersurface condition. A real analytic coordinate change flattens the initial hypersurface, and adjoining derivatives through order $m-1$ reduces the problem to a first-order real analytic system. Uniqueness here is in the real analytic class; it does not assert Hadamard well-posedness in smooth or Sobolev norms.

For a linear differential operator with real analytic coefficients, a classical solution with zero Cauchy data on a real analytic non-characteristic hypersurface vanishes locally near that hypersurface. The solution itself need not be real analytic. A proof solves the real analytic inhomogeneous formal adjoint problem against polynomial sources, annihilates all polynomial moments using boundary jets, and uses polynomial density to force the solution to vanish.

For the first-order quasilinear equation $a(x,u)\cdot \nabla u=b(x,u)$, suppose the coefficients and initial data $u=g$ on a smooth hypersurface $\Sigma$ are $C^1$. At an initial point $x_0$, require $a(x_0,g(x_0))\cdot n(x_0)\ne 0$ for a normal $n$ to $\Sigma$; this is precisely the condition that $\Sigma$ be a non-characteristic hypersurface. Then in a neighborhood of $x_0$ there is a unique $C^1$ solution taking the given data. Smooth coefficients and data give corresponding higher regularity.

The method of characteristics solves $\dot{x}=a(x,u)$, $\dot{u}=b(x,u)$ from each initial point. The non-characteristic condition makes the map from the initial coordinates and characteristic time to $x$ locally invertible, so these characteristic solutions define the asserted function.

The characteristics solve ${\dot{x}}_j=a_j(x)$ and $\dot{u}=b(x,u)$, starting with $x$ on the initial hypersurface and $u=\phi (x)$. The non-characteristic hypersurface condition at a point is $a\cdot n\ne 0$, where $n$ is any nonzero normal there. It makes the map from initial-surface coordinates and characteristic time to $x$ locally invertible. Local uniqueness for the ordinary differential equations and the inverse function theorem then give a unique local $C^1$ solution near that point, for the stated $C^1$ coefficients and data.

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.

For a constant-coefficient differential operator $P(D)={\sum }_{|\beta |\le m}{a}_{\beta }{D}^{\beta }$, a hypersurface with nonzero normal covector $\nu$ is a non-characteristic hypersurface precisely when its principal symbol $P_m(\nu )={\sum }_{|\beta |=m}{a}_{\beta }{\nu }^{\beta }$ is nonzero. Multiplying the symbol by the conventional factor $i^m$ does not affect this criterion. For the initial line here the normal is $(0,1)$.

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.

Write the order-$m$ principal symbol as $p_m(x,y;\xi ,\eta )={\sum }_{|\alpha |=m}{a}_{\alpha }(x,y){\xi }^{{\alpha }_1}{\eta }^{{\alpha }_2}$. The conormal of the plane is $(1,0)$. Therefore the non-characteristic hypersurface hypothesis is exactlyIn particular its value at the origin is nonzero, and continuity provides a uniform positive lower bound for its modulus in a sufficiently small neighborhood. No lower-order coefficient enters this condition.

The order-$m$ principal symbol of the formal adjoint is $p_m^{\ast }=(-1{)}^m{p}_m$. Derivatives falling on coefficients contribute only lower-order terms. Thus the original plane remains a non-characteristic hypersurface.

For the curved face use the defining function ${\rho }_{\epsilon }=x+y^2-\epsilon$, whose conormal is $(1,2y)$. On the face with $x>0$, $|y|<\sqrt{\epsilon }$ and $0<x\le \epsilon$. Therefore $(x,y,1,2y)$ tends uniformly to $(0,0,1,0)$ as $\epsilon \downarrow 0$. Since $p_m^{\ast }(0,0;1,0)\ne 0$, continuity givesThe same argument supplies a common neighborhood where the transformed coefficient of the highest normal derivative is nonzero, permitting the real analytic normal form used below.

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.