Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 105 / 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.

For the heat equation, viewed as a second-order equation in all $(t,x)$ variables, the principal symbol is $|\xi {|}^2$. It vanishes on $dt$, so $t=0$ is a characteristic hypersurface andWriting $u_t={\Delta }_{x}u$ does not repair its hypotheses: the right side has second spatial derivatives, exceeding the first-order normal-form allowance. Indeed real analytic initial data need not produce a jointly real analytic solution through time zero. In one dimension, $u_0(x)=(1+x^2{)}^{-1}$ would forceThe time Taylor coefficients $(-1{)}^k(2k)!/k!$ have zero radius of convergence.

Nevertheless the forward heat equation has Hadamard well-posedness in standard $L^2$ or Sobolev space classes for $t\ge 0$; its Fourier multiplier $e^{-t|\xi {|}^2}$ gives stable smoothing. Backward evolution has the growing multiplier $e^{|t||\xi {|}^2}$ and is unstable. Neither forward stability nor backward instability is decided simply by whether the real analytic theorem applies.

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.

Repeated integration by parts against a compactly supported test function gives the bilinear formal adjointThere is no boundary remainder because $v$ has compact support. The product rule makes this an order-at-most-$m$ linear differential operator with real analytic coefficients. The displayed integral pairing has no complex conjugation, so this is the formal transpose; for a sesquilinear pairing the coefficients would be conjugated.

For uniqueness, suppose $Q$ is the difference of two candidates. Then $\int uQv=0$ for every smooth compactly supported $u$ and $v$, so $Qv=0$ for every such $v$. At an interior point, a polynomial times a cutoff can prescribe every derivative of $v$ through order $m$ independently. Hence each coefficient of $Q$ is zero. This proves uniqueness as a differential operator, not only uniqueness of its integrated action.

Taken literally, the printed condition is insufficient: the choice of which factor vanishes may depend on the multi-index. For a counterexample already at $m=2$, let $P={\partial }_x^2$, let $\Omega =(0,a)\times (-b,b)$ be a small rectangle compactly contained in the unit ball, and choose $\chi \in C_c^{\infty }((-b,b))$ nonnegative with positive integral. PutOn the vertical edges $u=0$, and on the horizontal edges $u$ and its derivatives vanish. Every first derivative of $v$ is zero. Thus for every $|\beta |\le 1$ the same-index alternative in the PDF holds at every boundary point. But

The valid complete boundary-jet condition for formal adjoints is that, at each boundary point, either all derivatives of $u$ through order $m-1$ vanish or all such derivatives of $v$ vanish. On a bounded piecewise smooth domain with smooth coefficients up to its closure, repeated integration by parts produces boundary terms of the form $({\partial }^{\gamma }u){\partial }^{\delta }(a_{\alpha }v)$, multiplied by a normal component, with $|\gamma |,|\delta |\le m-1$. Expanding the second factor shows that every summand vanishes under this stronger whole-jet alternative. Hence the desired identity holds under that condition. The later cap argument satisfies this stronger condition face by face, so it remains valid despite the literal error in this subpart. Boundary regularity and finite integrals are also needed for a general noncompact-support integration formula; the caps used below have them.

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.

The coordinate change has inverse $x=\tilde{x}-{\tilde{y}}^2+\epsilon$, $y=\tilde{y}$. Thus the coefficient function needed after changing variables isThe PDF's displayed formula has an undefined subscript $f_{\alpha }$ and writes the forward change in the argument. It is not the coefficient pullback by the stated coordinates; the inverse expression above is the one used in transforming the equation. The forward polynomial composition, if intended instead, has the same type of uniform bounds, but is not that pullback.

For $|(\tilde{x},\tilde{y})|\le 1/2$ and $0\le \epsilon \le 1/32$, the inverse image lies in the fixed compact subset specified by $|x|\le 25/32$, $|y|\le 1/2$, whose Euclidean radius is less than one. A real analytic function extends holomorphically to a complex neighborhood of this compact set: local extensions agree on overlaps by uniqueness, and a finite covering gives a common smaller neighborhood. The polynomial inverse maps a fixed small complex neighborhood of the closed half-ball into that neighborhood, uniformly in $\epsilon$. The extensions are bounded there by a common constant $A$. Applying the multivariable Cauchy estimate on equal-radius polydiscs givesThis is uniform analyticity under polynomial coordinate changes. The argument gives a compact margin and a common complex radius, not merely separate analyticity for each parameter.

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.

For each polynomial $g$, take the real analytic adjoint solution on the common cap from the preceding part. On the flat boundary face the full boundary jet of $u$ vanishes; on the curved face that of $v$ vanishes. The complete boundary-jet condition for formal adjoints, proved under its corrected hypothesis, therefore applies. The two corner points have zero boundary measure, and the cap is a bounded piecewise smooth domain. It follows thatBy the Weierstrass approximation theorem, polynomials uniformly approximate $\bar{u}$ on the compact cap closure. Taking the limit gives ${\int }_{{\omega }_{\epsilon }}|u{|}^2=0$. Continuity then gives $u=0$ pointwise in the cap. For real $u$, simply approximate $u$ itself.

A positive cap alone is not a neighborhood of the origin. Apply the same argument after reflecting $x\mapsto -x$; the reflected operator remains real analytic and non-characteristic. Shrinking the two parameters to their minimum proves vanishing onThis set is open and contains the origin, including the flat face where the value data vanish. The argument is the parabolic-cap proof of Holmgren uniqueness, and it establishes Holmgren uniqueness theorem for nonanalytic $C^m$ solutions here.