Exercise 31, § 3

Let $L$ denote the differential operator

of order 1 on $\mathbf{R}^3$, where $(x_1, x_2, x_3)$ is the dual basis of the canonical basis of $\mathbf{R}^3$, the $x_i$ being regarded as polynomials on $\mathbf{R}^3$ (“Lewy operator”).

a) Let U be an open neighbourhood of 0 in $\mathbf{R}^3$. There exists a function $\varphi \in \mathscr{D}(U)$ such that $\varphi (0) = 1$ and such that $L(\varphi )$ is zero in a neighbourhood V of 0 contained in U. (Use the Cauchy–Kowalewski theorem.)

b) Let $u\in \mathscr{C}^{\infty}(U)$ be a function such that $L(u) = 0$ and $u(0) = 0$. Suppose that there exists $\delta >0$ such that $\mathscr{R}(u(x))<-2\delta$ for $x\in U$ - V. Put $v_t=\varphi$ exp($t(u+\delta )$). Then $v_t\in \mathscr{C}^{\infty}(U)$ and $L(v_t)$ tends to 0 in $\mathscr{D}(U)$ when $t\rightarrow +\infty$.

c) Let F be the space of functions $f\in \mathscr{C}^{\infty}(U)$ such that $\partial^{\alpha}f$ is bounded on U for every $\alpha \in \mathbf{N}^3$. The space F endowed with the seminorms $q_{\alpha}(f) =$ sup$_{x\in U}|\partial^{\alpha}f|$ is a Fréchet space.

d) Let $\psi \in \mathscr{D}(\mathbf{R}^3)$. Denote by $f_t$ the restriction to U of the function $x\mapsto t^3\psi (tx)$. Then $f_t\in F$ for $t\geqslant 1$, and there exists a real number $c\geqslant 0$ such that $q_{\alpha}(f_t)\leqslant ct^{|\alpha|+3}$ for every $\alpha \in \mathbf{N}^3$ and every $t\geqslant 1$.

e) For every $t\geqslant 1$, let $\lambda_t$ denote the linear form $f\mapsto \int_Uf v_t$ on F. It is continuous and, for $t$ sufficiently large, one has

$\lambda_t(f_t) =e^{\delta t}(\int_U\psi (x)$ exp$(\sum_{j=1}^3\xi_jx_j)dx_1dx_2dx_3+\varepsilon (t))$

where $\xi_j=\partial_ju(0)$ and $\varepsilon (t)\rightarrow 0$ when $t\rightarrow +\infty$.

f ) Suppose that $(\xi_j)\not = 0$; there exists $f\in F$ such that the integrals $\int_Uf v_t$ are not bounded when $t\rightarrow +\infty$. (If the assertion were false, obtain a contradiction by using the preceding question and the Banach– Steinhaus theorem.)

g) The functions $v_t$ do not converge to 0 in $\mathscr{D}'(U)$ when $t\rightarrow +\infty$. h) Let $\xi = (0,0, \xi_3)\in \mathbf{R}^3$ be such that $\xi_3<0$. There exists a neighbourhood U of 0 in $\mathbf{R}^3$ and a function $u\in \mathscr{C}^{\infty}(U)$ such that $L(u) = 0$ and $u(0) = 0$, and satisfying the conditions of questions b) and f). (Use the Cauchy–Kowalewski theorem.)

i) Let $(v_t)$ be functions in $\mathscr{D}(U)$ such that $L(v_t)\rightarrow 0$, and let $f\in \mathscr{D}(U)$ be such that $\int_Uf v_t$ is not bounded when $t\rightarrow +\infty$ (question g)) ; the equation $Lv=f$ has no solution $v\in \mathscr{D}'(U)$ ("Lewy's theorem"). (Note that the formal adjoint of L coincides with L.)