We are given $g : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }$ continuous and 1-periodic in each of its arguments, of class $\mathscr{C}^1$. We assume $x \neq 0$ (but close to 0), and that a solution $h$ of equation (4) exists (1-periodic, $\mathscr{C}^1$, zero average). We set $\tilde{\alpha}(t) = \alpha(t) + x h(\alpha(t))$ where $\alpha$ satisfies $\alpha'(t) = \omega + x g(\alpha(t))$.
Show that there exists a function $\varepsilon : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }$ such that $$\tilde { \alpha } ^ { \prime } ( t ) = \omega + x \nu + x \varepsilon ( x , t )$$ and $\sup _ { t \in \mathbb { R } } \| \varepsilon ( x , t ) \| \rightarrow 0$ when $x \rightarrow 0$.

We are given $g : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }$ continuous and 1-periodic in each of its arguments, of class $\mathscr{C}^1$. We assume $x \neq 0$ (but close to 0), and that a solution $h$ of equation (4) exists (1-periodic, $\mathscr{C}^1$, zero average). We set $\tilde{\alpha}(t) = \alpha(t) + x h(\alpha(t))$ where $\alpha$ satisfies $\alpha'(t) = \omega + x g(\alpha(t))$, $\alpha(0)=(0,0)$. From question 15b, $\tilde{\alpha}'(t) = \omega + x\nu + x\varepsilon(x,t)$.
Let $T > 0$ be fixed. Show that there exists a function $\eta : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }$ such that $$\alpha ( t ) = ( \omega + x \nu ) t + x ( h ( 0,0 ) - h ( \omega t ) ) + x \eta ( x , t )$$ and $\sup _ { t \in [ 0 , T ] } \| \eta ( x , t ) \| \rightarrow 0$ when $x \rightarrow 0$.

We assume that $m = 0$, that is $$\mathcal{M} = \left(\begin{array}{lll} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right)$$ We assume that $G \in C^{2}(\mathbb{R}, \mathbb{R}^{3})$ satisfies $$\forall x \in \mathbb{R}, G^{\prime\prime}(x) = \frac{1}{2}\left(G(x)\right) \wedge G^{\prime}(x)$$ and that moreover there exists $\lambda > 0$ such that $$G(0) = (0, 0, 2\lambda), \quad G^{\prime}(0) = (1, 0, 0)$$
Show that $G \in C^{\infty}(\mathbb{R}, \mathbb{R}^{3})$.

We denote $L$ the operator that associates to a function $f : \mathbb { R } \rightarrow \mathbb { R }$ of class $\mathscr { C } ^ { 2 }$, the function $Lf$ defined by $$\forall x \in \mathbb { R } , \quad L f ( x ) = \frac { 1 } { 2 } f ^ { \prime \prime } ( x ) - x f ^ { \prime } ( x )$$ We recall that the measure $\mu$ is defined by $\mu ( x ) = \frac { 1 } { \sqrt { \pi } } e ^ { - x ^ { 2 } }$.
3a. Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be of class $\mathscr { C } ^ { 2 }$. Show that $L f = \frac { 1 } { 2 \mu } \left( \mu f ^ { \prime } \right) ^ { \prime }$.
3b. Let $h _ { 1 } , h _ { 2 }$ be two functions in $\mathscr { C } _ { b } ^ { 2 }$. Show that $$\int h _ { 1 } ( x ) \left( L h _ { 2 } \right) ( x ) \mu ( x ) d x = - \frac { 1 } { 2 } \int h _ { 1 } ^ { \prime } ( x ) h _ { 2 } ^ { \prime } ( x ) \mu ( x ) d x$$ after having justified the existence of each term of the formula.

We consider a function $f \in \mathscr { C } _ { b } ^ { 0 }$. We define for $( t , x ) \in \mathbb { R } ^ { 2 }$ $$\Phi _ { f } ( t , x ) = \int f ( x \cos t + y \sin t ) \mu ( y ) d y$$ where $\mu ( x ) = \frac { 1 } { \sqrt { \pi } } e ^ { - x ^ { 2 } }$, and $Lf(x) = \frac{1}{2} f''(x) - x f'(x)$. We assume that $f \in \mathscr { C } _ { b } ^ { 2 }$.
5a. Show that, on $\mathbb { R } ^ { 2 } , \Phi _ { f }$ is of class $\mathscr { C } ^ { 1 }$ and $\partial _ { x x } \Phi _ { f }$ is well defined, continuous and bounded.
5b. Let $( t , x ) \in \mathbb { R } ^ { 2 }$. Find a relation between $\partial _ { x } \Phi _ { f } ( t , x )$ and $\Phi _ { f ^ { \prime } } ( t , x )$.
5c. Show that for all $( t , x ) \in \mathbb { R } ^ { 2 }$, we have $\partial _ { t } \Phi _ { f } ( t , x ) \cos t = L \Phi _ { f } ( t , x ) \sin t$.
5d. Show that for all $t \in \mathbb { R }$, we have $\int \Phi _ { f } ( t , x ) \mu ( x ) d x = \int f ( x ) \mu ( x ) d x$.

Let $f : \mathbb { R } \rightarrow \mathbb { R } _ { + }$ be a positive function in $\mathscr { C } _ { b } ^ { 0 }$. We define for $t \in \mathbb { R }$ $$J ( t ) = \int h \left( \Phi _ { f } ( t , x ) \right) \mu ( x ) d x$$ where $\Phi _ { f } ( t , x ) = \int f ( x \cos t + y \sin t ) \mu ( y ) d y$, $\mu ( x ) = \frac { 1 } { \sqrt { \pi } } e ^ { - x ^ { 2 } }$, and $h(x) = x\ln(x)$ for $x > 0$, $h(0) = 0$. We assume throughout this question that $f \in \mathscr { C } _ { b } ^ { 2 }$ and that there exists $\delta > 0$ such that $$\forall x \in \mathbb { R } , \quad f ( x ) \geqslant \delta .$$ We denote $g = \left( f ^ { \prime } \right) ^ { 2 } / f$.
7a. Show that $J$ is then of class $\mathscr { C } ^ { 1 }$ on $\mathbb { R }$ and that $$\forall t \in \mathbb { R } , \quad J ^ { \prime } ( t ) \cos t = - \frac { \sin t } { 2 } \int \frac { \left( \partial _ { x } \Phi _ { f } ( t , x ) \right) ^ { 2 } } { \Phi _ { f } ( t , x ) } \mu ( x ) d x$$
7b. Let $( t , x ) \in \mathbb { R } ^ { 2 }$. Show that $$\Phi _ { f ^ { \prime } } ( t , x ) ^ { 2 } \leqslant \Phi _ { f } ( t , x ) \Phi _ { g } ( t , x )$$
7c. Conclude that $$\int h ( f ( x ) ) \mu ( x ) d x - h \left( \int f ( y ) \mu ( y ) d y \right) \leqslant \frac { 1 } { 4 } \int g ( x ) \mu ( x ) d x$$

We keep all the notations from Part I and we assume that hypotheses (H1), (H2), (H3), (H4) and (H5) are all satisfied. Let $(w_1, w_2)$ be a characterizing pair of $G$ (satisfying properties (A), (B) and (C) of question 12). For any $\lambda \in \mathbb{R}$, we consider the following problem, denoted $\mathcal{P}_\lambda$: $$\text{Find } u \in G \text{ such that: } \forall v \in G, (u \mid v) - \lambda (T(u) \mid T(v)) = 0$$ and we denote by $H_\lambda$ the set of solutions $u$ of this problem.
(a) Show that if $(\mathcal{P}_\lambda)$ admits a solution $u \neq 0_E$, then necessarily $\lambda > 0$.
(b) Let $u \in G$. Show that $u$ is a solution of $(\mathcal{P}_\lambda)$ if and only if $$\left(\operatorname{Id}_E + \lambda T^2\right)(u) \in G^\perp$$ Deduce that there exist two real numbers $\alpha$ and $\beta$ such that: $$u = \alpha \left(\operatorname{Id}_E + \lambda T^2\right)^{-1} T(w_1) + \beta \left(\operatorname{Id}_E + \lambda T^2\right)^{-1} T(w_2)$$ (c) Show that the problem $(\mathcal{P}_\lambda)$ admits a non-zero solution if and only if $$Q_1(\lambda) \cdot Q_2(\lambda) = 0$$ where for $i \in \{1,2\}$, $Q_i$ is the polynomial $$Q_i(X) = \sum_{k=0}^{m-1} (-1)^k \left(T^{2k+1}(w_i) \mid T(w_i)\right) X^k$$ (d) Suppose that $\lambda$ is a root of the product polynomial $Q_1 Q_2$. Show that $\operatorname{dim}(H_\lambda) = 2$ if $\lambda$ is a common root of $Q_1$ and $Q_2$, and $\operatorname{dim}(H_\lambda) = 1$ otherwise.
(e) Show that $$\forall i \in \{1,2\}, Q_i(X) = \sum_{k=0}^{m-1} (-1)^k S\left(w_i, T^{2k+1}(w_i)\right) X^k$$

We place ourselves in the particular case where $E = \mathbb{R}_{2m}[X]$, with $m \geq 2$ a fixed natural integer. This vector space is equipped with the scalar product $$\forall (P,Q) \in E^2, \quad (P \mid Q) = \int_{-1}^{1} P(t)Q(t)\,dt$$ The two endomorphisms $T$ and $M$ of $E$ are defined by $$\forall P \in \mathbb{R}_{2m}[X], \quad T(P) = P' \text{ and } M(P) = P^*$$ where $P^*(X) = P(-X)$.
Show that $T$ and $M$ satisfy hypotheses (H1), (H2), (H3) and (H4).

We place ourselves in the particular case where $E = \mathbb{R}_{2m}[X]$, with $m \geq 2$ a fixed natural integer. This vector space is equipped with the scalar product $$\forall (P,Q) \in E^2, \quad (P \mid Q) = \int_{-1}^{1} P(t)Q(t)\,dt$$ The endomorphisms $T(P) = P'$ and $M(P) = P^*$ are defined as before. We set $$\mathbb{R}_{2m-1}^0[X] = \{P \in \mathbb{R}_{2m-1}[X] \mid P(-1) = 0 \text{ and } P(1) = 0\}$$ The polynomials $L_n$ are defined by $L_n(X) = \frac{1}{2^n n!} R_n^{(n)}(X)$ where $R_n(X) = (X^2-1)^n$.
Let $\lambda \in \mathbb{R}$. We consider the problem: find $P \in \mathbb{R}_{2m-1}^0[X]$ such that $$\forall Q \in \mathbb{R}_{2m-1}^0[X], \int_{-1}^{1} P(t)Q(t)\,dt - \lambda \int_{-1}^{1} P'(t)Q'(t)\,dt = 0$$ Show that this problem admits a non-identically zero solution $P$ if and only if $\lambda$ is a root of the polynomial $$K(X) = \left(\sum_{k=0}^{m-1} (-1)^k L_{2m-1}^{(2k+1)}(1) X^k\right) \cdot \left(\sum_{k=0}^{m-1} (-1)^k L_{2m}^{(2k+1)}(1) X^k\right)$$

Give a non-constant function belonging to $\mathcal{H}(U)$. Is the product of two harmonic functions a harmonic function?

Give a non-constant function belonging to $\mathcal{H}(U)$. Is the product of two harmonic functions a harmonic function?

Let $f$ be a function from $\mathbb{R}$ to $\mathbb{C}$, continuous and integrable on $\mathbb{R}$. The Fourier transform is defined as $\mathcal{F}(f) : \left\lvert\, \begin{aligned} & \mathbb{R} \rightarrow \mathbb{C} \\ & \xi \mapsto \int_{-\infty}^{+\infty} f(x) \exp(-\mathrm{i} 2\pi \xi x) \mathrm{d}x \end{aligned}\right.$. Show that $\mathcal{F}(f)$ is continuous on $\mathbb{R}$.

Show that if $f$ is positive, then $u$ is also positive.

We are given two functions $u$ and $v$, of class $\mathcal{C}^2$ on $\mathbb{R}$, not identically zero, and we set $$\forall (x,y) \in \mathbb{R}^2, \quad f(x,y) = u(x)v(y)$$ We assume that $f$ is harmonic on $\mathbb{R}^2$. Show that there exists a real constant $\lambda$ such that $u$ and $v$ are solutions respectively of the equations $$z'' + \lambda z = 0 \quad \text{and} \quad z'' - \lambda z = 0$$

We seek to determine the non-zero harmonic functions on $\mathbb{R}^2$ with separable variables, that is, functions $f$ that can be written in the form $f(x,y) = u(x)v(y)$. We are given two functions $u$ and $v$, of class $\mathcal{C}^2$ on $\mathbb{R}$, not identically zero, and we set $$\forall (x,y) \in \mathbb{R}^2, \quad f(x,y) = u(x)v(y)$$ We assume that $f$ is harmonic on $\mathbb{R}^2$.
Show that there exists a real constant $\lambda$ such that $u$ and $v$ are solutions respectively of the equations $$z'' + \lambda z = 0 \quad \text{and} \quad z'' - \lambda z = 0$$

Let $f$ be a function from $\mathbb{R}$ to $\mathbb{C}$, of class $\mathcal{C}^{1}$. We assume that $f$ and its derivative $f^{\prime}$ are integrable on $\mathbb{R}$. Show that $f$ tends to zero at $+\infty$ and at $-\infty$.

Give, depending on the sign of $\lambda$, the form of harmonic functions with separable variables.

We seek to determine the non-zero harmonic functions on $\mathbb{R}^2$ with separable variables, that is, functions $f$ that can be written in the form $f(x,y) = u(x)v(y)$. We are given two functions $u$ and $v$, of class $\mathcal{C}^2$ on $\mathbb{R}$, not identically zero, and we set $$\forall (x,y) \in \mathbb{R}^2, \quad f(x,y) = u(x)v(y)$$ We assume that $f$ is harmonic on $\mathbb{R}^2$.
Give, depending on the sign of $\lambda$, the form of harmonic functions with separable variables.

Let $f$ be a function from $\mathbb{R}$ to $\mathbb{C}$, of class $\mathcal{C}^{1}$. We assume that $f$ and its derivative $f^{\prime}$ are integrable on $\mathbb{R}$. Show that, for any real $\xi$, $\mathcal{F}\left(f^{\prime}\right)(\xi) = 2\mathrm{i}\pi\xi \mathcal{F}(f)(\xi)$.

Let $f$ be a real function of class $\mathcal{C}^2$ on $\mathbb{R}^2 \setminus \{(0,0)\}$. We set, for all $(r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R}$, $$g(r,\theta) = f(r\cos(\theta), r\sin(\theta))$$ Show that $f$ belongs to $\mathcal{H}\left(\mathbb{R}^2 \setminus \{(0,0)\}\right)$ if and only if, for all $(r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R}$, $$r^2 \frac{\partial^2 g}{\partial r^2}(r,\theta) + \frac{\partial^2 g}{\partial \theta^2}(r,\theta) + r\frac{\partial g}{\partial r}(r,\theta) = 0$$

Let $f$ be a real function of class $\mathcal{C}^2$ on $\mathbb{R}^2 \setminus \{(0,0)\}$. We set, for all $(r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R}$, $$g(r,\theta) = f(r\cos(\theta), r\sin(\theta))$$ Show that $f$ belongs to $\mathcal{H}(\mathbb{R}^2 \setminus \{(0,0)\})$ if and only if, for all $(r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R}$, $$r^2 \frac{\partial^2 g}{\partial r^2}(r,\theta) + \frac{\partial^2 g}{\partial \theta^2}(r,\theta) + r\frac{\partial g}{\partial r}(r,\theta) = 0$$

Determine the radial harmonic functions of $\mathbb{R}^2 \setminus \{(0,0)\}$, that is, the functions $f$ belonging to $\mathcal{H}\left(\mathbb{R}^2 \setminus \{(0,0)\}\right)$ such that $(r,\theta) \mapsto f(r\cos(\theta), r\sin(\theta))$ is independent of $\theta$.

Let $f$ be a real function of class $\mathcal{C}^2$ on $\mathbb{R}^2 \setminus \{(0,0)\}$. We set, for all $(r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R}$, $$g(r,\theta) = f(r\cos(\theta), r\sin(\theta))$$ Determine the radial harmonic functions of $\mathbb{R}^2 \setminus \{(0,0)\}$, that is, the functions $f$ belonging to $\mathcal{H}(\mathbb{R}^2 \setminus \{(0,0)\})$ such that $(r,\theta) \mapsto f(r\cos(\theta), r\sin(\theta))$ is independent of $\theta$.

Let $a, b, r_1$ and $r_2$ be four real numbers such that $0 < r_1 < r_2$. Determine a function $f$ of class $\mathcal{C}^2$ on $\mathbb{R}^2 \setminus \{(0,0)\}$ such that $$\begin{cases} \Delta f = 0 \\ f(x,y) = a & \text{if } \|(x,y)\| = r_1 \\ f(x,y) = b & \text{if } \|(x,y)\| = r_2 \end{cases}$$

Let $a, b, r_1$ and $r_2$ be four real numbers such that $0 < r_1 < r_2$. Determine a function $f$ of class $\mathcal{C}^2$ on $\mathbb{R}^2 \setminus \{(0,0)\}$ such that $$\begin{cases} \Delta f = 0 \\ f(x,y) = a & \text{if } \|(x,y)\| = r_1 \\ f(x,y) = b & \text{if } \|(x,y)\| = r_2 \end{cases}$$