Let $f$ be a function of class $C^2$ from $\mathbb{R}^n$ to itself. If $i$, $j$ and $k$ are three integers in $\llbracket 1, n \rrbracket$, the second partial derivative of $f_k$ at $x$ with respect to the variables $x_i$ and $x_j$ is denoted $\mathrm{D}_{i,j} f_k(x)$ or $\frac{\partial^2 f_k}{\partial x_i \partial x_j}(x)$, or also $f_{i,j,k}(x)$.
Justify that, for all $x$ in $\mathbb{R}^n$ and all $i$, $j$ and $k$ in $\llbracket 1, n \rrbracket$, we have $f_{i,j,k}(x) = f_{j,i,k}(x)$.

Let $f$ be a function of class $C^2$ from $\mathbb{R}^n$ to itself. If $i$, $j$ and $k$ are three integers in $\llbracket 1, n \rrbracket$, the second partial derivative of $f_k$ at $x$ with respect to the variables $x_i$ and $x_j$ is denoted $f_{i,j,k}(x)$.
We assume that the Jacobian matrix $J_f(x)$ is antisymmetric for all $x$ in $\mathbb{R}^n$.
Show that for all $x$ in $\mathbb{R}^n$, and all $i$, $j$ and $k$ in $\llbracket 1, n \rrbracket$, $f_{i,j,k}(x) = -f_{i,k,j}(x)$.

Let $f$ be a function of class $C^2$ from $\mathbb{R}^n$ to itself. If $i$, $j$ and $k$ are three integers in $\llbracket 1, n \rrbracket$, the second partial derivative of $f_k$ at $x$ with respect to the variables $x_i$ and $x_j$ is denoted $f_{i,j,k}(x)$.
We assume that the Jacobian matrix $J_f(x)$ is antisymmetric for all $x$ in $\mathbb{R}^n$.
Deduce that, for all $x$ in $\mathbb{R}^n$ and all $i$, $j$ and $k$ in $\llbracket 1, n \rrbracket$, we have $f_{i,j,k}(x) = 0$.

Let $\lambda \in \mathbb{R}^*$ and $(x_0, y_0) \in \mathbb{R}^2$ be fixed. We define: $$\Omega_{x_0, y_0, \lambda} = \left\{ \lambda(x,y) + (x_0, y_0) \mid (x,y) \in \Omega \right\}$$ Let $g : \Omega_{x_0, y_0, \lambda} \rightarrow \mathbb{R}$ be a harmonic application.
Show that the application $(x,y) \mapsto g\left(\lambda(x,y) + (x_0, y_0)\right)$ is harmonic on $\Omega$.

Show that the applications $$h_1 : \left|\begin{array}{rll} \mathbb{R}^2 \backslash \{(0,0)\} & \rightarrow & \mathbb{R} \\ (x,y) & \mapsto & \ln(x^2 + y^2) \end{array}\right. \quad \text{and} \quad h_2 : \left|\begin{array}{rll} \mathbb{R}^2 \backslash \{(0,0)\} & \rightarrow & \mathbb{R} \\ (x,y) & \mapsto & \dfrac{1}{x^2 + y^2} \end{array}\right.$$ are harmonic.

Deduce that, for all $t \in \mathbb{R}$, the application $(x,y) \mapsto \dfrac{1 - \left((x + \cos t)^2 + (y + \sin t)^2\right)}{x^2 + y^2}$ is harmonic on $\mathbb{R}^2 \backslash \{(0,0)\}$.

For $(x,y) \in D(0,1)$ fixed, we define the complex number $z = x + iy$ and we set for $t$ real (when the expression makes sense): $$\mathrm{N}(x,y,t) = \frac{1 - |z|^2}{|z - e^{it}|^2} = \frac{1 - (x^2 + y^2)}{(x - \cos t)^2 + (y - \sin t)^2}$$
Show that, for all $t \in \mathbb{R}$, the application $$\mathrm{N}_t : \left|\begin{array}{rll} D(0,1) & \rightarrow & \mathbb{R} \\ (x,y) & \mapsto & \mathrm{N}(x,y,t) \end{array}\right.$$ is harmonic.
One may use question II.B.3.

Let $f : C(0,1) \rightarrow \mathbb{R}$ be a continuous application. We define: $$\mathrm{N}_f(x,y) = \frac{1}{2\pi} \int_0^{2\pi} \mathrm{N}(x,y,t) f(\cos t, \sin t)\, \mathrm{d}t$$ on $D(0,1)$, and $$u(x,y) = \begin{cases} \mathrm{N}_f(x,y) & \text{if } (x,y) \in D(0,1) \\ f(x,y) & \text{if } (x,y) \in C(0,1) \end{cases}$$ on $\bar{D}(0,1)$.
a) Show that $\mathrm{N}_f$ admits a second-order partial derivative $\partial_{11} \mathrm{N}_f$ with respect to $x$.
Similarly, one can show that $\mathrm{N}_f$ admits second-order partial derivatives with respect to all its variables, continuous on $D(0,1)$. This result is admitted for the rest.
Express, for all $(x,y) \in D(0,1)$, for all $(i,j) \in \{1,2\}^2$, $\partial_{ij} \mathrm{N}_f(x,y)$ in terms of $\partial_{ij} \mathrm{N}(x,y,t)$.
b) Deduce that $u$ is harmonic on $D(0,1)$.