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 $D(0,1)$.
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 $D(0,1)$.