Elliptic PDEs

Derive the weak formulation of the partial differential equationwith the initial value $u(x,0)=u_0(x)$.Please provide a detailed ans easy to follow solution. 

Use the mean value formulas to prove that (for $n\geq 3$) u(0) = \frac{1}{n \alpha(n)r^{n-1}}\int_{\partial B\left(0,r\right)} g \, dS + \frac{1}{n\left(n-2\right) \alpha \left(n\right)} \int_{B\left(...

Let $E=(E^1, E^2, E^3)$ be the electric field and $B=(B^1, B^2,B^3)$ be the magnetic field satifying the Maxwell's equations: Show that the components of the electric and magnetic fields satisfy the...

Let $\Omega$ be a bounded domain in $\mathbb R^n$ with smooth boundary. Assume that $u \in C^2(\bar \Omega) \cap H_0^1 (\Omega)$ be a strong solution to with $u=0$ on $\partial \Omega$. Show that $u \...

Let $\Omega$ be a bounded open subset of $\mathbb{R}^ n$ . Prove that there exists a constant $C$, depending only on $\Omega$, such that \max |u| \leq C (\max_{\partial \Omega }|g|+\max_{\overline{\Om...