All Questions

Suppose we are given a sequence $\phi_k$ of traces (i.e. functions defined on boundary $\partial B_1$) such that $$ \phi_k \rightarrow 0 \;\mbox{in $L^{\infty}(\partial B_1)$} $$ (one can consider $C^{...

$\newcommand{\avint}{⨍}$ Let $B_r$ be a call of radius $r$ and centre origin and $k<1$.$w$ satisfy the following PDE: $$ \begin{cases} -\Delta w = 0 \qquad \mbox{in $B_r\setminus B_{kr}$}\\ w=0 \...

Let $B=B_1(0)\subset \mathbb R^N$ and let $u\geq 0$ solve the PDE $$ -\Delta_p u=1\,\,\mbox{in $B$} $$ Let another function $f$ be such that $$ \begin{cases} -\Delta_p f =1 \;\;\mbox{in $B$}\\ f=0 \...