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Let $\Omega\subset \mathbb R^N$ be an open, smooth and bounded subset. Given a $N\times N$, bounded and elliptic matrix of Hölder continuous functions. That is, $A(x)= \{a_{ij}(x)\}_{N\times N}$, $a_{ …

given a symmetric matrix of Holder continuous functions $A(x)$ such that $$ \frac{1}{C} |\xi|^2 \leq \langle A(x)\xi,\xi \rangle \leq C |\xi|^2 $$ find a vector field $\Phi$ such that $$ D \Phi(x)^t D …

It is well known that for any non negative Harmonic function w ($\Delta w=0$, $w\geq 0$) in a ball, $B_1(0)$, $\exists$, C>0 such that $\forall y\in B_{1/2}(0)$ $$ Cw(0)\leq w (y) $$ It is a clear i …

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 …

Suppose we have a second order elliptic differential operator $$ L(v) = -\text{div}(A(x) \nabla v) $$ $A(x)$ is a bounded and strictly positive definite matrix with Hölder continuous entries. And supp …