Consider the annulus $\mathcal A:= B(0,2)\setminus B(0,1)$ in $\mathbb R^n$, $n\geq2$ and the divergence form elliptic operator $\operatorname{div}(A\nabla \cdot)$ on $\mathcal A$ where $A$ is a uniformly elliptic matrix with $L^\infty$ coefficients.
My question is: given any linear function $L\cdot x$ with $\lvert L\rvert\leq1$ (in whatever reasonable vector norm you want to consider), does there exist a continuous function $u$ on $\overline{\mathcal A}$ satisfying
- $u\rvert_{\partial B_1} = L\cdot x$,
- $u\rvert_{\partial B_2} = 0$,
- $\lvert\operatorname{div}(A\nabla u)\rvert(\overline{\mathcal A}) \leq C$ where the LHS is understood as a measure and the constant $C$ only depends on the ellipticity and $L^\infty$ bounds of the matrix $A$?
If $A$ is the identity (or very regular) the answer is easy as any $C^2$ extension will work, but I wonder what happens with more general operators….
| |work just fine as norm delimiters, but, every so often, TeX needs to know explicitly which is an opening delimiter and which is a closing delimiter. Note $|\operatorname{div}(A\nabla u)|$|\operatorname{div}(A\nabla u)|versus $\lvert\operatorname{div}(A\nabla u)\rvert$\lvert\operatorname{div}(A\nabla u)\rvert. I edited accordingly. (While I was there, I also changed $B(0, 2) \backslash B(0, 1)$B(0, 2) \backslash B(0, 1)to $B(0, 2) \setminus B(0, 1)$B(0, 2) \setminus B(0, 1).) $\endgroup$