I have some questions on solvability of the following elliptic PDE: in $\mathbb{R}^2$, for $f\in L^2$, $$a^{ij} u_{x^i x^j} +b^i u_{x^i} + c u = f.$$ Here $\{a^{ij}(x)\}_{i,j=1,2}$ is symmetric matrix and there is a $\kappa>0$ such that $$ \kappa^{-1} |\xi|^2 \geq a^{ij} \xi^i \xi^j \geq \kappa |\xi|^2$$ for all $\xi \in \mathbb{R}^2$. Here I used the Einstein summation convention. Also, assume that $a^{ij}, b^i, c$ are bounded and measurable. No further regularity assumption is needed (e.g. uniformly continuous, VMO condition since we work on $L^2$ and $\mathbb{R}^2$)
Let us denote $L = a^{ij} D_{ij} +b^i D_{i} +c$.
According to some literature(arXiv:1404.5647), if $a^{ij}$ are merely measurable, then there is a $W^{2,2}$-estimate for $Lu-\lambda u=f$ for large $\lambda>0$ in whole domain $\mathbb{R}^2$. They give references as follows:
- Serge Bernstein. Sur la g´en´eralisation du probl`eme de Dirichlet. Math. Ann., 69(1):82–136, 1910.
- Giorgio Talenti. Equazioni lineari ellittiche in due variabili. Matematiche (Catania), 21:339– 376, 1966.
But I fail to access these references due to language problem (French, and mathematics language problem). I cannot access the second paper in online and offline in my country.
Is their any book which give a standard proof of this question? I found some exercises from Krylov's book. But I want to find another reference.
Thanks you for in advance.
The better behavior of elliptic PDE with measurable coefficients in two dimensions is explained by their relation with quasiconformal maps. I know two book sources that develop this relation.