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Elliptic equations for measures: regularity and global bounds of densities. (English) Zbl 1206.35242
For given an elliptic operator $Lf=L_{A,b}f:=\sum_{i,j=1}^{n}a^{ij}\partial_{x_{i}}\partial_{x_{j}}f+ \sum_{i=1}^{n}b^{i}\partial_{x_{i}}f,$ where $$x\mapsto A(x)=(a^{ij}(x))$$ is a Borel mapping with values in the space of positive symmetric matrices and $$x\mapsto b(x)=(b^{i}(x))$$ is a Borel measurable vector field, we say that a measure $$\mu$$ on $$\mathbb R^n$$ is a solution to equation $$L^{*}\mu=\nu$$ if $$L\phi\in L^1(|\mu|)$$ for all $$\phi\in C_{0}^{\infty}(\mathbb R^n)$$, $$\nu\in W^{p,-1}$$ and one has $$\int L\phi\,d\mu=\langle\nu,\phi\rangle$$. Here $$W^{p,r}(\mathbb R^n)$$ denote the Sobolev class of functions that belong to $$L^{p}(\mathbb R^n)$$ along with their generalized partial derivatives of order up to $$r$$, and dual to $$W^{p,1}(\mathbb R^n)$$ is denoted by $$W^{p',-1}(\mathbb R^n)$$, $$p':=p(1-p)^{-1}$$.
The authors also consider divergence form operators $Lf={\mathcal L}_{A,b}f :=\sum_{i,j=1}^{n}\partial_{x_{i}}(a^{ij}\partial_{x_{j}}f) + \sum_{i=1}^{n}b^{i}\partial_{x_{i}}f.$ The membership of solution in the Sobolev classes $$W^{p,1}(\mathbb R^n)$$ is established. If $$A$$ is locally Hölder continuous and nondegenerate, then $$\mu$$ has a density $$\rho\in L^{r}_{\text{loc}}(\mathbb R^n)$$ for any $$r<n/(n-1)$$. Bounds of the form $$\rho(x)\leq C\Phi(x)^{-1}$$ for the corresponding densities are obtained. Here $$\Phi\in W^{1,1}_{\text{loc}}(\mathbb R^n)$$ be some positive function.

##### MSC:
 35R05 PDEs with low regular coefficients and/or low regular data 35J15 Second-order elliptic equations 60J60 Diffusion processes
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