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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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