Let A = (aij) be a matrix-valued Borel mapping on a domain Ω ⊂ ℝd, let b = (bi) be a vector field on Ω, and let LA, bφ = aij∂xi∂xjφ + bi∂xiφ. We study Borel measures μ on Ω that satisfy the elliptic equation L*A, bμ = 0 in the weak sense: ∫ LA, bφ dμ = 0 for all φ ∈ C0∞(Ω). We prove that, under mild conditions, μ has a density. If A is locally uniformly nondegenerate, A ∈ Hlocp, 1 and b ∈ Llocp for some p > d, then this density belongs to Hlocp, 1. Actually, we prove Sobolev regularity for solutions of certain generalized nonlinear elliptic inequalities. Analogous results are obtained in the parabolic case. These results are applied to transition probabilities and invariant measures of diffusion processes.
Bibliographical noteFunding Information:
We are grateful to W. Stannat for useful discussions. This work has been supported in part by the Russian Foundation of Fundamental Research projects 97–01–00932 and 00–15–99267, the INTAS Grant 94– 378, the INTAS-RFBR Grant 95-0099, the DFG Grant 436 RUS 113/ 343/0(R), the SFB–343, NSF Grant DMS–9876586, and the EU–TMR– Project ERB–FMRX–CT96–0075.