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Elliptic operators with unbounded drift coefficients and Neumann boundary condition. (English) Zbl 1046.35025
The authors study the realization $$A_N$$ of the linear elliptic operator ${\mathcal A}=\tfrac12 \Delta -\langle DU,D\cdot\rangle$ in $$L^2(\Omega,\mu)$$ with Neumann boundary condition. Here $$\Omega$$ is a possibly unbounded convex open set in $${\mathbb R}^N,$$ $$U$$ is a convex unbounded function with $$DU(x)$$ being the element with minimal norm in the subdifferential of $$U$$ at $$x$$ and $$\mu(dx)=c \exp(-2U(x))dx$$ is a probability measure, infinitesimally invariant for $${\mathcal A}.$$
The main result is that $A_N\colon\quad \left\{u\in H^2(\Omega,\mu)\colon\;\partial u/\partial n=0,\;{\mathcal A}u \in L^2(\Omega,\mu)\right\}$ is a self-adjoint dissipative operator. The technique of proofs involves a penalization method applied to the family of operators ${\mathcal A}_\varepsilon u(x)={1\over 2}\Delta u(x) -\langle DU_\varepsilon(x),Du(x)\rangle,\quad x\in {\mathbb R}^N$ with $U_\varepsilon(x)=U(x)+{1\over{2\varepsilon}}(\text{ dist\,}(x,\Omega))^2.$ An additional assumption on the convexity of $$U-\omega| x| ^2/2$$ for some $$\omega>0$$ allows the authors to show that the measure $$\mu$$ satisfies Poincaré and $$\log$$-Sobolev inequalities, whence smoothing properties and asymptotic behavior of the semigroup generated by $$A_N$$ follow.

##### MSC:
 35J25 Boundary value problems for second-order elliptic equations 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 47D07 Markov semigroups and applications to diffusion processes
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