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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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[1] Bakry, PD., L’hypercontractivité et son utilisation en théorie des semi-groupes, Lectures on probability theory, lecture notes in mathematics, Vol. 1581, (1994), Springer Berlin
[2] Bogachev, V.I.; Krylov, N.V.; Röckner, M., On regularity of transition probabilities and invariant measures of singular diffusions under minimal conditions, Comm. partial differential equations, 26, 2037-2080, (2001) · Zbl 0997.35012
[3] Brézis, H., Opérateurs maximaux monotones, (1973), North-Holland Amsterdam
[4] Cerrai, S., Second order PDE’s in finite and infinite dimensions, A probabilistic approach, Lecture notes in mathematics, Vol. 1762, (2001), Springer Berlin
[5] G. Da Prato, Finite dimensional convex gradient systems perturbed by noise, Lecture Notes in Pure and Applied Mathematics, Dekker Lecture Notes n. 234, in: G. Goldstein, R. Nagel and S. Romanelli (Eds.), (2003) 129-145. · Zbl 1053.47044
[6] Da Prato, G., Some properties of monotone gradient systems, Dynamic cont. discrete imp. systems ser. A, 8, 401-414, (2001) · Zbl 0997.47031
[7] Eberle, A., Uniqueness and non-uniqueness of singular diffusion operators, Lecture notes in mathematics, Vol. 1718, (1999), Springer Berlin · Zbl 0957.60002
[8] Kolmogorov, A.N., Uber die analytischen methoden in der wahrscheinlichkeitsrechnung, Math. ann., 104, 415-458, (1931) · JFM 57.0613.03
[9] Krylov, N.V., Introduction to the theory of diffusion processes, (1995), American Mathematical Society Providence, RI · Zbl 0844.60050
[10] Ladyzhenskaja, O.A.; Ural’ceva, N.N., Linear and quasilinear elliptic equations, (1968), Academic Press New York
[11] Lunardi, A., Schauder theorems for linear elliptic and parabolic problems with unbounded coefficients in \(R\^{}\{n\}\), Studia math., 128, 171-198, (1998) · Zbl 0899.35014
[12] Lunardi, A.; Vespri, V., Optimal L∞ and Schauder estimates for elliptic and parabolic operators with unbounded coefficients, (), 217-239 · Zbl 0887.47034
[13] A. Lunardi, V. Vespri, Generation of strongly continuous semigroups by elliptic operators with unbounded coefficents in \(L\^{}\{p\}(R\^{}\{n\})\), Rend. Istit. Mat. Univ. Trieste 28 (1997), Special issue dedicated to the memory of Pierre Grisvard, 251-279. · Zbl 0899.35027
[14] Metafune, G.; Pallara, D.; Wacker, M., Feller semigroups on \(R\^{}\{N\}\), Sem. forum, 65, 159-205, (2002) · Zbl 1014.35050
[15] G. Metafune, J. Prüss, A. Rhandi, R. Schnaubelt, The domain of the Ornstein-Uhlenbeck operator on an Lp space with invariant measure, Ann. Scuola. Norm. Sup. Pisa (2002) 471-485. · Zbl 1170.35375
[16] G. Metafune, A. Rhandi, J. Prüss, R. Schnaubelt, Lp regularity for elliptic operators with unbounded coefficients, preprint.
[17] Stannat, W., (nonsymmetric) Dirichlet operators on L1existence, uniqueness and associated Markov processes, Ann. scuola norm. sup. Pisa ser. 4, 28, 99-140, (1999) · Zbl 0946.31003
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