zbMATH — the first resource for mathematics

Sobolev regularity for a class of second order elliptic PDE’s in infinite dimension. (English) Zbl 1328.35291
The paper is concerned with the differential equation \[ \lambda u-\frac{1}{2}\text{Tr}[D^2u]-\langle Ax-DU(x),Du\rangle=f(1) \] in the infinite-dimensional separable Hilbert space \(H\) (with norm \(\|\cdot\|\) and inner product \(\langle\cdot,\cdot\rangle\)), where \(A:D(A)\subset H\to H\) is a linear self-adjoint negative operator and the operator \(A^{-1}\) is of trace class, \(U:H\to\mathbb{R}\cup\{+\infty\}\) is a convex, proper, lower bounded and lower semicontinuous operator. Here, \(\lambda>0\) and \(f:H\to\mathbb{R}\) are given, \(Du\) and \(D^2u\) represent the first and the second derivatives of the unknown function \(u\), and \(\text{Tr}[D^2u]\) is the trace of \(D^2u\). Under some assumptions, the authors prove that for \(\lambda>0\) and \(f\in L^2(H,\nu)\), the weak solution \(u\) of (1) belongs to the Sobolev space \(W^{2,2}(H,\nu)\) (where \(\nu\) is the log-concave probability measure of the system), and that it has other maximal regularity properties. Some perturbations of (1) with unbounded operators are also investigated. Applications of the obtained results to reaction-diffusion Kolmogorov equations and to Cahn-Hilliard stochastic PDEs are finally presented.

35R15 PDEs on infinite-dimensional (e.g., function) spaces (= PDEs in infinitely many variables)
35R60 PDEs with randomness, stochastic partial differential equations
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
37L40 Invariant measures for infinite-dimensional dissipative dynamical systems
35B65 Smoothness and regularity of solutions to PDEs
Full Text: DOI Euclid arXiv
[1] Albeverio, S. and Röckner, M. (1991). Stochastic differential equations in infinite dimensions: Solutions via Dirichlet forms. Probab. Theory Related Fields 89 347-386. · Zbl 0725.60055 · doi:10.1007/BF01198791
[2] Barbu, V. (2010). Nonlinear Differential Equations of Monotone Types in Banach Spaces . Springer, New York. · Zbl 1197.35002
[3] Bogachev, V. I. (1998). Gaussian Measures . Amer. Math. Soc., Providence, RI. · Zbl 0938.28010
[4] Brézis, H. (1973). Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert . North-Holland, Amsterdam. · Zbl 0252.47055
[5] Chojnowska-Michalik, A. and Gołdys, B. (1995). Existence, uniqueness and invariant measures for stochastic semilinear equations on Hilbert spaces. Probab. Theory Related Fields 102 331-356. · Zbl 0859.60057 · doi:10.1007/BF01192465
[6] Davies, E. B. (1989). Heat Kernels and Spectral Theory . Cambridge Univ. Press, Cambridge. · Zbl 0699.35006
[7] Da Prato, G. (2004). Kolmogorov Equations for Stochastic PDEs . Birkhäuser, Basel. · Zbl 1066.60061
[8] Da Prato, G. (2006). An Introduction to Infinite-Dimensional Analysis . Springer, Berlin. · Zbl 1109.46001
[9] Da Prato, G. and Debussche, A. (1996). Stochastic Cahn-Hilliard equation. Nonlinear Anal. 26 241-263. · Zbl 0838.60056 · doi:10.1016/0362-546X(94)00277-O
[10] Da Prato, G., Debussche, A. and Goldys, B. (2002). Some properties of invariant measures of nonsymmetric dissipative stochastic systems. Probab. Theory Related Fields 123 355-380. · Zbl 1087.60049 · doi:10.1007/s004400100188
[11] Da Prato, G., Flandoli, F., Priola, E. and Röckner, M. (2013). Strong uniqueness for stochastic evolution equations in Hilbert spaces perturbed by a bounded measurable drift. Ann. Probab. 41 3306-3344. · Zbl 1291.35455 · doi:10.1214/12-AOP763
[12] Da Prato, G. and Röckner, M. (2002). Singular dissipative stochastic equations in Hilbert spaces. Probab. Theory Related Fields 124 261-303. · Zbl 1036.47029 · doi:10.1007/s004400200214
[13] Da Prato, G. and Zabczyk, J. (1992). Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications 44 . Cambridge Univ. Press, Cambridge. · Zbl 0761.60052 · doi:10.1017/CBO9780511666223
[14] Da Prato, G. and Zabczyk, J. (1996). Ergodicity for Infinite-Dimensional Systems. London Mathematical Society Lecture Note Series 229 . Cambridge Univ. Press, Cambridge. · Zbl 0849.60052 · doi:10.1017/CBO9780511662829
[15] Da Prato, G. and Zabczyk, J. (2002). Second Order Partial Differential Equations in Hilbert Spaces. London Mathematical Society Lecture Notes 293 . Cambridge Univ. Press, Cambridge. · Zbl 1012.35001
[16] Elezović, N. and Mikelić, A. (1991). On the stochastic Cahn-Hilliard equation. Nonlinear Anal. 16 1169-1200. · Zbl 0729.60057 · doi:10.1016/0362-546X(91)90204-E
[17] Haussmann, U. G. and Pardoux, É. (1986). Time reversal of diffusions. Ann. Probab. 14 1188-1205. · Zbl 0607.60065 · doi:10.1214/aop/1176992362
[18] Jona-Lasinio, G. and Sénéor, R. (1991). On a class of stochastic reaction-diffusion equations in two space dimensions. J. Phys. A 24 4123-4128. · Zbl 0745.60058 · doi:10.1088/0305-4470/24/17/028
[19] Lunardi, A., Metafune, G. and Pallara, D. (2005). Dirichlet boundary conditions for elliptic operators with unbounded drift. Proc. Amer. Math. Soc. 133 2625-2635. · Zbl 1210.35134 · doi:10.1090/S0002-9939-05-08068-8
[20] Lunardi, A. and Vespri, V. (1998). Optimal \(L^{\infty}\) and Schauder estimates for elliptic and parabolic operators with unbounded coefficients. In Reaction Diffusion Systems ( Trieste , 1995) (G. Caristi and E. Mitidieri, eds.). Lecture Notes in Pure and Applied Mathematics 194 217-239. Dekker, New York. · Zbl 0887.47034
[21] Phelps, R. R. (1978). Gaussian null sets and differentiability of Lipschitz map on Banach spaces. Pacific J. Math. 77 523-531. · Zbl 0396.46041 · doi:10.2140/pjm.1978.77.523
[22] Röckner, M. (1999). \(L^{p}\)-analysis of finite and infinite-dimensional diffusion operators. In Stochastic PDE’s and Kolmogorov Equations in Infinite Dimensions ( Cetraro , 1998) (G. Da Prato, ed.). Lecture Notes in Math. 1715 65-116. Springer, Berlin. · doi:10.1007/BFb0092418
[23] Shigekawa, I. (1992). Sobolev spaces over the Wiener space based on an Ornstein-Uhlenbeck operator. J. Math. Kyoto Univ. 32 731-748. · Zbl 0777.60047
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.