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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.

MSC:
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
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