arXiv subjects
Publications and source records attributed to Alessandra Lunardi.
We compare the perimeter measure with the Airault-Malliavin surface measure and we prove that all open convex subsets of abstract Wiener spaces have finite perimeter. By an explicit counter-example, we show that in general this is not true for compact convex domains.
We consider a family of self-adjoint Ornstein--Uhlenbeck operators $L_α $ in an infinite dimensional Hilbert space H having the same gaussian invariant measure $μ$ for all $α\in [0,1]$. We study the Dirichlet problem for the equation $λϕ- L_αϕ= f$ in a closed set K, with $f\in L^2(K, μ)$. We first prove that the variational solution, trivially provided by the Lax---Milgram theorem, can be represented, as expected, by means of the transition semigroup stopped to K. Then we address two problems: 1) the regularity of the solution $φ$ (which is by definition in a Sobolev space $W^{1,2}_α(K,μ)$) of the Dirichlet problem; 2) the meaning of the Dirichlet boundary condition. Concerning regularity, we are able to prove interior $W^{2,2}_α$ regularity results; concerning the boundary condition we consider both irregular and regular boundaries. In the first case we content to have a solution whose null extension outside K belongs to $W^{1,2}_α(H,μ)$. In the second case we exploit the Malliavin's theory of surface integrals which is recalled in the Appendix of the paper, then we are able to give a meaning to the trace of $ϕ$ at the boundary of K and to show that it vanishes, as it is natural.
We consider a class of second order linear nonautonomous parabolic equations in R^d with time periodic unbounded coefficients. We give sufficient conditions for the evolution operator G(t,s) be compact in C_b(R^d) for t>s, and describe the asymptotic behavior of G(t,s)f as t-s goes to infinity in terms of a family of measures mu_s, s in R, solution of the associated Fokker-Planck equation.
In this paper we investigate a class of nonautonomous linear parabolic problems with time depending Ornstein-Uhlenbeck operators. We study the asymptotic behavior of the associated evolution operator and evolution semigroup in the periodic and non-periodic situation. Moreover, we show that the associated evolution operator is hypercontractive.
We characterize the domain of the realization of the linear parabolic operator Gu := u_t + L(t)u (where, for each real t, L(t) is an Ornstein-Uhlenbeck operator), in L^2 spaces with respect to a suitable measure, that is invariant for the associated evolution semigroup. As a byproduct, we obtain optimal L^2 regularity results for evolution equations with time-depending Ornstein-Uhlenbeck operators.