Maximal \(L^2\) regularity for Dirichlet problems in Hilbert spaces. (English) Zbl 1293.35359
The authors study the Dirichlet problem
\[
\lambda U- \mathcal{L}U=F \text{ in } \mathcal{O}, \qquad U=0 \text{ on } \partial \mathcal{O}.
\]
Here \( \mathcal{O}\) is an open subset of a Hilbert space \(X\), \(F\in L^2 (\mathcal{O},\mu)\), \(\mu\) is a non-degenerate centered Gaussian measure on \(X\), and \(\mathcal{L}\) is an Ornstein-Uhlenbeck operator.
The main results of the paper concern the problem whether the weak solution \(U\) belongs to the Sobolev space \(W^{2,2} (\mathcal{O},\mu)\).
Previous results show that this question has a positive answer in the case \(\mathcal{O}=X\), but on general domains \(\mathcal{O}\) an explicit basis of \( L^2 (\mathcal{O},\mu)\) formed by eigenfunctions is not available, that could play the role of the Hermite polynomials and Wiener chaos decomposition used in the case \(\mathcal{O}=X\). The authors follow a completely different approach that consists of two steps:
Step 1. Dimension-free \(W^{2,2}\) estimates for finite-dimensional approximations.
Step 2. Approximate weak solutions by cylindrical functions that solve the finite-dimensional problems.
Both steps are rather delicate.
The final result gives a sufficient condition in terms of geometric properties of the boundary \(\partial \mathcal{O}\). These conditions are defined via properties of a function that has \(\mathcal{O}\) as a level set. Examples discussed include half-spaces, regions below a graph, spheres, and the special case of \(X=L^2(0,1)\).
The main results of the paper concern the problem whether the weak solution \(U\) belongs to the Sobolev space \(W^{2,2} (\mathcal{O},\mu)\).
Previous results show that this question has a positive answer in the case \(\mathcal{O}=X\), but on general domains \(\mathcal{O}\) an explicit basis of \( L^2 (\mathcal{O},\mu)\) formed by eigenfunctions is not available, that could play the role of the Hermite polynomials and Wiener chaos decomposition used in the case \(\mathcal{O}=X\). The authors follow a completely different approach that consists of two steps:
Step 1. Dimension-free \(W^{2,2}\) estimates for finite-dimensional approximations.
Step 2. Approximate weak solutions by cylindrical functions that solve the finite-dimensional problems.
Both steps are rather delicate.
The final result gives a sufficient condition in terms of geometric properties of the boundary \(\partial \mathcal{O}\). These conditions are defined via properties of a function that has \(\mathcal{O}\) as a level set. Examples discussed include half-spaces, regions below a graph, spheres, and the special case of \(X=L^2(0,1)\).
Reviewer: Dirk Blömker (Augsburg)
MSC:
| 35R15 | PDEs on infinite-dimensional (e.g., function) spaces (= PDEs in infinitely many variables) |
Keywords:
PDEs in infinite dimensions; Ornstein-Uhlenbeck operators; Dirichlet problems; maximal Sobolev regularity; cylindrical functions; finite-dimensional approximationReferences:
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