Abstract
We survey some recent results of existence, uniqueness and regularity about elliptic equations with infinitely many variables \(x_h,\;h\in \mathbb N\), where x h = 〈x, e h 〉 and (e h ) is an orthonormal basis in a separable Hilbert space H.
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Notes
- 1.
Which is not meaningful in L 2(H, N Q ) because N Q (Q 1∕2(H)) = 0.
- 2.
The integral is intended in the strong sense that is for any fixed x ∈ H.
- 3.
C b (H) represents the space of all real, bounded and continuous functions in H.
- 4.
That is μ << ν and ν << μ.
- 5.
B b (H) is the space of all \(\varphi :H\to \mathbb {R}\) bounded and Borel.
- 6.
\(C^\infty _b(H)\) is the space of all real mappings in \(\mathbb {R}\) which are infinitely many differentiable with bounded derivatives of all order.
- 7.
One can show that μ is unique, see e.g. [3, Theorem 2.34].
- 8.
Note that the function φ h belongs to \(D(\mathcal {L})\) if and only if h ∈ D(A ∗).
- 9.
For the definition of Γ and H γ , γ ∈ Γ, see the Appendix.
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Prato, G.D. (2018). Elliptic Operators with Infinitely Many Variables. In: Rassias, T. (eds) Current Research in Nonlinear Analysis. Springer Optimization and Its Applications, vol 135. Springer, Cham. https://doi.org/10.1007/978-3-319-89800-1_5
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