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A Dirichlet Poisson smoothing index

Definition

Assume Countable Choice. Let $Ω$ be a bounded domain and let $1 / 2 < s \leq 1$ . Say that $Ω$ has a Dirichlet Poisson smoothing index $s$ with constant $C_{r e g}$ if, for every $g \in L^{2} (Ω)$ , there is a unique $ζ_{g} \in H_{0}^{1} (Ω)$ satisfying $\int_{Ω} \nabla ζ_{g} \cdot \nabla v d x = \int_{Ω} g v d x (v \in H_{0}^{1} (Ω)),$ and $ζ_{g} \in H^{1 + s} (Ω)$ with $∥ ζ_{g} ∥_{H^{1 + s} (Ω)} \leq C_{r e g} ∥ g ∥_{L^{2} (Ω)} .$ This is a hypothesis about the continuous adjoint Poisson problem. It is not asserted for every bounded domain.

Depends on

Used by

L2 error estimate under a Dirichlet Poisson smoothing index Theorem

Dependency tree · two levels

19 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Jean-Luc Guermond, MATH 661 Fall 2025, Part VIII, Chapter 36, Crouzeix–Raviart approximation (standard reference, not scraped)