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The Bogovskii integral on a star-shaped set

Definition

Let $U \subset R^{d}$ be bounded and open, star-shaped with respect to a closed ball $\overline{B} (a, ρ) \subset U$ . Choose $ω \in C_{c}^{\infty} (B (a, ρ))$ with $\int ω = 1$ . For $f \in C_{c}^{\infty} (U)$ , initially define the vector-valued integral $(B_{U} f) (x) = \int_{U} f (y) (x - y) \int_{0}^{1} ω ⁣ (y + \frac{x - y}{t}) \frac{d t}{t^{d + 1}} d y .$ The inner integral is first interpreted for $t > 0$ ; its locally integrable kernel, support and divergence identity are proved in the following lemma. All data and fields here are real. This definition makes no finite-quadrature or numerical-algorithm claim.

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Sources

Ricardo G. Durán, An elementary proof of the continuity from L2_0(Omega) to H1_0(Omega)^n of Bogovskii’s right inverse of the divergence (standard reference, not scraped)
Daniele Boffi, Franco Brezzi, Michel Fortin, Mixed Finite Element Methods and Applications (complete textbook) (standard reference, not scraped)