Sentence examples for measure density condition from inspiring English sources

In the first one, Theorem 1, we prove that if the Sobolev embedding theorem holds in Ω, in any of all the possible cases, then Ω satisfies the measure density condition.

Theorem 4.3 Suppose that the complement of Ω satisfies the measure density condition (11).

Here, we assume that the complement of Ω satisfies the measure density condition, it means that there exists a positive constant C > 0 such that | Ω c ∩ B | ≥ C | B |, (11).

Let f = ∑ I f I 1 e I, then f ≥ ψ in Ω. Theorem 3.6 Suppose that the complement of Ω satisfies the measure density condition (11), and let u ∈ W loc 1, p ( Ω, C ℓ n k ) be a weak solution to the obstacle problem (14), where f ∈ W 1, p + δ ( Ω, C ℓ n k ) for some δ > 0.

end{aligned} Then, using the measure density condition (2.8) and the geometry of Reifenberg flatness (4.18), we conclude that begin{aligned} biglvert bigl{ xinOmega_{5r_{y_{i}}} y_{i}): vert Duvert ^{2} > ( A lambda)^{frac{p^{p(x)}} bigr} bigrvert le C_{4} varepsilonbiglvert Omega_{r_{y_{i}}} y_{i}) bigrvert.

We consider solutions to degenerate parabolic equations with measurable coefficients, having on the right-hand side a measure satisfying a suitable density condition; we prove integrability results for the gradient in the Marcinkiewicz scale.

The computational results suggest that maintaining a uniform KOH concentration in the electrolyte (for example, at design point of 7 M) be an effective measure to increase the limiting current density condition.

However, to achieve the embedding theorem, one needs the density condition on the measure, namely begin{aligned} mu(B x,r geq Cr^{omega} end{aligned} (1.7) for any (xin X) and (r>0).

Newer cheap and portable scanners use ultrasound, and portable X-ray measures measure density in the heel.

Because XPCT can penetrate surfaces much more deeply, it is used to measure density.

For the operating conditions in these experiments, the density is 1.51 ± 0.15 g cm−3; see the Supporting Information for a description of the interferometric technique employed to measure density and film growth rates.