Sentence examples for bounded domain such that from inspiring English sources

Let Ω ⊂ ℝ2 be a bounded domain such that the boundary, ∂ Ω, is Lipschitz continuous.

Let (Omega ) be a conformally bounded domain such that (0in partial Omega ).

(Chern and Lin [16]) Let (Omega ) be a smooth bounded domain such that (0in partial Omega ).

For further considerations, we suppose that has smooth boundary (e.g., ), and the crack (smooth) can be extended to an arbitrary smooth, simply connected, closed curve enclosing a bounded domain such that the normal vector on coincides with the outward normal vector on which we again denote by.

In this case the group (Aut ({mathcal {D}})) contains an infinite cocompact subgroup, so it is natural to look first at domains which have a big group of automorphisms, especially at bounded homogeneous domains, i.e., bounded domains such that the group ({mathrm {Aut}}(Omega )) of biholomorphisms of (Omega ) acts transitively.

If Ω 1 ⊂ Ω 2 are bounded domains, then λ 1 ( Ω 1 ) ≥ λ 1 ( Ω 2 ) ≥ 0. Thus, if Ω 1 ⊂ Ω 2 ⊂ ⋯ ⊂ M are bounded domains such that ⋃ Ω i = M, then the limit λ 1 ( M ) = lim i → ∞ λ 1, p ( Ω i ) ≥ 0. exists, and it is independent of the choice of { Ω i }.

Here, is a bounded domain of such that its Lebesgue's measure, and.

Let (Omega ) be a bounded smooth domain such that (0in Omega ) and assume (0le sle 2).

Let (Omega ) be a bounded smooth domain such that (0in partial Omega ) and assume (0le sle 2).

Assume F is a map from Ω into Rn where Ω is a bounded domain in Rn such that |Dƒ| ∈ Ln(logL)−s with 0≥ s ≥ 1, i.e., ∫Ω |Dƒ|n[log(1 +|Dƒ|]−s ≤ ∞, then J ∈ L log L 1−s(K) for any compact subset K ⊂ Ω, where J=det(Dƒ).

In this spirit we consider the slightly more general problem of finding or characterizing the maximal function (u Dto {mathbb{R}}) defined in the closure of a bounded convex plane domain, such that (u(x leqpsi(x)) for all (xinpartial D), for a given function ψ on ∂D.