On ε-Friedrichs inequalities and its application
Journal Title: Математичні Студії - Year 2019, Vol 51, Issue 1
Abstract
Let n∈N be a fixed number, Ω be a bounded domain in Rn, L2(Ω), L∞(Ω) be the Lebesgue spaces, H1(Ω) and H10(Ω) be the Sobolev spaces, Πℓ(α):=⊗k=1n(αk;αk+ℓ),α=(α1,…,αn)∈Rn,ℓ>0. There are proved the following assertions about ε-Friedrichs inequality (Theorem 1) in the space H1(Ω) (Theorem 1) and H10(Ω) (Theorem 2) : for every ε>0 there exist Nε∈N and ω1,…,ωNε∈L∞(Ω) such that the inequality ∫Ω|v(x)|2dx≤ε∫Ω|∇v(x)|2dx+∑j=1Nε(∫Ωv(x)ωj(x)dx)2 holds for every v∈H1(Ω) (Theorem 1) and v∈H10(Ω) (Theorem 2), where Ω is a bounded domain in Rn for which there exist numbers ℓ>0, m∈N, and α1,…,αm∈Rn satisfying the following conditions 1) Ω¯¯¯¯=Πℓ(α1)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯∪…∪Πℓ(αm)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯; 2) for every i,j∈{1,…,m} with i≠j we obtain: Πℓ(αi)∩Πℓ(αj)=∅.
Authors and Affiliations
O. M. Buhrii
Keywords
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- EP ID EP584883
- DOI 10.15330/ms.51.1.19-24
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