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
Related Articles
- EP ID EP584883
- DOI 10.15330/ms.51.1.19-24
- Views 47
- Downloads 0
On generalized preopen sets
Firstly in this paper, we find some conditions under which μ-preopen sets of a GTS or μ-space X may be equivalent to μ-open in X. Finally, we obtain some characterizations of generalized paracompactness of a GTS or μ-spa...
Periodic words connected with the Tribonacci-Lucas numbers
We introduce periodic words that are connected with the Tribonacci-Lucas numbers and investigate their properties.
Prime ends in the Sobolev mapping theory on Riemann surfaces
We prove criteria for continuous and homeomorphic extension to the boundary of mappings with finite distortion between domains on the Riemann surfaces by prime ends of Caratheodory.
Two remarks on the book by Roman Duda “Pearls from a Lost City. The Lvov School of Mathematics”
We consider the murder of Lviv professors and students in July 1941, mentioned in the book by Roman Duda “Pearls from a Lost City. The Lvov School of Mathematics".
The regularity of quotient paratopological groups
Let H be a closed subgroup of a regular Abelian paratopological group G. The group reflexion G♭ of G is the group G endowed with the strongest group topology, which is weaker than the original topology of G. We show that...