A sharp angle inequalities for pairs of elliptic operators in the case of domain with a conical point

A sharp angle inequalities for pairs of elliptic operators in the case of domain with a conical point

A sharp angle inequality is used in a proof of solvability of a general nonlinear elliptic problem with applying of topological methods. Our proof is founded at reducing to the inequality in the case of a domain with a smooth boundary.

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Datum:1999
1. Verfasser: Dzhafarov, R.M.
Format: Artikel
Sprache:English
Veröffentlicht: Інститут прикладної математики і механіки НАН України 1999
Schriftenreihe:Нелинейные граничные задачи
Online Zugang:http://dspace.nbuv.gov.ua/handle/123456789/169270
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Zitieren:A sharp angle inequalities for pairs of elliptic operators in the case of domain with a conical point / R.M. Dzhafarov // Нелинейные граничные задачи: сб. науч. тр. — 1999. — Т. 9. — С. 40-45. — Бібліогр.: 2 назв. — англ.

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spelling irk-123456789-1692702020-06-10T01:26:20Z A sharp angle inequalities for pairs of elliptic operators in the case of domain with a conical point Dzhafarov, R.M. A sharp angle inequality is used in a proof of solvability of a general nonlinear elliptic problem with applying of topological methods. Our proof is founded at reducing to the inequality in the case of a domain with a smooth boundary. 1999 Article A sharp angle inequalities for pairs of elliptic operators in the case of domain with a conical point / R.M. Dzhafarov // Нелинейные граничные задачи: сб. науч. тр. — 1999. — Т. 9. — С. 40-45. — Бібліогр.: 2 назв. — англ. 0236-0497 http://dspace.nbuv.gov.ua/handle/123456789/169270 en Нелинейные граничные задачи Інститут прикладної математики і механіки НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
description A sharp angle inequality is used in a proof of solvability of a general nonlinear elliptic problem with applying of topological methods. Our proof is founded at reducing to the inequality in the case of a domain with a smooth boundary.
format Article
author Dzhafarov, R.M.
spellingShingle Dzhafarov, R.M.
A sharp angle inequalities for pairs of elliptic operators in the case of domain with a conical point
Нелинейные граничные задачи
author_facet Dzhafarov, R.M.
author_sort Dzhafarov, R.M.
title A sharp angle inequalities for pairs of elliptic operators in the case of domain with a conical point
title_short A sharp angle inequalities for pairs of elliptic operators in the case of domain with a conical point
title_full A sharp angle inequalities for pairs of elliptic operators in the case of domain with a conical point
title_fullStr A sharp angle inequalities for pairs of elliptic operators in the case of domain with a conical point
title_full_unstemmed A sharp angle inequalities for pairs of elliptic operators in the case of domain with a conical point
title_sort sharp angle inequalities for pairs of elliptic operators in the case of domain with a conical point
publisher Інститут прикладної математики і механіки НАН України
publishDate 1999
url http://dspace.nbuv.gov.ua/handle/123456789/169270
citation_txt A sharp angle inequalities for pairs of elliptic operators in the case of domain with a conical point / R.M. Dzhafarov // Нелинейные граничные задачи: сб. науч. тр. — 1999. — Т. 9. — С. 40-45. — Бібліогр.: 2 назв. — англ.
series Нелинейные граничные задачи
work_keys_str_mv AT dzhafarovrm asharpangleinequalitiesforpairsofellipticoperatorsinthecaseofdomainwithaconicalpoint
AT dzhafarovrm sharpangleinequalitiesforpairsofellipticoperatorsinthecaseofdomainwithaconicalpoint
first_indexed 2025-07-15T04:01:43Z
last_indexed 2025-07-15T04:01:43Z
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fulltext A SHARP ANGLE INEQUALITIES FOR PAIRS OF ELLIPTIC OPERATORS IN THE CASE OF DOMAIN WITH A CONICAL POINT. c© R.M. Dzhafarov A sharp angle inequality is used in a proof of solvability of a general nonlinear elliptic problem with applying of topological methods. Our proof is founded at reducing to the inequality in the case of a domain with a smooth boundary [2]. Denotations. G is a bounded domain with the conical point in the origin of coordinates. ∂G is a smooth boundary everywhere without of the origin of coordinates. ρ is a diam G. Gk = {x ∈ G : ρ 2k+1 ≤ r ≤ ρ 2k , r = |x|} Ωk is a domain with the smooth boundary wich contain Gk. ℵ0,λ 2m(A,B, Ω) is a class of linear elliptic with the constant A in Ω operators of order 2m with coefficients bounded in the norm ‖ · ‖C0,λ(Ω) by B. φk are functions of the unity partition: ∞∑ k=0 φ2 k(x) = 1 ∀x ∈ G, supp φk ⊂ Ωk; φk ≤ √ N, x ∈ Gk. Denote |Dα(φkrκ)| ≤ const · rκ−|α| (i1) W l κ(G) is weighting space with norm ‖u‖W l κ(G) =   ∫ G ∑ |α|≤k rκ−2(l−|α|)|Dαu|2dx   1 2 ◦ W l κ(G) is a closure C∞0 (G) in the norm ‖ · ‖W l κ(G). Lemma 1. For operators L ∈ ℵ0,λ 2m(A,B, Ωk), Mk ∈ ℵ0,λ 2m(1, C̃2,Ωk) (where C̃2 is some constant) and functions φk ∈ C∞(Ωk), u ∈ W 2m κ (G) ∩ ◦ W m κ (G) following inequality is true Re ∫ Ωk rκφ2 k(x)Lu ·Mkudx ≥ ≥ Re ∫ Ωk L(φkur κ 2 ) ·Mk(φkur κ 2 )dx− −C1‖u‖W 2m κ (Ωk)‖u‖W 2m−1 κ−2 (Ωk) (1) Lemma 2. Let L ∈ ℵ0,λ 2m(A, B, Ωk). One can designate the constants C̃1, C̃2 and operator Mk(x,D) ∈ ℵ0,λ 2m(1, C̃2, Ωk) [2] such that ∀u ∈ W 2m 2 (Ωk)∩ ◦ W m 2 (Ω) the relation Re ∫ Ω Lu ·Mkudx ≥ ≥ C̃1 ∫ Ωk ∑ |α|≤2m ( ρ 2k )2|α|−4m |Dαu|2dx− C̃2 ∫ Ωk ( ρ 2k )−4m |u|2dx (2) is valid. Lemma 3. If u ∈ W 2m κ (G) then φkr κ 2 u ∈ W 2m 2 (G). Lemma 4. (Interpolating inequality.) For a fixed k and for any function u from W l κ(G) the inequality ‖u‖2 W j κ−2(l−j)(Ωk) ≤ ε1‖u‖2W l κ(Ωk) + C(ε1)‖u‖2W 0 κ−2l(Ωk), (3) where j < l, ε1 > 0 is fulfiled. Those lemmas enable prove a theorem. Theorem 1. Let L ∈ ℵ0,λ 2m(A,B,G). There exist such constants C2, C3, C̃2 depended of known parameter A, B, λ, m, G and such operator M = ∞∑ k=0 φ2 k(x)Mk(x,D), Mk ∈ ℵ0,λ 2m(1, C̃2, G) that ∀u ∈ ◦ W m κ−2m−2l(G) ∩W 2m κ (G) the inequality Re ∫ G rκL(x,D)u ·M(x,D)udx ≥ ≥ C2‖u‖2W 2m κ (G) − C3‖u‖2W 0 κ−4m(G) (4) is valid. The lemma 3 enables substitute (2) into (1) Re ∫ Ω φ2 k(x)rκLu ·Mkudx ≥ ≥ C4 ∫ Ωk ∑ |α|≤2m ( ρ 2k )2|α|−4m |Dα(φkur κ 2 |2dx− −C5 ∫ Ωk ( ρ 2k ) |φkr κ 2 u|2dx− −C3‖u‖W 2m−1 κ−2 (Ωk)‖u‖W 2m κ (Ωk . (5) Consider first addend of the right member of (5). ∫ Ωk ∑ |α|≤2m ( ρ 2k )2|α|−4m |Dα(φkr κ 2 )|2dx = = ∫ Ωk ∑ |α|≤2m ( ρ 2k )2|α|−4m |φkr κ 2 Dαu+ + ∑ β<α; β+γ=α cβγDβuDγ(φkr κ 2 )|2dx ≥ ≥ ∫ Ωk ∑ |α|≤2m ( ρ 2k )2|α|−4m |φkr κ 2 Dαu|2− − 4 ∫ Ωk ∑ |α|≤2m ( ρ 2k )2|α|−4m |φkr κ 2 Dαu|∗ ∗| ∑ β<α; β+γ=α cβγDβuDγ(φkr κ 2 )|dx ≥ ≥ N · ∫ Gk ∑ |α|≤2m rκ−2(2m−|α|)|Dαu|2dx− − ε2 ∫ Ωk ∑ |α|≤2m rκ−2(2m−|α|)|Dαu|2dx− −C(ε2) ∫ Ωk ∑ |α|≤2m r2|α|−4m ∑ β<α; β+γ=α rκ−2γ |Dβu|2dx ≥ ≥ N · ‖u‖2W 2m κ (Gk)− − ε2 · ‖u‖2W 2m κ (Ωk)− −C6 ∫ Ωk ∑ |α|≤2m r2|α|−4m ∑ β<α; β+γ=α rκ−2α+2β+2α−4m|Dβu|2dx ≥ ≥ N · ‖u‖2W 2m κ (Gk) − ε2 · ‖u‖2W 2m κ (Ωk) − C7‖u|2W 2m−1 κ−2 (Ωk) ≥ ≥ N · ‖u‖2W 2m κ (Gk) − ε3 · ‖u‖2W 2m κ (Ωk) − C8‖u|2W 0 κ−4m(Ωk) . Here Caushy inequality and the interpolating inequality were used. Due to this inequal- ities we can estimate third addend of the right member of (5) by means of − ε4‖u‖2W 2m κ (Ωk) − C9‖u‖2W 0 κ−4m(Ωk). Now we can write down (5) as Re ∫ Ωk φ2 k(x)rκLu ·Mkudx ≥ ≥ C10‖u‖2W 2m κ (Gk) − ε5‖u‖2W 2m κ (Ωk) − C11‖u‖2W 0 κ−2m(Ωk) To the left it is possible the substitution Ωk at G. Sum with respect to k and substitute Ωk at Gk−1 ∩Gk ∩Gk+1 to the right. Thereby we are getting (4). We introduce ℵl,λ 2m(A,B, G) as a class of a linear elliptic operators with the constant of ellipticity A and with coefficients having the derivatives of the order d (d ≤ l) bounded by B ·r−d. Thereto the derivatives of the order l are Holder continuous with the constant λ and bounded by B · r−l−λ. Let [u, v]W l κ(G) = ∫ G ∑ |α|,|β|≤l cα,β(x)rκ−2l+|α|+|β|DαuDβvdx, (6) where cαβ(x) are real infinity differentiable functions of following construction cαβ(x) = ∞∑ k=0 c̃αβ ( x · 2k ρ ) · r−(κ−2l+|α|+|β|) k = ∞∑ k=0 ck αβ(x), where rk = r ρ/2k . Here c̃αβ are real infinity differentiable functions satisfying to the conditions C12‖ũ‖2W l 2(Ω′) ≤ ∫ Ω′ ∑ |α|,|β|≤l c̃αβ(x′)DαũDβũdx′ ≤ ≤ C13‖ũ‖2W l 2(Ω′), where Ω′ = 2k ρ Ωk; c̃αβ have finite support and ∀Ω” ⊂ Rn \ Ω′ ∫ Ω” ∑ |α|,|β|≤l c̃αβ(x′)DαũDβũdx′ ≥ 0. It can shou that cαβ satisfy to the conditions C14‖u‖2W l κ(G) ≤ [u, u]W l κ(G) ≤ C15‖u‖W l κ(G). Lemma 5. For operators L ∈ ℵl,λ 2m(A,B, Ωk), Mk ∈ ℵl,λ 2m(1, C̃2,Ωk) and functions φk ∈ C∞(Ωk), u ∈ W 2m+l κ (Ωk) ∩ ◦ W m κ−2m−2l(Ωk) the relation Re[Lu, φ2 kMku]W l κ(Ωk) ≥≥ Re ∫ Ωk ∑ |α|,|β|≤l cαβ(x)r|α|+|β|−2lDαL(x,D)(φkr κ 2 u)∗ ∗DβMk(x,D)(φkr κ 2 u)dx − C16‖u‖W l+2m κ (Ωk)‖u‖W l+2m−1 κ−2 (Ωk) (7) is valid. Lemma 6. Let L ∈ ℵl,λ 2m(A,B, Ωk). There exist such constants C ′1, C ′ 2 > 0 and operator Mk(x, D) ∈ ℵl,λ 2m(1, C ′2, Ωk) that ∀u ∈ W 2m+l 2 (Ωk) ∩ ◦ W m 2 (Ωk) following inequality is fulfiled. Re ∫ Ωk ∑ |α|,|β|≤l ck αβ(x) ( ρ 2k )|α|+|β|−2l DαLuDβMkudx ≥ ≥ C ′1 ∫ Ωk ∑ |α|≤2m+l ( ρ 2k )2|α|−4m−2l |Dαu|2dx− −C ′2 ∫ Ωk ( ρ 2k )−4m−2l |u|2dx (8) Theorem 2. Let L ∈ ℵl,λ 2m(A,B, G). There exist such real infinity differentiable func- tions cαβ (|α|, |β| ≤ l) satisfying to the condition C14‖u‖2W l κ(G) ≤ [u, u]W l κ(G) ≤ C15‖u‖2W l κ(G), the positive constants K1, K2, C ′2 depending of known parameter only and such operator M = ∑∞ k=0 φ2 k(x)Mk(x, D), Mk(x,D) ∈ ℵl,λ 2m(1, C ′2, G) that for any function u ∈ W 2m+l κ (G) ∩ ◦ W m κ−2m−2l(G) the inequality Re[Lu,Mu]W l κ(G) ≥ K1‖u‖2W 2m+l κ (G) − K2‖u‖2W 0 κ−4m−2l(G) (9) is valid. On the lemma 3 φkr κ 2 u ∈ W 2m+l κ (G). Therefore Re[Lu, φ2 kMku]W l κ(G) ≥ ≥ C17 ∫ Ωk ∑ |α|≤2m+l ( ρ 2k )2|α|−4m−2l |Dα(φkr κ 2 u)|2dx− −C18 ∫ Ωk ( ρ 2k )−4m−2l |φkr κ 2 u|2dx− −C19‖u‖W 2m+l κ (Ωk)‖u‖W 2m+l−1 κ−2 (Ωk) (10) Consider the first addend of the right member of (10). We shall be use interpolating inequality and inequality |a + b| ≥ |a|2 − 4|a| · |b|. ∫ Ωk ∑ |α|≤2m+l ( ρ 2k )2|α|−4m−2l |Dα(φkr κ 2 u)|2dx = = ∫ Ωk ∑ |α|≤2m+l ( ρ 2k )2|α|−4m−2l |φkr κ 2 Dαu− − ∑ γ<α; γ+δ=α cγδD δ(φkr κ 2 )Dγu|2dx ≥ ≥ N · C20 ∫ Gk ∑ |α|≤2m+l rκ−2(2m+l−|α|)|Dαu|2dx− −C21 ∫ Ωk ∑ |α|≤2m+l r2|α|−4m−2l ∑ γ<α; γ+δ=α { r κ 2−|α|+|γ||Dγu| } ∗ ∗{ r κ 2 |Dαu|} dx ≥ ≥ N · C20‖u‖2W 2m+l κ (Gk) − − ε6‖u‖2W 2m+l κ (Ωk) − C22‖u‖2W 2m+l−1 κ−2 (Ωk) ≥ N · C20‖u‖2W 2m+l κ (Gk) − − ε7‖u‖2W 2m+l κ (Ωk) − −C23‖u‖2W 0 κ−4m−2l(Ωk) (11) We apply Caushy inequality and inequality (3) to third addend of the right member of (10). Thereby we get Re[Lu, φ2 kMku]W l κ(Ωk) ≥ ≥ N · C20‖u‖2W 2m+l κ (Gk) − ε8‖u‖2W 2m+l κ (Ωk) − C24‖u‖2W 0 κ−4m−2l(Ωk) (12) from (10). In the left member we substitute Ωk at G and in the right member substitute Ωk at Gk−1 ∪Gk ∪Gk+1. Summing with respect to k gives (9). References 1. Kondrat’ev V.A., Boundary value problems for elliptic equations in domains with conical or angular points, Trudy Moskovskogo Mat. Obschestva 16 (1967), 219-292. 2. Skrypnik I.V., Nonlinear elliptic boundary value problems, Leipzig: B.G. Teubner Verlagsges, 1986.

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