By Mikhail Borsuk (auth.)
The target of this publication is to enquire the habit of vulnerable options of the elliptic transmission challenge in a local of boundary singularities: angular and conic issues or edges. This challenge is mentioned for either linear and quasilinear equations. A imperative new function of this e-book is the distinction of our estimates of vulnerable options of the transmission challenge for linear elliptic equations with minimum tender coeciffients in n-dimensional conic domain names. in simple terms few works are dedicated to the transmission challenge for quasilinear elliptic equations. as a result, we examine the vulnerable strategies for normal divergence quasilinear elliptic second-order equations in n-dimensional conic domain names or in domain names with edges. the root of the current paintings is the strategy of integro-differential inequalities. Such inequalities with precise estimating constants let us identify attainable or very best estimates of recommendations to boundary worth difficulties for elliptic equations close to singularities at the boundary. a brand new Friedrichs–Wirtinger sort inequality is proved and utilized to the research of the habit of vulnerable recommendations of the transmission challenge. All effects are given with whole proofs. The e-book should be of curiosity to graduate scholars and experts in elliptic boundary worth difficulties and applications.
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Extra resources for Transmission Problems for Elliptic Second-Order Equations in Non-Smooth Domains
1−κ)2−j For all κ ∈ (0, 1) we deﬁne sets G(j) ≡ G0 , j = 0, 1, 2, . .. It is easy κ to verify that G0 ≡ G(∞) ⊂ · · · ⊂ G(j+1) ⊂ G(j) ⊂ · · · ⊂ G(0) ≡ G10 . Now we consider the sequence of cut-oﬀ functions ζj (x ) ∈ C∞ (G(j) ) such that 0 ≤ ζj (x ) ≤ 1 in G(j) and ζj (x ) ≡ 1 in G(j+1) , ζj (x ) ≡ 0 for |x | > κ + 2−j (1 − κ); 2j+1 1−κ |∇ ζj | ≤ κ + 2−j−1 (1 − κ) < |x | < κ + 2−j (1 − κ). for j n , j = 0, 1, 2, . .. 19) replacing ζ(|x |) by ζj (x ) and t by tj ; then taking tj -th root, we obtain v tj+1 ,G(j+1) C 1−κ ≤ 2/tj 2p j · 4 tj · (tj ) p−n · t1 v j tj ,G(j) .
Let assumptions (a)–(e) be satisﬁed with A(r) Dini-continuous at zero. 2) , if s < λ holds for all x ∈ Gd0 . 2. Local estimate at the boundary 37 2 holds. 5) if s < λ. 2 Local estimate at the boundary We derive here a result asserting the local boundedness (near the conical point) of the weak solution of problem (L). 4. Let u(x) be a weak solution of the problem (L) and assumptions (a)–(c) be satisﬁed. 1) ∈ (0, d), where C > 0 is a constant depending holds for any t > 0, κ ∈ (0, 1) and only on n, ν∗ , μ∗ , t, p, κ, f |ai (x)|2 .
5) if s < λ. 2 Local estimate at the boundary We derive here a result asserting the local boundedness (near the conical point) of the weak solution of problem (L). 4. Let u(x) be a weak solution of the problem (L) and assumptions (a)–(c) be satisﬁed. 1) ∈ (0, d), where C > 0 is a constant depending holds for any t > 0, κ ∈ (0, 1) and only on n, ν∗ , μ∗ , t, p, κ, f |ai (x)|2 . Lp/2 (G) i=1 Proof. We apply the Moser iteration method. First we assume that t ≥ 2. We consider the integral identity (II) and make the coordinate transformation x = x .
Transmission Problems for Elliptic Second-Order Equations in Non-Smooth Domains by Mikhail Borsuk (auth.)