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Asymptotic approximation for the solution to a semi-linear elliptic problem in a thin aneurysm-type domain

100%
Mel’nyk T. A., Klevtsovskiy A. V.
Open Mathematics
|
2017
|
tom 15
|
nr 1
1351-1370
EN
A semi-linear boundary-value problem with nonlinear Robin boundary conditions is considered in a thin 3D aneurysm-type domain that consists of thin curvilinear cylinders that are joined through an aneurysm of diameter 𝓞(ε). Using the multi-scale analysis, the asymptotic approximation for the solution is constructed and justified as the parameter ε → 0. Namely, we derive the limit problem (ε = 0) in the corresponding graph, define other terms of the asymptotic approximation and prove energetic and uniform pointwise estimates. These estimates allow us to observe the impact of the aneurysm on some properties of the solution.
2
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Degenerate triply nonlinear problems with nonhomogeneous boundary conditions

100%
Ammar K.
Open Mathematics
|
2010
|
tom 8
|
nr 3
548-568
EN
The paper addresses the existence and uniqueness of entropy solutions for the degenerate triply nonlinear problem: b(v)t − div α(v, ▽g(v)) = f on Q:= (0, T) × Ω with the initial condition b(v(0, ·)) = b(v 0) on Ω and the nonhomogeneous boundary condition “v = u” on some part of the boundary (0, T) × ∂Ω”. The function g is continuous locally Lipschitz continuous and has a flat region [A 1, A 2,] with A 1 ≤ 0 ≤ A 2 so that the problem is of parabolic-hyperbolic type.
3
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Topological tools for the prescribed scalar curvature problem on S n

100%
Abuzaid D., Ben Mahmoud R., Chtioui H., Rigane A.
Open Mathematics
|
2014
|
tom 12
|
nr 12
1829-1839
EN
In this paper, we consider the problem of the existence of conformal metrics with prescribed scalar curvature on the standard sphere S n, n ≥ 3. We give new existence and multiplicity results based on a new Euler-Hopf formula type. Our argument also has the advantage of extending well known results due to Y. Li [16].
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