Displaying similar documents to “Extensions from the Sobolev spaces H 1 satisfying prescribed Dirichlet boundary conditions”

Nonuniqueness for some linear oblique derivative problems for elliptic equations

Gary M. Lieberman (1999)

Commentationes Mathematicae Universitatis Carolinae

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It is well-known that the “standard” oblique derivative problem, Δ u = 0 in Ω , ∂ u / ∂ ν - u = 0 on ∂ Ω ( ν is the unit inner normal) has a unique solution even when the boundary condition is not assumed to hold on the entire boundary. When the boundary condition is modified to satisfy an obliqueness condition, the behavior at a single boundary point can change the uniqueness result. We give two simple examples to demonstrate what can happen.

Homogenization in polygonal domains

David Gérard-Varet, Nader Masmoudi (2011)

Journal of the European Mathematical Society

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We consider the homogenization of elliptic systems with ε -periodic coefficients. Classical two-scale approximation yields an O ( ε ) error inside the domain. We discuss here the existence of higher order corrections, in the case of general polygonal domains. The corrector depends in a non-trivial way on the boundary. Our analysis substantially extends previous results obtained for polygonal domains with sides of rational slopes.

On singular perturbation problems with Robin boundary condition

Henri Berestycki, Juncheng Wei (2003)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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We consider the following singularly perturbed elliptic problem ϵ 2 Δ u - u + f ( u ) = 0 , u > 0 in Ω , ϵ ∂ u ∂ ν + λ u = 0 on ∂ Ω , where f satisfies some growth conditions, 0 ≤ λ ≤ + ∞ , and Ω ⊂ ℝ N ( N > 1 ) is a smooth and bounded domain. The cases λ = 0 (Neumann problem) and λ = + ∞ (Dirichlet problem) have been studied by many authors in recent years. We show that, there exists a generic constant λ * > 1 such that, as ϵ → 0 , the least energy solution has a spike near the boundary if λ ≤ λ * , and has an interior spike near the innermost part of the domain if λ > λ * . Central...

Reduction of a Schwartz-type boundary value problem for biharmonic monogenic functions to Fredholm integral equations

Serhii V. Gryshchuk, Sergiy A. Plaksa (2017)

Open Mathematics

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We consider a commutative algebra over the field of complex numbers with a basis e1, e2 satisfying the conditions [...] (e12+e22)2=0,e12+e22≠0. ( e 1 2 + e 2 2 ) 2 = 0 , e 1 2 + e 2 2 ≠ 0 . Let D be a bounded simply-connected domain in ℝ2. We consider (1-4)-problem for monogenic -valued functions Φ(xe1 + ye2) = U1(x, y)e1 + U2(x, y)i e1 + U3(x, y)e2 + U4(x, y)i e2 having the classic derivative in the domain Dζ = xe1 + ye2 : (x, y) ∈ D: to find a monogenic in Dζ function Φ, which is continuously extended to the boundary ∂Dζ, when...

Boundary regularity and compactness for overdetermined problems

Ivan Blank, Henrik Shahgholian (2003)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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Let D be either the unit ball B 1 ( 0 ) or the half ball B 1 + ( 0 ) , let f be a strictly positive and continuous function, and let u and Ω ⊂ D solve the following overdetermined problem: Δ u ( x ) = χ Ω ( x ) f ( x ) in D , 0 ∈ ∂ Ω , u = | ∇ u | = 0 in Ω c , where χ Ω denotes the characteristic function of Ω , Ω c denotes the set D ∖ Ω , and the equation is satisfied in the sense of distributions. When D = B 1 + ( 0 ) , then we impose in addition that u ( x ) ≡ 0 on { ( x ' , x n ) | x n = 0 } . We show that a fairly mild thickness assumption on Ω c will ensure enough compactness on...

Large time regular solutions to the MHD equations in cylindrical domains

Wisam Alame, Wojciech M. Zajączkowski (2011)

Applicationes Mathematicae

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We prove the large time existence of solutions to the magnetohydrodynamics equations with slip boundary conditions in a cylindrical domain. Assuming smallness of the L₂-norms of the derivatives of the initial velocity and of the magnetic field with respect to the variable along the axis of the cylinder, we are able to obtain an estimate for the velocity and the magnetic field in W ₂ 2 , 1 without restriction on their magnitude. Then the existence follows from the Leray-Schauder fixed point theorem. ...

Asymptotic analysis of the initial boundary value problem for the thermoelastic system in a perforated domain

M. Sango (2003)

Colloquium Mathematicae

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We study the initial boundary value problem for the system of thermoelasticity in a sequence of perforated cylindrical domains Q T ( s ) , s = 1,2,... We prove that as s → ∞, the solution of the problem converges in appropriate topologies to the solution of a limit initial boundary value problem of the same type but containing some additional terms which are expressed in terms of quantities related to the geometry of Q T ( s ) . We give an explicit construction of that limit problem.

A nonlocal elliptic equation in a bounded domain

Piotr Fijałkowski, Bogdan Przeradzki, Robert Stańczy (2004)

Banach Center Publications

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The existence of a positive solution to the Dirichlet boundary value problem for the second order elliptic equation in divergence form - ∑ i , j = 1 n D i ( a i j D j u ) = f ( u , ∫ Ω g ( u p ) ) , in a bounded domain Ω in ℝⁿ with some growth assumptions on the nonlinear terms f and g is proved. The method based on the Krasnosel’skiĭ Fixed Point Theorem enables us to find many solutions as well.