Nonhomogeneous Neumann Problem Research Articles

Finite element approximations in a non-Lipschitz domain: Part II

In a paper by R. Durán, A. Lombardi, and the authors (2007) the finite element method was applied to a non-homogeneous Neumann problem on a cuspidal domain Ω ⊂ R 2 \Omega \subset \mathbb {R}^2 , and quasi-optimal order error estimates in the energy norm were obtained for certain graded meshes. In this paper, we study the error in the L 2 L^2 norm obtaining similar results by using graded meshes of the type considered in that paper. Since many classical results in the theory Sobolev spaces do not apply to the domain under consideration, our estimates require a particular duality treatment working on appropriate weighted spaces. On the other hand, since the discrete domain Ω h \Omega _h verifies Ω ⊂ Ω h \Omega \subset \Omega _h , in the above-mentioned paper the source term of the Poisson problem was taken equal to 0 0 outside Ω \Omega in the variational discrete formulation. In this article we also consider the case in which this condition does not hold and obtain more general estimates, which can be useful in different problems, for instance in the study of the effect of numerical integration, or in eigenvalue approximations.

Saint-Venant’s Problem for Compound Piezoelectric Beams

The solution of the Saint-Venant’s Problem for a slender compound piezoelectric beam presented in this paper generalizes the recent solution by the authors and E. Harash (J. Appl. Mech. 11:1–10, 2007) for a homogeneous piezoelectric beam and the solution for a compound elastic beam developed by O. Rand and the first author (Analytical Methods in Anisotropic Elasticity with Symbolic Computational Tools, Birkhauser, Boston, 2005). Justification for this approximation emerges from the St. Venant’s Principle. The stress, strain and (electrical) displacement components (“solution hypothesis”) are presented as a set of initially assumed expressions involving twelve tip loading parameters, six unknown weight coefficients, and three pairs of torsion/bending functions of two variables. Each pair of functions satisfies the so-called coupled non-homogeneous Neumann problem (CNNP) in the cross-sectional domain. The work develops concepts of the torsion/bending functions, the torsional rigidity and piezoelectric shear center, the tip coupling matrix, for a compound piezoelectric beam. Examples of exact and approximate solutions for rectangular laminated beams made of transtropic materials are presented.

Existence of solutions for p(x)-Laplacian nonhomogeneous Neumann problems with indefinite weight

In this paper we study the nonhomogeneous Neumann problem of p ( x ) -Laplacian equations, where the weighted function V ( x ) is indefinite and the potential is only locally Lipschitz. By the theory of the generalized Lebesgue–Sobolev space W 1 , p ( x ) ( Ω ) , a nonsmooth Mountain Pass theorem and the Weierstrass theorem, we obtain the existence results of the problem.

Periodic unfolding and nonhomogeneous Neumann problems in domains with small holes

We consider elliptic problems in periodically perforated domains in R N , with nonhomogeneous Neumann conditions on the boundary of the holes. We are interested in the asymptotic behavior of the solutions as the period ε goes to zero. In a first case all the holes are “small”, i.e., are of size r ( ε ) with r ( ε ) / ε → 0 . In the second case, there are again small holes but also holes of size ε. We use the periodic unfolding method introduced in Cioranescu et al. (2002), which allows us to study second order operators with highly oscillating coefficients and so, to generalize here the results of Conca and Donato (1988). In both cases, if r ( ε ) = exp ( N / N − 1 ) , an additional term appears in the right-hand side of the limit equation. To cite this article: A. Ould Hammouda, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

Bloch wave homogenization of a non-homogeneous Neumann problem

In this paper, we use the Bloch wave method to study the asymptotic behavior of the solution of the Laplace equation in a periodically perforated domain, under a non-homogeneous Neumann condition on the boundary of the holes, as the size of the holes goes to zero more rapidly than the domain period. This method allows to prove that, when the hole size exceeds a given threshold, the non-homogeneous boundary condition generates an additional term in the homogenized problem, commonly referred to as “the strange term” in the literature.

Finite Element Approximations in a NonLipschitz Domain

In this paper we analyze the approximation by standard piecewise linear finite elements of a nonhomogeneous Neumann problem in a cuspidal domain. Since the domain is not Lipschitz, many of the results on Sobolev spaces, which are fundamental in the usual error analysis, do not apply. Therefore, we need to work with weighted Sobolev spaces and to develop some new theorems on traces and extensions. We show that, in the domain considered here, suboptimal order can be obtained with quasi‐uniform meshes even when the exact solution is in $H^2$, and we prove that the optimal order with respect to the number of nodes can be recovered by using appropriate graded meshes.

On the nonhomogeneous Neumann problem with weight and with critical nonlinearity in the boundary

We consider the problem: − div ( p ( x ) ∇ u ) = λ u + f ( x ) in Ω , ∂ u ∂ ν = Q ( x ) | u | 2 N − 2 u on ∂ Ω , where Ω is a bounded smooth domain in R N , N ≥ 3 . Under some conditions on ∂ Ω , p , Q , f , λ and the mean curvature at some point x 0 , we prove the existence of solutions of the above problem. We use variational arguments, namely Ekeland’s variational principle, the min–max principle and the mountain pass theorem.

Multiple positive solutions for a quasilinear nonhomogeneous Neumann problems with critical Hardy exponents

In this paper, we study the existence, nonexistence and multiplicity of positive solutions for nonhomogeneous Neumann boundary value problems of the type { − Δ p u + λ u p − 1 = u q | x | s x ∈ Ω , u > 0 x ∈ Ω , | ∇ u | p − 2 D γ u = ϕ x ∈ ∂ Ω ∖ { 0 } , where Ω is a bounded domain in R n with smooth boundary, 0 ∈ ∂ Ω , 2 ≤ p < n , Δ p u = div ( | ∇ u | p − 2 ∇ u ) is the p -Laplacian operator, p − 1 < q ≤ p ∗ ( s ) − 1 , 0 ≤ s < p − 1 , p ∗ ( s ) = ( n − s ) p n − p , ϕ ∈ C α ( Ω ̄ ) , 0 < α < 1 , ϕ ( x ) ≥ 0 , ϕ ( x ) ≢ 0 and λ is a real constant.

A Fully Nonlinear Nonhomogeneous Neumann Problem

Let Ω be an open bounded set in ℝN, N≥3, with connected Lipschitz boundary ∂Ω and let a(x,ξ) be an operator of Leray–Lions type (a(⋅,∇u) is of the same type as the operator |∇u|p−2∇u, 1<p<N). If τ is the trace operator on ∂Ω, [φ] the jump across ∂Ω of a function φ defined on both sides of ∂Ω, the normal derivative ∂/∂νa related to the operator a is defined in some sense as 〈a(⋅,∇u),ν〉, the inner product in ℝN, of the trace of a(⋅,∇u) on ∂Ω with the outward normal vector field ν on ∂Ω. If β and γ are two nondecreasing continuous real functions everywhere defined in ℝ, with β(0)=γ(0)=0, f∈L1(ℝN), g∈L1(∂Ω), we prove the existence and the uniqueness of an entropy solution u for the following problem, $$\mathbf{(P)}\quad \left\{\begin{array}{l}-\mathrm{div}[a(x,\nabla u)]+\beta(u)=f\quad \mbox{in}\ \mathbb{R}^{N}\setminus \partial \Omega,\cr \bigl[\frac{\partial u}{\partial \nu_{a}}\big]+\gamma(\tau u)=g\quad \hbox{on}\ \partial \Omega,\cr [u]=0\end{array}\right.$$ in the sense that, if Tk(r)=max {−k,min (r,k)}, k>0, r∈ℝ, ∇u is the gradient by means of truncation (∇u=DTku on the set {|u| 0}, then \(u\in \mathcal{T}^{1,p}(\mathbb{R}^{N})\) and u satisfies, $$\int_{\mathbb{R}^{N}}\langle a(\cdot,Du),DT_{k}(u-\varphi)\rangle \leqslant \int_{\mathbb{R}^{N}}(f-\beta(u))T_{k}(u-\varphi)+\int_{\partial \Omega}(g-\gamma(u))T_{k}(\tau u-\tau\varphi),$$ for every k>0 and every \(\varphi\in \mathcal{T}^{1,p}(\mathbb{R}^{N})\cap L^{\infty }(\mathbb{R}^{N})\) .

Nonhomogeneous Neumann problem for the Poisson equation in domains with an external cusp

In this work we analyze the existence and regularity of the solution of a nonhomogeneous Neumann problem for the Poisson equation in a plane domain Ω with an external cusp. In order to prove that there exists a unique solution in H 1 ( Ω ) using the Lax–Milgram theorem we need to apply a trace theorem. Since Ω is not a Lipschitz domain, the standard trace theorem for H 1 ( Ω ) does not apply, in fact the restriction of H 1 ( Ω ) functions is not necessarily in L 2 ( ∂ Ω ) . So, we introduce a trace theorem by using weighted Sobolev norms in Ω. Under appropriate assumptions we prove that the solution of our problem is in H 2 ( Ω ) and we obtain an a priori estimate for the second derivatives of the solution.

Multiplicity of positive solutions for a class of quasilinear nonhomogeneous Neumann problems

In this paper we study the existence, nonexistence and multiplicity of positive solutions for nonhomogeneous Neumann boundary value problem of the type - Δ p u + λ u p - 1 = u q in Ω , u > 0 in Ω , | ∇ u | p - 2 ∂ u ∂ η = ϕ on ∂ Ω , where Ω is a bounded domain in R n with smooth boundary, 1 < p < n , Δ p u = div ( | ∇ u | p - 2 ∇ u ) is the p -Laplacian operator, p - 1 < q ⩽ p * - 1 , p * = np / ( n - p ) , ϕ ∈ C α ( Ω ¯ ) , 0 < α < 1 , ϕ ≢ 0 , ϕ ( x ) ⩾ 0 and λ is a real parameter. The proofs of our main results rely on different methods: lower and upper solutions and variational approach.

The Neumann problem in the half-space

We study here the nonhomogeneous Neumann problem in the half-space R N + with N⩾2. We give in L p theory, with 1< p<∞, a basic existence and regularity results in weighted Sobolev spaces. To cite this article: C. Amrouche, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 151–156.

Non-homogeneous Neumann problems in domains with small holes

On etudie le comportement limite des solutions de problemes de Neumann non homogenes dans des ouverts finement perfores. Des estimations a priori detaillees exprimees en fonction des parametres e et r(e) des trous donnent l'ordre de grandeur exact de la norme H 1 des solutions

A cosine operator approach to modelingL 2(0,T; L 2 (?))?Boundary input hyperbolic equations

This paper investigates the regularity properties of the solution of a second-order hyperbolic equation defined over a bounded domain ź with boundary Γ, under the action of a boundary forcing term inL2(0,T; L2(Γ)). Both Dirichlet and Neumann nonhomogeneous cases are considered. A functional analytic model based on cosine operator functions is presented, which provides an input-solution formula to be interpreted in appropriate topologies. With the help of this model, it is shown, for example, that the solution of the nonhomogeneous Dirichlet problem is inL2(0,T; L2(ź)), when ź is either a parallelepiped or a sphere, while the solution of the nonhomogeneous Neumann problem is inL2(0,T; H3/4-e(ź)) when ź is a parallelepiped and inL2(0,T; H2/3(ź) when ź is a sphere. The Dirichlet case for general domains is studied by means of pseudodifferential operator techniques.

The Rate of Convergence for the Finite Element Method

The character of the proper refinement of the elements (mesh) around the boundary is studied. It is shown that it is possible to obtain the highest (optimal) rate of convergence by this refinement.