Neumann-type Problem Research Articles

Homogenization of integrals with pointwise gradient constraints via the periodic unfolding method

The pointwise gradient constrained homogenization process, for Neumann and Dirichlet type problems, is analyzed by means of the periodic unfolding method recently introduced in [21]. Classically, the proof of the homogenization formula in presence of pointwise gradient constraints relies on elaborated measure theoretic arguments. The one proposed here is elementary: it is based on weak convergence arguments in L p spaces, coupled with suitable regularization techniques. Keywords: Homogenization, Gradient constrained problems, Periodic unfolding method Mathematics Subject Classification (2000): 49J45, 35B27, 74Q05

Weighted estimates for elliptic systems on Lipschitz domains

Let Ω be a bounded Lipschitz domain in R n , n ≥ 3. Let ω λ (Q) = IQ - Q 0 | λ , where Q 0 is a fixed point on ∂Ω. For a second order elliptic system with constant coefficients on Ω, we study boundary value problems with boundary data in the weighted space L 2 (∂Ω, ω λ dσ), where dσ denotes the surface measure on ∂Ω. We show that there exists e > 0 such that the Dirichlet problem is uniquely solvable for -min(2+e, n-1) < A < e, and the Neumann type problem is uniquely solvable if -e < A < min(2+e, n-1). The regularity for the Dirichlet problem with data in the weighted Sobolev space L 2 1 (∂Ω, ωλ dσ) is also considered.

Boundary value problems in Morrey spaces for elliptic systems on Lipschitz domains

Let Ω be a bounded Lipschitz domain in [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /], n ≥ 3. Let [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /] be a second order elliptic system with constant coefficients satisfying the Legendre-Hadamard condition. We consider the Dirichlet problem [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /] = 0 in Ω, u = f on ∂Ω with boundary data f in the Morrey space L 2,λ (∂Ω). Assume that 0 ≤ λ &lt; 2 + ε for n ≥ 4 where ε &gt; 0 depends on Ω, and 0 ≤ λ ≤ 2 for n = 3. We obtain existence and uniqueness results with nontangential maximal function estimate ║( u )*║ L 2,λ(∂Ω) ≤ C ║ f ║ L 2,λ(∂Ω) . If [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="04i" /] satisfies the strong elliptic condition and 0 ≤ λ &lt; min ( n -1, 2+ε), we show that the Neumann type problem [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="05i" /] = 0 in Ω, ∂ u / ∂ν = g ∈ H 2,λ (∂Ω) on ∂Ω, ║(∇ u )*║ H 2,λ (∂Ω) &lt; ∞ has a unique solution. Here H 2,λ (∂Ω) is an atomic space with the property ( H 2,λ (∂Ω))* = L 2,λ (∂Ω). The invertibility of layer potentials on L 2,λ (∂Ω) and H 2,λ (∂Ω) is also obtained. Finally we study the Dirichlet problem for the biharmonic equation. We establish a similar estimate in L 2,λ for the biharmonic equation, in which case the range 0 ≤ λ &lt; 2 + ε is sharp for n = 4 or 5.

Infinitely Many Critical Points of Non-Differentiable Functions and Applications to a Neumann-Type Problem Involving the p-Laplacian

For a family of functionals in a Banach space, which are possibly non-smooth and depend also on a positive real parameter, the existence of a sequence of critical points (according to Motreanu and Panagiotopoulos (“Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities,” Nonconvex Optimization Applications, Vol. 29, Kluwer, Dordrecht, 1998, Chap. 3)) is established by mainly adapting a new technique due to Ricceri (2000, J. Comput. Appl. Math.113, 401–410). Two applications are then presented. Both of them treat the Neumann problem for an elliptic variational–hemivariational inequality with p-Laplacian.

Which sequences of holes are admissible for periodic homogenization with Neumann boundary condition?

In this paper we give a general presentation of the homogenization of Neumann type problems in periodically perforated domains, including the case where the shape of the reference hole varies with the size of the period (in the spirit of the construction of self-similar fractals). We shows that H 0 -convergence holds under the extra assumption that there exists a bounded sequence of extension operators for the reference holes. The general class of Jones-domains gives an example where this result applies. When this assumption fails, another approach, using the Poincar e{Wirtinger inequality is presented. A corresponding class where it applies is that of John-domains, for which the Poincar e{ Wirtinger constant is controlled. The relationship between these two kinds of assumptions is also claried.

Navier-Stokes equations on Lipschitz domains in Riemannian manifolds

The Navier-Stokes equations are a system of nonlinear evolution equations modeling the flow of a viscous, incompressible fluid. One ingredient in the analysis of this system is the stationary, linear system known as the Stokes system, a boundary value problem (BVP) that will be described in detail in the next section. Layer potential methods in smoothly bounded domains in Euclidean space have proven useful in the analysis of the Stokes system, starting with work of Odqvist and Lichtenstein, and including work of Solonnikov and many others. See the discussion in Chapter III of [10] and in [17], for the case of flow in regions with smooth boundary. A treatment based on the modern language of pseudodifferential operators can be found in [18]. In 1988, E. Fabes, C. Kenig and G. Verchota [6], extended this classical layer potential approach to cover Lipschitz domains in Euclidean space. In [6] the main result concerning the (constant coefficient) Stokes system on Lipschitz domains with connected boundary in Euclidean space, is the treatment of the L-Dirichlet boundary value problem (and its regular version). To achieve this, the authors solve certain auxiliary Neumann type problems and then exploit the duality between these and the original BVP’s at the level of boundary integral operators. P. Deuring and W. von Wahl [4] made use of the analysis in [6] to demonstrate the short-time existence of solutions to the Navier-Stokes equations in bounded Lipschitz domains in threedimensional Euclidean space. It was necessary in [4] to include the hypothesis that the boundary be connected. The hypothesis that the boundary be connected pervaded much work on the application of layer potentials to analysis on Lipschitz domains. It was certainly natural to speculate that this restriction was an artifact of the methods used and not ∗Partly supported by NSF grant DMS-9870018. †Partially supported by NSF grant DMS-9877077. 1991Mathematics Subject Classification. Primary 35Q30, 76D05, 35J25; Secondary 42B20, 45E05.

The generalized neumann problem for yang-mills connections

The purpose of this paper is defining a new boundary value problem for Yang-Mills connections, which is the most general in the context of Neumann-type problems for forms. We achieve this by reflecting the base manifold across the boundary, and lifting this action non-trivially to the bundle. This way we obtain a twisted boundary value problem in which the boundary conditions are mixed, of Dirichlet type on some of the Lie-algebra components of the connection A, of Neumann type on others. This problem arises naturally and it can be viewed in the context of generalizing non-linear Hodge theory for connections. We prove a good gauge theorem for this problem. We give an application.

A Nonlinear Neumann-Type Problem of a System of High Order Hyperbolic Integro-Differential Equations

The paper concerns a nonlinear Neumann-type boundary value problem for a system of hyperbolic integro-differential equations of order 2pwith two independent variables. The problem is reduced to a system of integro-functional equations and hence the existence and uniqueness of a local solution is proved by using the Banach fixed point theorem.

On boundary value problems of neumann type for the dirac operator in a half space of

In this paper Neumann type problems are investigated for monogenic functions and for solutions of nonhomogenwus Dirac equations in the upper half space of , in both the classical sense and LP sense. For each Neumann type problem, a representation of the solution, unique up to a Clifford constant, is obtained.

Oscillation Problems in Thin Plates with Transverse Shear Deformation

In this paper, the authors present a modern theory of thin elastic plates with transverse shear deformation where the disturbance is represented by a train of harmonic waves. Dirichlet- and Neumann-type problems are formulated together with appropriate radiation conditions (in the case of the exterior domain). The paper shows that uniqueness for exterior problems is guaranteed for a range of flexural waves. In the interior problems, the presence of eigenfrequencies means that there is no general uniqueness result. The paper also indicates how corresponding results can be proved for micropolar plates.

Fundamental sequences of functions in the approximation of solutions to exterior problems in the bending of thin micropolar plates

In this paper we show how to construct fundamental sequences for approximate solutions to exterior Dirichlet and Neumann-type problems in the bending of micropolar plates. The sequences are based on singular solutions of the governing equilibrium equations. These singular solutions are, however, unbounded at infinity. This leads to difficulties when applying the usual methods for proving linear independence and completeness. By decomposing the sequences into divergent and convergent parts we show that they can be accommodated in a more general framework developed in previous work. This allows us to overcome the difficulties mentioned above. Kupradze's method of generalised Fourier series is then modified and used to construct approximations which converge uniformly to the corresponding exact solutions

A nonlinear variational Neumann's type problem

In this work I consider a nonlinear Neumann's problem involving a maximal monotone operator. The existence of a H2-solution was proved in ([4]). Here I prove a weak existence theorem, the uniqueness of a weak continuous solution, the continuous dependence on the data and I state a priori estimate. At the end by applying the finite element method, I obtain approximate solutions, by means of a perturbed variational problem and an estimate of the discretization error. An algorithm is explained.

The neumann problem for equations of monge‐ampère type

AbstractWe prove the existence of a classical solution to a Neumann type problem for nonlinear elliptic equations of Monge‐Ampère type. The methods depend upon the establishment of a priori derivative estimates up to order two and yield a sharp result for the equation of prescribed Gauss curvature. We also present applications to asymptotic problems which approach the non‐unique case and maximum principles for linear equations of Aleksandrov‐Bakelman type.

Bending, extension, and torsion of naturally twisted rods

Saint-Venant /1/ established that the spatial problem of linear elasticity theory of the deformation of straight rods with a load-free side surface allows of practically complete investigation: the extension problem is solved exactly (if the boundary layer is ignored), and the bending and torsion problems reduce to Neumann problems for the Laplace equation in the region of the rod cross-section (see /2, 3/). It is shown below that an analogous situation holds for a naturally twisted rod: the spatial problem is successfully reduced to a Neumann-type problem for a certain system of second-order elliptic equations in the cross-section. It is essential that this can be done for an arbitrary value of the rod twist. For zero twist the problem in the section reduces to the Saint-Venant problem. In the case of centrally-symmetric sections, the problem decomposes into two independent problems, on bending and on extensiontorsion. Variational principles and certain bilateral estimates of the extension and torsion stiffness are constructed for the latter, and the case of oblong sections is investigated. The extension-torsion problem for naturally twisted rods was examined earlier in /4/. The difference from this research is discussed in Sect.4.

A criterion for the neumann type problem over a differential complex on a strongly pseudo convex domain

A criterion for the neumann type problem over a differential complex on a strongly pseudo convex domain

Stable iterative methods for solving neumann type problems

Stable iterative methods for solving neumann type problems

Biharmonic functions and Brownian motion

Let R be a bounded open subset of N-dimensional Euclidean space EN,N ≧ 1, let {xt: t ≧ 0} be a separable Brownian motion starting at a point x ɛ R, and let τ = τR be the first time the motion hits the complement of R. It is known [1] that if g is a bounded measurable function on the boundary ∂R of R, then h(x) = Ex[g(xτ)] is a harmonic function of x ɛ R which “solves” the Dirichlet problem for the boundary function g; i.e., Δh = 0 on R, where Δ is the Laplacian. In elastic plate problems, one must solve the biharmonic equation subject to certain boundary conditions. For the more important applications, these boundary conditions involve the values of u and the normal derivative of u at points of ∂R. Even though a treatment of this Neumann type problem is not available at this time, some things can be said about biharmonic functions and their relationship to Brownian motion. We will show, in fact, that u(x)= Ex[τ(xτ)] is a biharmonic function on R which “satisfies” the boundary conditions (i) u=0 on ∂R and (ii) Δu= −2g on ∂R, provided g satisfies certain hypotheses. More generally, we will show that u(x)=Ex[Δkg(XΔ)] is polyharmonic of order k + 1 on R (i.e., Δk + 1u = Δ(Δku) = 0 on R) and that it satisfies certain boundary conditions. A treatment of the special case g ≡ 1 on ∂R can be found in [3].

Biharmonic functions and Brownian motion

Let R be a bounded open subset of N-dimensional Euclidean space EN, N ≧ 1, let {xt : t ≧ 0} be a separable Brownian motion starting at a point x ɛ R, and let τ = τR be the first time the motion hits the complement of R. It is known [1] that if g is a bounded measurable function on the boundary ∂R of R, then h(x) = Ex [g(xτ )] is a harmonic function of x ɛ R which “solves” the Dirichlet problem for the boundary function g; i.e., Δh = 0 on R, where Δ is the Laplacian. In elastic plate problems, one must solve the biharmonic equation subject to certain boundary conditions. For the more important applications, these boundary conditions involve the values of u and the normal derivative of u at points of ∂R. Even though a treatment of this Neumann type problem is not available at this time, some things can be said about biharmonic functions and their relationship to Brownian motion. We will show, in fact, that u(x)= Ex[τ(xτ)] is a biharmonic function on R which “satisfies” the boundary conditions (i) u=0 on ∂R and (ii) Δu= −2g on ∂R, provided g satisfies certain hypotheses. More generally, we will show that u(x)=Ex[Δk g(X Δ)] is polyharmonic of order k + 1 on R (i.e., Δk + 1 u = Δ(Δk u) = 0 on R) and that it satisfies certain boundary conditions. A treatment of the special case g ≡ 1 on ∂R can be found in [3].