Homogenization of foliated annuli

Let \(\Omega = \Omega _0 \backslash \bar \Omega _1\) be a regular annulus inR ^N and \(\phi :\bar \Omega \to R\) be a regular function such that φ=0 on ?Ω₀, φ=1 on ?Ω₁ and ▽φ ≠ 0. Let K_n be the subset of functions v ε W^1,p (Ω) such that v=0 on ?Ω₀, v=1 on ?Ω₁, v=(unprescribed) constant on n given level surfaces of φ. We study the convergence of sequences of minimization problems of the type $$Inf\left\{ {\int\limits_\Omega {\frac{1}{{a_n \circ \phi }}G(x,(a_n \circ \phi )\nabla v)dx;v \in K_n } } \right\},$$ where a_n ε L^∞ (0,1) and G: (x, ζ) ε Ω × R^N → G(x, ζ εR is convex with respect to ξ and verifies some standard growth conditions.     

L p Solutions of One-Dimensional Backward Stochastic Differential Equations with Continuous Coefficients

In this article, we study one-dimensional backward stochastic differential equations with continuous coefficients. We show that if the generator f is uniformly continuous in (y, z), uniformly with respect to (t, ω), and if the terminal value ξ ∈L ^p (Ω, ? _T , P) with 1 < p ≤ 2, the backward stochastic differential equation has a unique L ^p solution.     

The structurally stable linear systems on the half-line are those with exponential dichotomies

Let A be a selft-adjoint operator on the Hilbert space L²Ω, ?) = {u ε L_loc²(Ω)|∫_Ω|² ?(x)dx < + ∞} defined by means of a closed, semibounded, sesquilinear form a(·, ·). We obtain a necessary and sufficuents condition for the spectrum of A to be discrete. We apply this result to a Sturm-Liouville problem for an elliptic operator with discontinuous coefficients defined on an unbounded domain and to the study of the spectrum of a Hamiltonian defined by a pseudodifferential operator.     

A combined hybrid mixed element method for incompressible miscible displacement problem with local discontinuous Galerkin procedure

In this article, we propose a combined hybrid discontinuous mixed finite element method for miscible displacement problem with local discontinuous Galerkin method. Here, to obtain more accurate approximation and deal with the discontinuous case, we use the hybrid mixed element method to approximate the pressure and velocity, and use the local discontinuous Galerkin finite element method for the concentration. Compared with other combined methods, this method can improve the efficiency of computation, deal with the discontinuous problem well and keep local mass balance. We study the convergence of this method and give the corresponding optimal error estimates in L^∞(L²) for velocity and concentration and the super convergence in L^∞(H¹) for pressure. Finally, we also present some numerical examples to confirm our theoretical analysis.     

Nonstationary Stokes system in weighted Sobolev spaces

We consider an initial‐boundary value problem for nonstationary Stokes system in a bounded domain Omega??³ with slip boundary conditions. We assume that Ω is crossed by an axis L. Let us introduce the following weighted Sobolev spaces with finite norms: and where ?(x) = dist{x, L}. We proved the result. Given the external force f∈L_{2, ?µ}(Ω^T), initial velocity v₀∈H(Ω), µ∈?₊\? there exist velocity v∈H(Ω^T) and the pressure p, ?p∈L_{2, ?µ}(Ω^T) and a constant c, independent of v, p, f, such that As we consider the Stokes system in weighted Sobolev spaces the following two things must be used:

• 1. the slip boundary condition and
• 2. the Helmholtz–Weyl decomposition.

Copyright © 2010 John Wiley & Sons, Ltd.     

H2 Convergence of Solutions of a Biharmonic Problem on a Truncated Convex Sector Near the Angle π

We consider a biharmonic problem Δ²u_ω = f_ω with Navier type boundary conditions u_ω = Δu_ω = 0, on a family of truncated sectors Ω_ω in ?² of radius r, 0 < r < 1 and opening angle ω, ω ∈ (2π/3, π] when ω is close to π. The family of right-hand sides (f_ω)_{ω∈(2π/3, π]} is assumed to depend smoothly on ω in L²(Ω_ω). The main result is that u_ω converges to u_π when ω → π with respect to the H²-norm. We can also show that the H²-topology is optimal for such a convergence result.

     

G-convergence and the weak operator topology

We show that a bounded sequence (a_n)_n of symmetric d × d-matrix valued functions is G-convergent if and only if ((ι^∗︁a_nι)⁻¹ )_n converges in the weak operator topology. Here ι: R(grad₀) ↪ L²(Ω)^d denotes the (canonical) embedding from the range of the weak gradient grad₀ defined on H¹₀(Ω) into L²(Ω)_d, where Ω ⊆ ℝ^d is open and bounded. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Asymptotic analysis for Cahn–Hilliard type phase‐field systems related to tumor growth in general domains

This article considers a limit system by passing to the limit in the following Cahn–Hilliard type phase‐field system related to tumor growth as β↘0: in a bounded or an unbounded domain with smooth‐bounded boundary. Here, , T > 0, α > 0, β > 0, p ≥ 0, B is a maximal monotone graph, and π is a Lipschitz continuous function. In the case that Ω is a bounded domain, p and ?Δ + 1 are replaced with p(φ_β) and ?Δ, respectively, and p is a Lipschitz continuous function; Colli, Gilardi, Rocca, and Sprekels (Discrete Contin Dyn Syst Ser S 2017; 10:37–54) have proved existence of solutions to the limit problem with this approach by applying the Aubin–Lions lemma for the compact embedding H¹(Ω)?L²(Ω) and the continuous embedding L²(Ω)?(H¹(Ω))^?. However, the Aubin–Lions lemma cannot be applied directly when Ω is an unbounded domain. The present work establishes existence of weak solutions to the limit problem along with uniqueness and error estimates in terms of the parameter β↘0. To this end, we construct an applicable theory by noting that the embedding H¹(Ω)?L²(Ω) is not compact in the case that Ω is an unbounded domain.     

On non-effective weights in Orlicz spaces

Given a weight w in Ω ⊂ ∝^N, |Ω| < ∞ and a Young function φ, we consider the weighted modular ∫_Ω ω(f(x))w(x)dx and the resulting weighted Orlicz space L_ω(w). For a Young function Ω ∉ Δ₂(∞) we present a necessary and sufficient conditions in order that L_ω(w) = L_ω(X_Ω) up to the equivalence of norms. We find a necessary and sufficient condition for ω in order that there exists an unbounded weight w such that the above equality of spaces holds. By way of applications we simplify criteria from [5] for continuity of the composition operator from L_ω into itself when ω Δ₂(∞) and obtain necessary and sufficient condition in order that the composition operator maps L_ω. continuously onto L_ω.     

On a wave equation with supercritical interior and boundary sources and damping terms

This article addresses nonlinear wave equations with supercritical interior and boundary sources, and subject to interior and boundary damping. The presence of a nonlinear boundary source alone is known to pose a significant difficulty since the linear Neumann problem for the wave equation is not, in general, well‐posed in the finite‐energy space H¹(Ω) × L₂(?Ω) with boundary data in L₂ due to the failure of the uniform Lopatinskii condition. Further challenges stem from the fact that both sources are non‐dissipative and are not locally Lipschitz operators from H¹(Ω) into L₂(Ω), or L₂(?Ω). With some restrictions on the parameters in the model and with careful analysis involving the Nehari Manifold, we obtain global existence of a unique weak solution, and establish exponential and algebraic uniform decay rates of the finite energy (depending on the behavior of the dissipation terms). Moreover, we prove a blow up result for weak solutions with nonnegative initial energy.     

Generalized Stokes resolvent equations in an infinite layer with mixed boundary conditions

In this paper we prove unique solvability of the generalized Stokes resolvent equations in an infinite layer Ω₀ = ℝ^{n –1} × (–1, 1), n ≥ 2, in L^q ‐Sobolev spaces, 1 < q < ∞, with slip boundary condition of on the “upper boundary” ∂Ω⁺₀ = ℝ^{n –1} × {1} and non‐slip boundary condition on the “lower boundary” ∂Ω^–₀ = ℝ^{n –1} × {–1}. The solution operator to the Stokes system will be expressed with the aid of the solution operators of the Laplace resolvent equation and a Mikhlin multiplier operator acting on the boundary. The present result is the first step to establish an L^q ‐theory for the free boundary value problem studied by Beale [9] and Sylvester [22] in L ²‐spaces. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Global attractors for 2‐D wave equations with displacement‐dependent damping

In this paper the long‐time behaviour of the solutions of 2‐D wave equation with a damping coefficient depending on the displacement is studied. It is shown that the semigroup generated by this equation possesses a global attractor in H(Ω) × L₂(Ω) and H²(Ω)∩H(Ω) × H(Ω). Copyright © 2009 John Wiley & Sons, Ltd.     

Existence and comparison results for nonlinear volterra integral equations in a banach space

Let L ⁰(ΩA,P) be the space of equivalent classes of random variables defined on a probability space (Ω,A,P). Let H be the closed subspace of L ⁰(Ω,A,P) spanned by a sequence of i.i.d. (independent and identically distributed) random variables having the symmetric nondegenerate law F. It is proved that H is linearly homeomorphic to l _p for 0<p≤2 if F belongs to the domain of normal attraction of symmetric stable law withexponent p.     

Small into-isomorphism from L∞ (A, μ) into L∞(B, υ)

In this paper we shall assert that if T is an isomorphism of L_∞(Ω₁, A, μ) into L_∞(Ω₂, B, υ) satisfying the condition ‖T‖·‖T ^?1‖?1+? for ?∈ $\left( {0,\frac{1}{5}} \right)$ , then $\frac{T}{{\parallel T\parallel }}$ is close to an isometry with an error less than 6ε in some conditions.     

Kähler structures on complex torus

Consider the complex torus T _C under the natural action of the compact real torus T. In this paper, we study T-invariant Kähler structures ω on T_C. For each ω, we consider the corresponding line bundleL on T _C. Namely, the Chern class ofL is [ω], and L is equipped with a connection ? whose curvature is ω. We construct a canonical T-invariant L ²-structure on the sections ofL,and let H _ω be the square-integrable holomorphic sections ofL.Then the Hilbert space H _ω is a unitary T-representation, and we study the multiplicity of the (l-dimensional) irreducible unitary T-representations in H_ω. We shall see that the multiplicity is controlled by the image of the moment map corresponding to the T-action preserving ω.     

Minimizing the L ∞ Norm of the Gradient with an Energy Constraint

ABSTRACT

The variational problem in L ^∞ considered is to minimize F(u) = ‖Du‖ _{L ^∞(Ω)} subject to ∈ t _Ω |Du|² dx ≤ E for given E > 0. It is proven that a constrained minimizer exists and satisfies an Aronsson-Euler equation in the viscosity sense which depends on a parameter Λ_∞ ≥ 0. This parameter splits Ω into two parts. In one part the minimizer satisfies the infinity laplace equation and in the remaining part the minimizer is the solution of the elasto-plastic torsion problem with constraint ‖Du‖ _{L ^∞}  ≤ Λ_∞.     

Estimates for the Torsion Function and Sobolev Constants

Let Ω be an open set in Euclidean space, and let u : Ω → ?^?+? be the expected lifetime of Brownian motion in Ω. It is shown that if u?∈?L ^p (Ω) for some p?∈?[1, ?∞?) then (i) u?∈?L ^q (Ω) for all q?∈?[p,?∞?], and (ii) \({trace}\left(e^{t\Delta_{\Omega}}\right)<\infty\) for all t?>?0, where ??Δ_Ω is the Dirichlet Laplacian acting in L ²(Ω). Pointwise bounds are obtained for u in terms of the first Dirichlet eigenfunction for Ω, assuming that the spectrum of ??Δ_Ω is discrete. It is shown that if Ω is open, bounded and connected in the plane and \(\partial\Omega\) has an interior wedge with opening angle α at vertex v then the first Dirichlet eigenfunction and u are comparable near v if and only if α?≥?π/2. Two sided estimates are obtained for the Sobolev constant

$ C_p(\Omega):= \inf\left\{\Vert \nabla u \Vert_2^2: u \in C_0^{\infty}(\Omega),\ \Vert u\Vert_p = 1\right\}, $

where 0?p?Ω satisfies a strong Hardy inequality, and the distance to the boundary function δ?∈?L ^2p/(2???p)(Ω).

     

On the Navier problem for the stationary Navier-Stokes equations

The Navier problem is to find a solution of the steady-state Navier-Stokes equations such that the normal component of the velocity and a linear combination of the tangential components of the velocity and the traction assume prescribed value a and s at the boundary. If Ω is exterior it is required that the velocity converges to an assigned constant vector u₀ at infinity. We prove that a solution exists in a bounded domain provided ‖a_‖L²(∂Ω) is less than a computable positive constant and is unique if ‖a_‖W1/2,2(∂Ω)+‖s_‖L²(∂Ω) is suitably small. As far as exterior domains are concerned, we show that a solution exists if ‖a_‖L²(∂Ω)+‖a−u₀⋅n_‖L²(∂Ω) is small.     

On the space of delta <Emphasis Type="Italic">m</Emphasis>-subharmonic functions

The aim of this paper is to introduce the vector space δF _m(Ω). Equipping this space by a norm defined using the Hessian measure, we prove that it is a Banach space and that the class F _m(Ω) is closed in it. Moreover we show that the topology defined by this norm is stronger than the convergence in Capm-capacity. At the end of this paper we prove a relationship between weak convergence and the convergence in capacity Cap_m,T in the class δSH _m ^loc (Ω).