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.     

An energy-decreasing rearrangement of level sets preserving the domain and volume constraints

Let Ω⊂R² be a bounded and regular domain, u∈C³(Ω) and V⊂Ω a domain where the subset K₀ of points where the curvature of the t-level sets of u is zero admits a regular t-parameterization. We exhibit a local correction of u in a neighborhood of a particular point x^∗∈K₀⊂V such that the volume ∫f(u) is preserved and the Dirichlet integral ∫²|∇u| decreases. Consequently, a certain monotonic property is deduced for constrained minimizers in H¹(Ω). Our result can be applied to classical variational and free-boundary problems.     

On the principal eigenvalue of some nonlocal diffusion problems

In this paper we analyze some properties of the principal eigenvalue λ₁(Ω) of the nonlocal Dirichlet problem (J∗u)(x)−u(x)=−λu(x) in Ω with u(x)=0 in RN?Ω. Here Ω is a smooth bounded domain of RN and the kernel J is assumed to be a C¹ compactly supported, even, nonnegative function with unit integral. Among other properties, we show that λ₁(Ω) is continuous (or even differentiable) with respect to continuous (differentiable) perturbations of the domain Ω. We also provide an explicit formula for the derivative. Finally, we analyze the asymptotic behavior of the decreasing function Λ(γ)=λ₁(γΩ) when the dilatation parameter γ>0 tends to zero or to infinity.     

Scalar conservation laws with general boundary condition and continuous flux function

We introduce a notion of entropy solution for a scalar conservation law on a bounded domain with nonhomogeneous boundary condition: ut+divΦ(u)=f on Q=(0,T)×Ω, u(0,⋅)=u₀ on Ω and “u=a on some part of the boundary (0,T)×∂Ω.” Existence and uniqueness of the entropy solution is established for any Φ∈C(R;RN), u₀∈L^∞(Ω), f∈L^∞(Q), a∈L^∞((0,T)×∂Ω). In the L¹-setting, a corresponding result is proved for the more general notion of renormalised entropy solution.     

Regularity of hyperbolic equations underL 2(0,T; L 2 (Γ))-Dirichlet boundary terms

With Ω an open bounded domain inR ⁿ with boundary Γ, letf(t; f ₀,f ₁;u) be the solution to a second order linear hyperbolic equation defined on Ω, under the action of the forcing termu(t) applied in the Dirichlet B.C., and with initial dataf ₀ ∈L ₂ (Ω) andf ₁ ∈H ^?1 (Ω). In a previous paper [6], we proved (among other things) that the mapu → f ? f _t , from the Dirichlet input into the solution is continuous fromL ₂(0,T; L ₂ (Γ)) intoL ₂(0,T; L ₂(Ω))?L₂ (0, T; H ^?1 (Ω)). Here, we make crucial use of this result to present the following marked improvement: the mapu → f ?f _t is continuous fromL ₂ (0, T; L ₂ (Γ)) intoC([0, T]; L ₂ (Ω))?C([0, T]; H ^?1 (Ω)). Our approach uses the cosine operator model introduced in [6], along with crucial relevant fact from cosine operator theory. A new trace theory result, on which we base our proof here, plays also a decisive role in the corresponding quadratic optimal control problem [7]. Whenu, instead, acts in the Neumann B. C. and Ω is either a sphere or a parallelepiped, the same approach leads to the same improvement over results obtained in [6] to the regularity int of the solution (i.e., fromL ₂ (0, T) toC[0, T]).     

Asymptotic behaviour of nonlocal reaction-diffusion equations

The existence of a global attractor in L²(Ω) is established for a reaction-diffusion equation on a bounded domain Ω in R^d with Dirichlet boundary conditions, where the reaction term contains an operator F:L²(Ω)→L²(Ω) which is nonlocal and possibly nonlinear. Existence of weak solutions is established, but uniqueness is not required. Compactness of the multivalued flow is obtained via estimates obtained from limits of Galerkin approximations. In contrast with the usual situation, these limits apply for all and not just for almost all time instants.     

On the Minimization of Dirichlet Eigenvalues of the Laplace Operator

We study variational problems of the form $$\inf\{\lambda_k(\Omega): \Omega\ \mbox{open in}\ \mathbb{R}^m,\ T(\Omega ) \le1 \},$$ where λ _k (Ω) is the k-th eigenvalue of the Dirichlet Laplacian acting in L ²(Ω), and where T is a non-negative set function defined on the open sets in ? ^m , which is invariant under isometries, additive on disjoint families of open sets, and is such that the ball with T(B)=1 is a minimizer for k=1. Upper bounds are obtained for the number of components of any bounded minimizer if T satisfies a scaling relation. For example, we show that if T is Lebesgue measure and if k≤m+1 then any bounded minimizer has at most 7 components. We also consider variational problems over open sets Ω in ? ^m involving the (m?1)-dimensional Hausdorff measure of ?Ω.     

A basic inequality for the Stokes operator related to the Navier boundary condition

We show that ‖Au+Δu_‖L²(Ωε)?C(ε‖∇u_‖L²(Ωε)+‖u_‖L²(Ωε)), where Ωε is a thin domain in R³ of depth ε, the vector field u belongs to the domain of A, which is the Stokes operator for divergence-free vector fields on Ωε satisfying the Navier boundary condition.     

Unitary Equivalence, Similarity and Calculation of K0-Group

In this paper, we are concerned with the classification of operators on complex separable Hilbert spaces, in the unitary equivalence sense and the similarity sense, respectively. We show that two strongly irreducible operators A and B are unitary equivalent if and only if W＊（A＋B）′≈M2（C）, and two operators A and B in B1（Ω） are similar if and only if A′（AGB）/J≈M2（C）. Moreover, we obtain V（H^∞（Ω,μ）≈N and Ko（H^∞（Ω,μ）≈Z by the technique of complex geometry, where Ω is a bounded connected open set in C, and μ is a completely non-reducing measure on Ω.     

ABSTRACT CAPACITY OF REGIONS AND COMPACT EMBEDDING WITH APPLICATIONS

The weighted Sobolev-Lions type spaces W pl,γ(Ω; E0, E) = W pl,γ(Ω; E) ∩ Lp,γ (Ω; E0) are studied, where E0, E are two Banach spaces and E0 is continuously and densely embedded on E. A new concept of capacity of region Ω∈ Rn in W pl,γ(; E0, E) is introduced. Several conditions in terms of capacity of region Ω and interpolations of E0 and E are found such that ensure the continuity and compactness of embedding operators. In particular, the most regular class of interpolation spaces Eα between E0 and E, depending of α and l, are found such that mixed differential operators Dα are bounded and compact from W pl,γ(Ω; E0, E) to Eα-valued Lp,γ spaces. In applications, the maximal regularity for differential-operator equations with parameters are studied.     

Spike-layer solutions to singularly perturbed semilinear systems of coupled Schrödinger equations

Let Ω be a bounded domain in RN(N?3), we are concerned with the interaction and the configuration of spikes in a double condensate by analyzing the least energy solutions of the following two couple Schrödinger equations in Ω

(S_ε)     

Topological stable rank of H ∞(Ω) for circular domains Ω

Let Ω be a circular domain, that is, an open disk with finitely many closed disjoint disks removed. Denote by H ^∞(Ω) the Banach algebra of all bounded holomorphic functions on Ω, with pointwise operations and the supremum norm. We show that the topological stable rank of H ^∞(Ω) is equal to 2. The proof is based on Suárez’s theorem that the topological stable rank of H ^∞( $ \mathbb{D} $ ) is equal to 2, where $ \mathbb{D} $ is the unit disk. We also show that for circular domains symmetric to the real axis, the Bass and topological stable ranks of the real-symmetric algebra H _? ^∞ (Ω) are 2.     

A lower bound on the Kobayashi metric near a point of finite type in ℂ n

Let Ω be a bounded domain in ? ⁿ andbΩ smooth pseudoconvex near z₀ ∈bΩ of finite type. Then there are constantsc>0 and ε′>0 such that the Kobayashi metric,K _Ω(z; X), satisfiesK _Ω(z; X)≥c|X|δ(z)^?t′ for allX ∈T _z ^1,0 ? ⁿ in a neighborhood ofz ₀. Here δ(z) denotes the distance fromz tobΩ. As an application, we prove the Hölder continuity of proper holomorphic maps onto pseudoconvex domains.     

Global existence and energy decay for a coupled system of Kirchhoff type equations with damping and source terms

This paper deals with the global existence and energy decay of solutions to some coupled system of Kirchhoff type equations with nonlinear dissipative and source terms in a bounded domain. We obtain the global existence by defining the stable set in H ₀ ¹ (Ω) × H ₀ ¹ (Ω), and the energy decay of global solutions is given by applying a lemma of V. Komornik.     

The Dirichlet-to-Neumann operator on rough domains

We consider a bounded connected open set Ω⊂Rd whose boundary Γ has a finite (d−1)-dimensional Hausdorff measure. Then we define the Dirichlet-to-Neumann operator D₀ on L₂(Γ) by form methods. The operator −D₀ is self-adjoint and generates a contractive C₀-semigroup S=(St)_t>0 on L₂(Γ). We show that the asymptotic behaviour of St as t→∞ is related to properties of the trace of functions in H¹(Ω) which Ω may or may not have.     

Boundary blow-up elliptic problems with nonlinear gradient terms

By Karamata regular variation theory and perturbation method, we show the exact asymptotical behaviour of solutions near the boundary to nonlinear elliptic problems Δu±q|∇u|=b(x)g(u), u>0 in Ω, u_|∂Ω=+∞, where Ω is a bounded domain with smooth boundary in RN, q?0, g∈C¹[0,∞),g(0)=0, g^′ is regularly varying at infinity with index ρ with ρ>0 and b is nonnegative nontrivial in Ω, which may be vanishing on the boundary.     

Boundary behavior in the Loewner-Nirenberg problem

Let Ω⊂Rⁿ be a bounded domain of class C^2+α, 0<α<1. We show that if n?3 and u_Ω is the maximal solution of equation Δu=n(n-2)u^(n+2)/(n-2) in Ω, then the hyperbolic radius is of class C^2+α up to the boundary. The argument rests on a reduction to a nonlinear Fuchsian elliptic PDE.