We prove that the best constant in the Sobolev inequality (WI,” c Lp* with $= f i and 1 c p < n) is achieved on compact Riemannian manifolds, or only complete under some hypotheses. We also establish stronger inequalities where the norms are to some exponent which seems optimal. 0 Elsevier, Paris

Interior a priori error estimates in Sobolev norms are derived from interior RitzGalerkin equations which are common to a class of methods used in approximating solutions of second order elliptic boundary value problems. The estimates are valid for a large class of piecewise polynomial subspaces used in practice, which are defined on both uniform and nonuniform meshes. It is shown that the erro...

Consider the degenerate elliptic operator Lδ := −∂2 x − δ x2 ∂ 2 y on Ω := (0, 1) × (0, l), for δ > 0, l > 0. We prove well-posedness and regularity results for the second-order degenerate elliptic equation Lδu = f in Ω, u|∂Ω = 0 using weighted Sobolev spaces Km a . In particular, by a proper choice of the parameters in the weighted Sobolev spaces Km a , we establish the existence and uniquenes...

We study the critical dissipative quasi-geostrophic equations in R with arbitrary H initial data. After showing certain decay estimate, a global well-posedness result is proved by adapting the method in [10]. A decay in time estimate for higher Sobolev norms of solutions is also discussed.

We study the critical dissipative quasi-geostrophic equations in R with arbitrary H initial data. After showing certain decay estimate, a global well-posedness result is proved by adapting the method in [11] with a suitable modification. A decay in time estimate for higher order homogeneous Sobolev norms of solutions is also discussed.

We establish a mortar boundary element scheme for hypersingular boundary integral equations representing elliptic boundary value problems in three dimensions. We prove almost quasi-optimal convergence of the scheme in broken Sobolev norms of order 1/2. Sub-domain decompositions can be geometrically non-conforming and meshes must be quasi-uniform only on sub-domains. Numerical results confirm th...

On the basis of a fully discrete trigonometric Galerkin method and two grid iterations we propose solvers for integral and pseudodifferential equations on closed curves which solve the problem with an optimal convergence order ‖uN − u‖λ ≤ cλ,μNλ−μ‖u‖μ, λ ≤ μ (Sobolev norms of periodic functions) in O(N log N) arithmetical operations.

This paper studies the global well-posedness of the incompressible magnetohydrodynamic (MHD) system with a velocity damping term. We establish the global existence and uniqueness of smooth solutions when the initial data is close to an equilibrium state. In addition, explicit large-time decay rates for various Sobolev norms of the solutions are also given.

Spherical splines are used to define approximate solutions to strongly elliptic pseudodifferential equations on the unit sphere. These equations arise from geodesy. The approximate solutions are found by using Galerkin method. We prove optimal convergence (in Sobolev norms) of the approximate solution by spherical splines to the exact solution. Our numerical results underlie the theoretical res...