In this paper we present integral conductor inequalities connecting the Lorentz p, q-(quasi)norm of a gradient of a function to a one-dimensional integral of the p, q-capacitance of the conductor between two level surfaces of the same function. These inequalities generalize an inequality obtained by the second author in the case of the Sobolev norm. Such conductor inequalities lead to necessary...

It is well known that finite element solutions for elliptic PDEs with Dirac measures as source terms converge, due to the fact that the solution is not in H1, suboptimal in classical norms. A standard remedy is to use graded meshes, then quasioptimality, i.e., optimal up to a log-factor, for low order finite elements can be recovered, e.g., in the L2-norm. Here we show for the lowest order case...

The Schatten quasi-norm was introduced tobridge the gap between the trace norm andrank function. However, existing algorithmsare too slow or even impractical for large-scale problems. Motivated by the equivalencerelation between the trace norm and its bilin-ear spectral penalty, we define two tractableSchatten norms, i.e. the bi-trace and tri-tracenorms, and prov...

We investigate quasi-Banach operator ideal products (A ◦ B,A ◦ B) which contain (L2,L2) as a factor. In particular, we ask for conditions which guarantee that A ◦ B is even a norm if each factor of the product is a 1-Banach ideal. In doing so, we reveal the strong influence of the existence of such a norm in relation to the accessibility of the product ideal and the structure of its factors.

Several inviscid models in hydrodynamics and geophysics such as the incompressible Euler vorticity equations, the surface quasi-geostrophic equation and the Boussinesq equations are not known to have even local well-posedness in the corresponding borderline Sobolev spaces. Here H is referred to as a borderline Sobolev space if the L∞-norm of the gradient of the velocity is not bounded by the H-...

This paper focuses on the delay-dependent robust stability of linear neutral delay systems. The systems under consideration are described by functional differential equations, with norm bounded time varying nonlinear uncertainties in the “state” and norm bounded time varying quasi-linear uncertainties in the delayed “state” and in the difference operator. The stability analysis is performed via...

In this paper we revisit and prove optimal order and mesh–independent convergence of an inexact Newton method where the linear Jacobian systems are solved with multigrid techniques. This convergence is shown using Banach spaces and the norm, max{‖ · ‖1, ‖ · ‖0,∞}, a stronger norm than is used in previous work. These results are valid for a class of second order, semi–linear, finite element, ell...

Lipschitzian and kernel aggregation operators with respect to the natural T indistinguishability operator ET and their powers are studied. A t-norm T is proved to be ET -lipschitzian, and is interpreted as a fuzzy point and a fuzzy map as well. Given an archimedean t-norm T with additive generator t, the quasi-arithmetic mean generated by t is proved to be the more stable aggregation operator w...

In this paper, we consider an interior penalty discontinuous Galerkin (DG) method for the time-dependent Maxwell’s equations in cold plasma. In Huang and Li (J. Sci. Comput., 42 (2009), 321–340), for both semi and fully discrete DG schemes, we proved error estimates which are optimal in the energy norm, but sub-optimal in the L2-norm. Here by filling this gap, we show that these schemes are opt...

Recently, using DiGiorgi-type techniques, Caffarelli and Vasseur [1] showed that a certain class of weak solutions to the drift diffusion equation with initial data in L gain Hölder continuity provided that the BMO norm of the drift velocity is bounded uniformly in time. We show a related result: a uniform bound on BMO norm of a smooth velocity implies uniform bound on the C norm of the solutio...