In this paper, motivated by the results published by R. Khalil and A. Saleh in 2005, we study the notion of k-smooth points and the notion of k-smoothness, which are dual to the notion of k-rotundity. Generalizing these notions and combining smoothness with the recently introduced notion of unitary, we study classes of Banach spaces for which the vector space, spanned by the state space corresp...

We construct an example of a real Banach space whose group of surjective isometries reduces to ± Id, but the group of surjective isometries of its dual contains the group of isometries of a separable infinite-dimensional Hilbert space as a subgroup. To do so, we present examples of extremely non-complex Banach spaces (i.e. spaces X such that ‖ Id+T ‖ = 1+‖T ‖ for every bounded linear operator T...

In this paper we prove an optimal local well-posedness result for the 1+2 dimensional system of nonlinear wave equations (NLW) with quadratic null-form derivative nonlinearities Qμν . The Cauchy problem for these equations is known to be ill-posed for data in the Sobolev space H with s ≤ 5/4 for all the basic null-forms, except Q0, thus leaving a gap to the critical regularity of sc = 1. Follow...

In this work we use algebraic dual spaces with a domain decomposition method to solve the Darcy equations. We define broken Sobolev and their finite dimensional counterparts. A global trace space is defined that connects solution between spaces. Use of results in sparse, metric-free representation incompressibility constraint, pressure gradient term, on continuity constraint sub domains. To dem...

We construct wavelet Riesz bases for the usual Sobolev spaces of divergence free functions on (0, 1)n that have vanishing normals at the boundary. We give a simultaneous space-time variational formulation of the instationary Stokes equations that defines a boundedly invertible mapping between a Bochner space and the dual of another Bochner space. By equipping these Bochner spaces by tensor prod...

In this paper, we generalize well-known results for the L2-norm a posteriori error estimation of finite element methods applied to linear elliptic problems in convex polygonal domains to the case where the polygons are nonconvex. An important factor in our analysis is the investigation of a suitable dual problem whose solution, due to the non-convexity of the domain, may exhibit corner singular...

This paper proposes a combination of contour deformation modelling in Sobolev spaces and the Condensation filter to track an object over a sequence of images. As Sobolev spaces are smoothness spaces this allows to control the smoothness of the contour deformation extending previous wavelet representations. We also introduce a probabilistic model for the wavelet deformation of the contour that i...

In this paper we analyze a class of projection operators with values in a subspace of polynomials. These projection operators are related to the Hubert spaces involved in the numerical analysis of spectral methods. They are, in the first part of the paper, the standard Sobolev spaces and, in the second part, some weighted Sobolev spaces, the weight of which is related to the orthogonality relat...