Local inverse and functional inequalities

Abstract

Inverse inequalities rely on the fact that all the norms are equivalent in finite-dimensional normed vector spaces. The term ‘inverse’ refers to the fact that high-order Sobolev (semi)norms are bounded by lower-order (semi)norms, but the constants involved in these estimates either tend to zero or to infinity as the meshsize goes to zero. Our purpose is then to study how the norm-equivalence constants depend on the local meshsize and the polynomial degree of the reference finite element. We also derive some local functional inequalities valid in infinite-dimensional spaces. All these inequalities are regularly invoked in this book. In the whole chapter, we consider a shape-regular sequence of affine meshes.