Abstract
The basic modern approach to boundary-value problems in differential equations of the type (0.1)–(0.2) is the so-called energy-method technique which took the name after a-priori estimates having sometimes physical analogies as bounds of an energy.1 This technique originated from modern theory of linear partial differential equations where, however, other approaches are efficient, too. On the abstract level, this method relies on relative weak compactness of bounded sets in reflexive Banach spaces, and either pseudomonotonicity or weak continuity of differential operators which are understood as bounded from one Banach space to another (necessarily different) Banach space. On the concrete-problem level, the main tool is a weak formulation of boundary-value problems in question, Poincaré and Hölder inequalities, and fine issues from the theory of Sobolev spaces.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2013 Birkhäuser Basel
About this chapter
Cite this chapter
Roubíček, T. (2013). Pseudomonotone or weakly continuous mappings. In: Nonlinear Partial Differential Equations with Applications. International Series of Numerical Mathematics, vol 153. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0513-1_2
Download citation
DOI: https://doi.org/10.1007/978-3-0348-0513-1_2
Published:
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-0512-4
Online ISBN: 978-3-0348-0513-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)