Algunos problemas de análisis sobre cúspides exteriores

In this thesis we study several problems of analysis on domains with external cusps. Mainly: the density of smooth functions on Sobolev spaces, the extension problem and Korn’s inequality. It is known that many classical analysis results on standard Sobolev spaces, which are necessary for the study of partial differential equations and for approximating their solutions using numerical methods, do not hold on singular domains, such as external cusps. Hence, it is necessary to work with weighted Sobolev spaces, where the weights are taken in a way that somehow compensates the singularity of the domain. The different notions of external cusp that we handle avoid any precise description of the domain’s boundary, even when they impose some restrictions. In the first place, we introduce the concept of normal cusp, which definition is based on the Whitney decomposition of the domain, and includes a sectional unifomity property that establishes some local regularity on the boundary. This notion allows us to prove the following results: • The density of smooth functions up to the boundary of the domain, in the Sobolev space W(Ω). • Extension theorems where we build operators of the form E : W(Ω) −→ W σ (R), being σ a proper weight. Afterwards, we introduce the notions of chains of rectangles and chains of quasi-rectangles. Chains of quasi-rectangles allow us to define a very general class of domains, that includes normal cusps, but admits more relaxed conditions on the boundary. For these chains, we prove: • Unweighted Poincaré inequalities. • Weighted Korn inequalities. In some cases we exhibit counterexamples that show that the obtained weights are optimal. We also study these problems considering that the original space is a weighted Sobolev space W k,p ω (Ω).