Cartesian Currents In The Calculus Of Variations Ii: Variational Integrals (ergebnisse Der Mathematik Und Ihrer Grenzgebiete. 3. Folge / A Series Of Modern Surveys In Mathematics)
by Mariano Giaquinta /
1998 / English / PDF
15.5 MB Download
Non-scalar variational problems appear in different fields. In
geometry, for in stance, we encounter the basic problems of
harmonic maps between Riemannian manifolds and of minimal
immersions; related questions appear in physics, for example in the
classical theory of a-models. Non linear elasticity is another
example in continuum mechanics, while Oseen-Frank theory of liquid
crystals and Ginzburg-Landau theory of superconductivity require to
treat variational problems in order to model quite complicated
phenomena. Typically one is interested in finding energy minimizing
representatives in homology or homotopy classes of maps, minimizers
with prescribed topological singularities, topological charges,
stable deformations i. e. minimizers in classes of diffeomorphisms
or extremal fields. In the last two or three decades there has been
growing interest, knowledge, and understanding of the general
theory for this kind of problems, often referred to as geometric
variational problems. Due to the lack of a regularity theory in the
non scalar case, in contrast to the scalar one - or in other words
to the occurrence of singularities in vector valued minimizers,
often related with concentration phenomena for the energy density -
and because of the particular relevance of those singularities for
the problem being considered the question of singling out a weak
formulation, or completely understanding the significance of
various weak formulations becames non trivial.
Non-scalar variational problems appear in different fields. In
geometry, for in stance, we encounter the basic problems of
harmonic maps between Riemannian manifolds and of minimal
immersions; related questions appear in physics, for example in the
classical theory of a-models. Non linear elasticity is another
example in continuum mechanics, while Oseen-Frank theory of liquid
crystals and Ginzburg-Landau theory of superconductivity require to
treat variational problems in order to model quite complicated
phenomena. Typically one is interested in finding energy minimizing
representatives in homology or homotopy classes of maps, minimizers
with prescribed topological singularities, topological charges,
stable deformations i. e. minimizers in classes of diffeomorphisms
or extremal fields. In the last two or three decades there has been
growing interest, knowledge, and understanding of the general
theory for this kind of problems, often referred to as geometric
variational problems. Due to the lack of a regularity theory in the
non scalar case, in contrast to the scalar one - or in other words
to the occurrence of singularities in vector valued minimizers,
often related with concentration phenomena for the energy density -
and because of the particular relevance of those singularities for
the problem being considered the question of singling out a weak
formulation, or completely understanding the significance of
various weak formulations becames non trivial.
