By Mariano Giaquinta, Guiseppe Modica, Jiri Soucek

This monograph (in volumes) bargains with non scalar variational difficulties coming up in geometry, as harmonic mappings among Riemannian manifolds and minimum graphs, and in physics, as good equilibrium configuations in nonlinear elasticity or for liquid crystals. The presentation is selfcontained and obtainable to non experts. subject matters are taken care of so far as attainable in an uncomplicated method, illustrating effects with easy examples; in precept, chapters or even sections are readable independently of the overall context, in order that elements should be simply used for graduate classes. Open questions are usually pointed out and the ultimate element of every one bankruptcy discusses references to the literature and occasionally supplementary effects. ultimately, a close desk of Contents and an in depth Index are of support to refer to this monograph

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**Example text**

An object F ∈ Ob(Tˆ ) is a contravariant functor from T to Ab. , an open set) of T , F (U ) is an abelian group and for φ φU → G in Tˆ , F (U ) −−→ G(U ) is a group homomorphism a morphism F − φU of abelian groups. Namely, a natural transformation φ (which will be called a morphism of presheaves) of presheaves F and G induces the group homomorphism φU over U from F (U ) to G(U ). 1) where 0G(U ) is a zero element of the abelian group G(U ). 2). Let aV ∈ ker φV ⊂ F (V ). Then ρVU (aV ) ∈ F (U ).

Note 7. 4). , ≈ → HomC C (ι lim Fi , F ). HomC (lim Fi , lim Fi ) − ←− ←− ←− For an identity morphism 1lim Fi on the left hand-side, there is ←− α ∈ HomC C (ι lim Fi , F ). 3). Next let −YF : ιY → F be a morphism in C C . 7) there exists a unique element hY ∈ HomC (Y, lim Fi ). 4). 8): C be a functor. 1) Set Then a representing object in C for the composed covariant functor F ◦ ι = HomC C (F, ι ·) from C to Set is the direct limit (or colimit) lim Fi of F . Namely, we have the −→ isomorphism of Cˆ = SetC ≈ lim Fi − → HomC C (F, ι ·).

3) is exact in B at F A , F A and F A . Namely, F φ is a monomorphism, ker F ψ = im F φ and F ψ is an epimorphism in B. , F ψ need not be an epimorphism, F is said to be a left exact functor. Similarly, when FA Fφ G FA Fψ G FA G0 is exact in B, F is said to be a right exact functor. 3)), F is said to be half-exact. 5 Injective Objects [Injective Objects] Let A be an abelian category. 6). Then the contravariant functor HomA (·, A) is a left exact functor from A to Ab. 2) where, for instance, φ∗ := HomA (φ, A).