In the history of the calculus of variations we can find some few books which have been as milestones in the development of the field. Certainly the book by L. Tonelli: “Fondamenti di calcolo delle variazioni”. (I: 1922; JFM 48.0581.09; II: 1923; JFM 49.0348.05) has been one of them, together with some few more. The last encyclopedical book in the field was probably the one by C. B. Morrey jr: “Multiple integrals in the calculus of variations (1966; Zbl 0142.38701)] which has been the reference for hundreds of papers in variational problems and elliptic partial differential equations.
Now, this new book by M. Giaquinta and S. Hildebrandt, in two volumes (cf. also the following review) will probably become the updated reference in the field for the next years. The choice to divide the matter into two volumes: the Lagrangian formalism and the Hamiltonian formalism, helps the reader to be guided into the field of variational problems, which has grown enormously during about three centuries. Moreover, the style adopted is clear and precise, and often small generalizations and weaker assumptions are sacrified for the sake of clarity.
The first volume, here under review, is divided into two parts: the first one starts from the classical critical points theory and the first variation of a functional of the form \[ {\mathcal F} (u) = \int_\Omega F(x,u,Du) dx, \] which leads to the Euler equation \[ \Bigl( - F_{z_i} + \sum_\alpha D_\alpha F_{p^i_\alpha} \Bigr) (x,u,Du) = 0 \] which is, under suitable convexity assumptions on the integrand \(F\), a second order elliptic system. The boundary conditions come out from the variational principle as natural boundary conditions. Other necessary optimality conditions are illustrated, like the Du Bois-Reymond equation and the Legendre-Hadamard conditions. In the case of nonsmooth minimizers, called broken extremals, the Weierstrass-Erdmann corner conditions are explained in detail. Also the case of variational problems of higher order, which occur for instance in the study of equilibria of thin elastic plates or beams, is presented.
The second part of the first volume is devoted to the study of the second variation and of sufficient conditions of optimality. The Weierstrass excess function is then considered and classical theories as the Jacobi field theory or the Weierstrass field theory are here illustrated in full generality.
The volume ends with a supplement on some topics from Differential Geometry about differential forms and curves and surfaces in \(\mathbb{R}^N\), with particular attention on the mean curvature and Gauss curvature.
Each section of the volume contains many interesting examples, classical and nonclassical, which illustrate the importance of every assumption, and make the topics easily understandable even for a nonspecialist reader. Moreover, the bibliography is very rich and the subject index is very carefully made.