Variational Methods for Engineers with Matlab

4Functional Spaces for Engineers

Functional spaces are Hilbert spaces formed by functions. For engineers, these spaces represent spaces of quantities having a spatial or temporal distribution, i.e. fields. For instance, the field of temperatures on a region of the space, the field of velocities of a continuous medium, the history of the velocities of a particle is functions. As previously observed, these spaces have some particularities connected to the fact that they are infinite dimensional.

From the beginning of the 20th Century, the works of Richard Courant, David Hilbert, Kurt Friedrichs and other mathematicians of Gottingen have pointed the necessity of a redefinition of derivatives and of the functional spaces involving derivatives.

Otto Nykodim introduced a class named “BL” (class of Bepo-Levi), involving (u, v) = ∫∇u.∇v dx [NIK 33]. He constructed a theory about this class and brought it to the framework of Hilbert spaces two years later [NIK 35], but the complete theory arrived later with the works of Serguei Sobolev, a Russian mathematician that introduced functional spaces fitting to variational methods.

Sobolev was closely connected to Jacques Hadamard, who faced the mathematical difficulties arising in fluid mechanics, when studying the Navier–Stokes equations [HAD 03a, HAD 03b]. Far from Hadamard, the Russian Nikolai Gunther, a former PhD student of Andrey Markov, was interested in potential theory and, particularly, in extensions of Kirchhoff formula and derivatives ...