Abstract
In this chapter we first present test functions, which are then used to introduce scalar distributions. The space \({\mathcal {D}}'(\varOmega )\) of distributions is analyzed in detail and some related applications are discussed: the interpretation of the density of a mass concentrated at a point by means of the Dirac distribution, solving the Poisson equation in \({\mathcal {D}}'(\varOmega )\), solving ordinary differential equations in \({\mathcal {D}}'(\mathbb {R})\), solving the equation of the vibrating string with non-smooth initial displacement function, and the boundary controllability for a problem associated with the same wave equation. We also introduce and discuss Sobolev spaces. In order to introduce vector distributions we shall present in a separate section the Bochner integral for vector functions. Vector distributions and W k, p(a, b; X) spaces are then presented. These will later be used in solving problems associated with parabolic and hyperbolic PDE’s.
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Notes
- 1.
Kurt Otto Friedrichs, German-American mathematician, 1901–1982.
- 2.
Paul Adrien Maurice Dirac, English theoretical physicist, 1902–1984.
- 3.
Israel M. Gel’fand, Russian mathematician, 1913–2009; Georgiy E. Shilov, Russian mathematician, 1917–1975.
- 4.
Pierre-Simon Laplace, French mathematician and astronomer, 1749–1827.
- 5.
George Green, British mathematical physicist, 1793–1841.
- 6.
Siméon Denis Poisson, French mathematician, engineer, and physicist, 1781–1840.
- 7.
Hermann Weyl, German mathematician, theoretical physicist, and philosopher, 1885–1955.
- 8.
Jean-Baptiste le Rond d’Alembert, French mathematician, mechanician, physicist, philosopher, and music theorist, 1717–1783.
- 9.
Sergei, L. Sobolev, Russian mathematician, 1908–1989.
- 10.
Vladimir I. Kondrashov, Russian mathematician, 1909–1971; Franz Relich, Austrian-German mathematician, 1906–1955.
- 11.
Mikhail V. Ostrogradski, Russian-Ukrainian mathematician, mechanician, and physicist, 1801–1862.
- 12.
Henri Poincaré, French mathematician, theoretical physicist, engineer, and philosopher of science, 1854–1912.
- 13.
Salomon Bochner, American mathematician, 1899–1982.
- 14.
Billy James Pettis, American mathematician, 1913–1979.
- 15.
Yukio Kōmura, Japanese mathematician, born 1931.
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Moroşanu, G. (2019). Distributions, Sobolev Spaces. In: Functional Analysis for the Applied Sciences. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-030-27153-4_5
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