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AuthorStruwe, Michael. author
TitleVariational Methods [electronic resource] : Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems / by Michael Struwe
ImprintBerlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 1990
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Descript XIV, 244 p. online resource


It would be hopeless to attempt to give a complete account of the history of the calculus of variations. The interest of Greek philosophers in isoperimetric problems underscores the importance of "optimal form" in ancient cultures, see Hildebrandt-Tromba [1] for a beautiful treatise of this subject. While variatioยญ nal problems thus are part of our classical cultural heritage, the first modern treatment of a variational problem is attributed to Fermat (see Goldstine [1; p.l]). Postulating that light follows a path of least possible time, in 1662 Ferยญ mat was able to derive the laws of refraction, thereby using methods which may already be termed analytic. With the development of the Calculus by Newton and Leibniz, the basis was laid for a more systematic development of the calculus of variations. The brothers Johann and Jakob Bernoulli and Johann's student Leonhard Euler, all from the city of Basel in Switzerland, were to become the "founding fathers" (Hildebrandt-Tromba [1; p.21]) of this new discipline. In 1743 Euler [1] subยญ mitted "A method for finding curves enjoying certain maximum or minimum properties", published 1744, the first textbook on the calculus of variations


I. The Direct Methods in the Calculus of Variations -- II. Minimax Methods -- III. Limit Cases of the Palais-Smale Condition -- Appendix A -- Sobolev Spaces -- Hรถlder Spaces -- Imbedding Theorems -- Density Theorem -- Trace and Extension Theorems -- Poincarรฉ Inequality -- Appendix B -- Schauder Estimates -- Weak Solutions -- A Regularity Result -- Maximum Principle -- Weak Maximum Principle -- Application -- Appendix C -- Frรฉchet Differentiability -- Natural Growth Conditions -- References

Mathematics Mathematical analysis Analysis (Mathematics) System theory Calculus of variations Statistical physics Dynamical systems Mathematics Systems Theory Control Calculus of Variations and Optimal Control; Optimization Analysis Statistical Physics Dynamical Systems and Complexity


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