One-dimensional Variational Problems: An IntroductionOne-dimensional variational problems are often neglected in favor of problems which use multiple integrals and partial differential equations, which are typically more difficult to handle. However, these problems and their associated ordinary differential equations do exhibit many of the same challenges and complexity of higher-dimensional problems, while being accessible to more students. This book for graduate students provides the first modern introduction to this subject. It emphasizes direct methods and provides an exceptionally clear view of the underlying theory. Except for standard material on measures, integration and convex functions, the book develops all of the necessary mathematical tools, including basic results for one-dimensional Sobolev spaces, absolutely continuous functions, and functions of bounded variation. |
Contents
Section 1 | 10 |
Section 2 | 10 |
Section 3 | 44 |
Section 4 | 46 |
Section 5 | 49 |
Section 6 | 50 |
Section 7 | 54 |
Section 8 | 82 |
Section 9 | 83 |
Section 10 | 104 |
Section 11 | 134 |
Section 12 | 156 |
Section 13 | 225 |
246 | |
Common terms and phrases
absolutely continuous functions assume assumptions Banach space Borel boundary conditions boundary values bounded variation Cı(I calculus of variations class C² classical Co(I compact consider converges weakly convex curve deduce defined denotes differential equations direct methods Dirichlet eigenfunction eigenvalue equibounded equivalent Euler equation F(uk F(un field on G finite Fp(x functional F Hıı Hamiltonian hence implies inequality integrand interval Lı(a Lagrangian Lagrangian F(x Lavrentiev phenomenon Lebesgue Lemma lim inf linear Lipschitz Lipschitz-continuous lower semicontinuous Math Mayer field meas measure minimizer of F minimizing sequence minimum problem Moreover obtain periodic solutions Proof Proposition prove satisfies Section semicontinuous with respect sequentially lower semicontinuous Sobolev spaces Springer Suppose Theorem theory Tonelli uniformly variational integral variational problems weak convergence weak derivative whence y₁ zero