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Global Solution Curves for Semilinear Elliptic Equations.
This book provides an introduction to the bifurcation theory approach to global solution curves and studies the exact multiplicity of solutions for semilinear Dirichlet problems, aiming to obtain a complete understanding of the solution set. This understanding opens the way to efficient computation...
| Clasificación: | Libro Electrónico |
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| Autor principal: | |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Singapore :
World Scientific,
2012.
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| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- Preface; Contents; 1. Curves of Solutions on General Domains; 1.1 Continuation of solutions; 1.2 Symmetric domains in R2; 1.3 Turning points and the Morse index; 1.4 Convex domains in R2; 1.5 Pohozaev's identity and non-existence of solutions for elliptic systems; 1.5.1 Non-existence of solutions in the presence of supercritical and lower order terms; 1.5.2 Non-existence of solutions for a class of systems; 1.5.3 Pohozhaev's identity for a version of p-Laplace equation; 1.6 Problems at resonance; 2. Curves of Solutions on Balls; 2.1 Preliminary results.
- 2.2 Positivity of solution to the linearized problem2.3 Uniqueness of the solution curve; 2.4 Direction of a turn and exact multiplicity; 2.5 On a class of concave-convex equations; 2.6 Monotone separation of graphs; 2.7 The case of polynomial f(u) in two dimensions; 2.8 The case when f(0) <0; 2.9 Symmetry breaking; 2.10 Special equations; 2.11 Oscillations of the solution curve; 2.11.1 Asymptotics of some oscillatory integrals; 2.11.2 Reduction to the oscillatory integrals; 2.12 Uniqueness for non-autonomous problems; 2.12.1 Radial symmetry for the linearized equation.
- 2.13 Exact multiplicity for non-autonomous problems2.14 Numerical computation of solutions; 2.14.1 Using power series approximation; 2.14.2 Application to singular solutions; 2.15 Radial solutions of Neumann problem; 2.15.1 A computer assisted study of ground state solutions; 2.16 Global solution curves for a class of elliptic systems; 2.16.1 Preliminary results; 2.16.2 Global solution curves for Hamiltonian systems; 2.16.3 A class of special systems; 2.17 The case of a "thin" annulus; 2.18 A class of p-Laplace problems; 3. Two Point Boundary Value Problems.
- 3.1 Positive solutions of autonomous problems3.2 Direction of the turn; 3.3 Stability and instability of solutions; 3.3.1 S-shaped curves of combustion theory; 3.3.2 An extension of the stability condition; 3.4 S-shaped solution curves; 3.5 Computing the location and the direction of bifurcation; 3.5.1 Sign changing solutions; 3.6 A class of symmetric nonlinearities; 3.7 General nonlinearities; 3.8 Infinitely many curves with pitchfork bifurcation; 3.9 An oscillatory bifurcation from zero: A model example; 3.10 Exact multiplicity for Hamiltonian systems; 3.11 Clamped elastic beam equation.
- 3.11.1 Preliminary results3.11.2 Exact multiplicity of solutions; 3.12 Steady states of convective equations; 3.13 Quasilinear boundary value problems; 3.13.1 Numerical computations for the prescribed mean curvature equation; 3.14 The time map for quasilinear equations; 3.15 Uniqueness for a p-Laplace case; Bibliography.