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Many physical problems are meaningfully formulated in a cylindrical domain. When the size of the cylinder goes to infinity, the solutions, under certain symmetry conditions, are expected to be identical in every cross-section of the domain. The proof of this, however, is sometimes difficult and almost never given in the literature. The present book partially fills this gap by providing proofs of the asymptotic behaviour of solutions to various important cases of linear and nonlinear problems in the theory of elliptic and parabolic partial differential equations. The book is a valuable resource for graduates and researchers in applied mathematics and for engineers. Many results presented here are original and have not been published elsewhere. They will motivate and enable the reader to apply the theory to other problems in partial differential equations.
Content Level »Research
Keywords »Elliptic equations - Sobolev space - differential equation - mathematics - model - partial differential equation
1. Introduction to Linear Elliptic Problems.- 1.1. The Lax—Milgram theorem.- 1.2. Elementary notions on Sobolev spaces.- 1.3. Applications to linear elliptic problems.- 2. Some Model Techniques.- 2.1. The case of lateral Dirichlet boundary conditions on a rectangle.- 2.2. The case of lateral Neumann boundary conditions on a rectangle.- 2.3. The case of lateral Dirichlet boundary conditions revisited.- 2.4. A different point of view.- Open problems.- 3. A General Asymptotic Theory for Linear Elliptic Problems.- 3.1. A general convergence result in H1 (S24,).- 3.2. A sharper rate of convergence.- 3.3. Convergence in higher Sobolev spaces.- Open problems.- 4. Nonlinear Elliptic Problems.- 4.1. Variational inequalities.- 4.2. Quasilinear elliptic problems.- 4.3. Strongly nonlinear problems.- Open problems.- 5. Asymptotic Behaviour of some Nonlinear Elliptic Problems.- 5.1. The case of variational inequalities.- 5.2. The case of quasilinear problems.- Open problems.- 6. Elliptic Systems.- 6.1. Some inequalities.- 6.2. Existence results for linear elliptic systems.- 6.3. Nonlinear elliptic systems.- Open problems.- 7. Asymptotic Behaviour of Elliptic Systems.- 7.1. The case of linear elliptic systems satisfying the Legendre condition.- 7.2. The system of elasticity.- Open problems.- 8. Parabolic Equations.- 8.1. Functional spaces for parabolic problems.- 8.2. Linear parabolic problems.- 8.3. Nonlinear parabolic problems.- 9. Asymptotic Behaviour of Parabolic Problems.- 9.1. The linear case.- 9.2. A nonlinear case.- Open problems.- Concluding Remark.