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| Preface | |
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| Introduction | |
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| Mathematical Modelling | |
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| Partial Differential Equations | |
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| Well Posed Problems | |
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| Basic Notations and Facts | |
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| Smooth and Lipschitz Domains | |
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| Integration by Parts Formulas | |
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| Diffusion | |
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| The Diffusion Equation | |
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| Introduction | |
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| The conduction of heat | |
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| Well posed problems (n = 1) | |
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| A solution by separation of variables | |
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| Problems in dimension n > 1 | |
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| Uniqueness | |
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| Integral method | |
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| Maximum principles | |
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| The Fundamental Solution | |
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| Invariant transformations | |
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| Fundamental solution (n = 1) | |
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| The Dirac distribution | |
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| Fundamental solution (n > 1) | |
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| Symmetric Random Walk (n = 1) | |
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| Preliminary computations | |
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| The limit transition probability | |
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| From random walk to Brownian motion | |
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| Diffusion, Drift and Reaction | |
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| Random walk with drift | |
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| Pollution in a channel | |
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| Random walk with drift and reaction | |
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| Multidimensional Random Walk | |
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| The symmetric case | |
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| Walks with drift and reaction | |
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| An Example of Reaction-Diffusion (n = 3) | |
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| The Global Cauchy Problem (n = 1) | |
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| The homogeneous case | |
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| Existence of a solution | |
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| The non homogeneous case. Duhamel's method | |
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| Maximum principles and uniqueness | |
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| An Application to Finance | |
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| European options | |
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| An evolution model for the price S | |
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| The Black-Scholes equation | |
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| The solutions | |
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| Hedging and self-financing strategy | |
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| Some Nonlinear Aspects | |
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| Nonlinear diffusion. The porous medium equation | |
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| Nonlinear reaction. Fischer's equation | |
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| Problems | |
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| The Laplace Equation | |
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| Introduction | |
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| Well Posed Problems. Uniqueness | |
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| Harmonic Functions | |
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| Discrete harmonic functions | |
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| Mean value properties | |
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| Maximum principles | |
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| The Dirichlet problem in a circle. Poisson's formula | |
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| Harnack's inequality and Liouville's theorem | |
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| A probabilistic solution of the Dirichlet problem | |
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| Recurrence and Brownian motion | |
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| Fundamental Solution and Newtonian Potential | |
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| The fundamental solution | |
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| The Newtonian potential | |
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| A divergence-curl system. Helmholtz decomposition formula | |
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| The Green Function | |
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| An integral identity | |
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| The Green function | |
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| Green's representation formula | |
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| The Neumann function | |
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| Uniqueness in Unbounded Domains | |
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| Exterior problems | |
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| Surface Potentials | |
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| The double and single layer potentials | |
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| The integral equations of potential theory | |
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| Problems | |
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| Scalar Conservation Laws and First Order Equations | |
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| Introduction | |
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| Linear Transport Equation | |
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| Pollution in a channel | |
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| Distributed source | |
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| Decay and localized source | |
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| Inflow and outflow characteristics. A stability estimate | |
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| Traffic Dynamics | |
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| A macroscopic model | |
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| The method of characteristics | |
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| The green light problem | |
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| Traffic jam ahead | |
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| Integral (or Weak) Solutions | |
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| The method of characteristics revisited | |
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| Definition of integral solution | |
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| The Rankine-Hugoniot condition | |
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| The entropy condition | |
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| The Riemann problem | |
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| Vanishing viscosity method | |
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| The viscous Burger equation | |
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| The Method of Characteristics for Quasilinear Equations | |
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| Characteristics | |
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| The Cauchy problem | |
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| Lagrange method of first integrals | |
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| Underground flow | |
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| General First Order Equations | |
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| Characteristic strips | |
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| The Cauchy Problem | |
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| Problems | |
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| Waves and Vibrations | |
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| General Concepts | |
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| Types of waves | |
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| Group velocity and dispersion relation | |
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| Transversal Waves in a String | |
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| The model | |
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| Energy | |
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| The One-dimensional Wave Equation | |
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| Initial and boundary conditions | |
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| Separation of variables | |
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| The d'Alembert Formula | |
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| The homogeneous equation | |
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| Generalized solutions and propagation of singularities | |
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| The fundamental solution | |
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| Non homogeneous equation. Duhamel's method | |
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| Dissipation and dispersion | |
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| Second Order Linear Equations | |
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| Classification | |
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| Characteristics and canonical form | |
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| Hyperbolic Systems with Constant Coefficients | |
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| The Multi-dimensional Wave Equation (n > 1) | |
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| Special solutions | |
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| Well posed problems. Uniqueness | |
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| Two Classical Models | |
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| Small vibrations of an elastic membrane | |
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| Small amplitude sound waves | |
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| The Cauchy Problem | |
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| Fundamental solution (n = 3) and strong Huygens' principle | |
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| The Kirchhoff formula | |
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| Cauchy problem in dimension 2 | |
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| Non homogeneous equation. Retarded potentials | |
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| Linear Water Waves | |
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| A model for surface waves | |
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| Dimensionless formulation and linearization | |
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| Deep water waves | |
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| Interpretation of the solution | |
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| Asymptotic behavior | |
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| The method of stationary phase | |
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| Problems | |
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| Elements of Functional Analysis | |
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| Motivations | |
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| Norms and Banach Spaces | |
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| Hilbert Spaces | |
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| Projections and Bases | |
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| Projections | |
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| Bases | |
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| Linear Operators and Duality | |
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| Linear operators | |
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| Functionals and dual space | |
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| The adjoint of a bounded operator | |
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| Abstract Variational Problems | |
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| Bilinear forms and the Lax-Milgram Theorem | |
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| Minimization of quadratic functionals | |
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| Approximation and Galerkin method | |
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| Compactness and Weak Convergence | |
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| Compactness | |
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| Weak convergence and compactness | |
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| Compact operators | |
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| The Fredholm Alternative | |
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| Solvability for abstract variational problems | |
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| Fredholm's Alternative | |
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| Spectral Theory for Symmetric Bilinear Forms | |
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| Spectrum of a matrix | |
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| Separation of variables revisited | |
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| Spectrum of a compact self-adjoint operator | |
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| Application to abstract variational problems | |
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| Problems | |
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| Distributions and Sobolev Spaces | |
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| Distributions. Preliminary Ideas | |
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| Test Functions and Mollifiers | |
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| Distributions | |
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| Calculus | |
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| The derivative in the sense of distributions | |
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| Gradient, divergence, lapacian | |
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| Multiplication, Composition, Division, Convolution | |
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| Multiplication. Leibniz rule | |
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| Composition | |
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| Division | |
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| Convolution | |
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| Fourier Transform | |
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| Tempered distributions | |
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| Fourier transform in S' | |
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| Fourier transform in L[superscript 2] | |
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| Sobolev Spaces | |
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| An abstract construction | |
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| The space H[superscript 1] ([Omega]) | |
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| The space H[superscript 1 subscript 0] ([Omega]) | |
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| The dual of H[superscript 1 subscript 0]([Omega]) | |
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| The spaces H[superscript m] ([Omega]), m > 1 | |
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| Calculus rules | |
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| Fourier Transform and Sobolev Spaces | |
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| Approximations by Smooth Functions and Extensions | |
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| Local approximations | |
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| Estensions and global approximations | |
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| Traces | |
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| Traces of functions in H[superscript 1] ([Omega]) | |
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| Traces of functions in H[superscript m] ([Omega]) | |
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| Trace spaces | |
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| Compactness and Embeddings | |
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| Rellich's theorem | |
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| Poincare's inequalities | |
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| Sobolev inequality in R[superscript n] | |
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| Bounded domains | |
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| Spaces Involving Time | |
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| Functions with values in Hilbert spaces | |
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| Sobolev spaces involving time | |
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| Problems | |
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| Variational Formulation of Elliptic Problems | |
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| Elliptic Equations | |
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| The Poisson Problem | |
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| Diffusion, Drift and Reaction (n = 1) | |
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| The problem | |
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| Dirichlet conditions | |
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| Neumann, Robin and mixed conditions | |
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| Variational Formulation of Poisson's Problem | |
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| Dirichlet problem | |
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| Neumann, Robin and mixed problems | |
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| Eigenvalues of the Laplace operator | |
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| An asymptotic stability result | |
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| General Equations in Divergence Form | |
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| Basic assumptions | |
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| Dirichlet problem | |
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| Neumann problem | |
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| Robin and mixed problems | |
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| Weak Maximum Principles | |
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| Regularity | |
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| Equilibrium of a plate | |
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| A Monotone Iteration Scheme for Semilinear Equations | |
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| A Control Problem | |
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| Structure of the problem | |
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| Existence and uniqueness of an optimal pair | |
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| Lagrange multipliers and optimality conditions | |
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| An iterative algorithm | |
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| Problems | |
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| Weak Formulation of Evolution Problems | |
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| Parabolic Equations | |
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| Diffusion Equation | |
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| The Cauchy-Dirichlet problem | |
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| Faedo-Galerkin method (I) | |
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| Solution of the approximate problem | |
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| Energy estimates | |
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| Existence, uniqueness and stability | |
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| Regularity | |
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| The Cauchy-Neuman problem | |
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| Cauchy-Robin and mixed problems | |
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| A control problem | |
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| General Equations | |
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| Weak formulation of initial value problems | |
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| Faedo-Galerkin method (II) | |
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| The Wave Equation | |
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| Hyperbolic Equations | |
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| The Cauchy-Dirichlet problem | |
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| Faedo-Galerkin method (III) | |
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| Solution of the approximate problem | |
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| Energy estimates | |
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| Existence, uniqueness and stability | |
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| Problems | |
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| Fourier Series | |
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| Fourier coefficients | |
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| Expansion in Fourier series | |
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| Measures and Integrals | |
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| Lebesgue Measure and Integral | |
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| A counting problem | |
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| Measures and measurable functions | |
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| The Lebesgue integral | |
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| Some fundamental theorems | |
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| Probability spaces, random variables and their integrals | |
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| Identities and Formulas | |
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| Gradient, Divergence, Curl, Laplacian | |
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| Formulas | |
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| References | |
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| Index | |