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Book 6B
Linear PDE: Characteristics, Waves and Regularity
The classical theory behind every estimate
First-order equations and shocks, the wave equation, representation formulas, Cauchy–Kovalevskaya, Perron’s method, Schauder and Lᵖ theory, De Giorgi–Nash–Moser, harmonic coordinates, linear parabolic theory, unique continuation and backward uniqueness, semigroups, hyperbolic systems, spectral asymptotics, and eigenvalue problems up to Perelman’s λ. Written in full detail, to learn from directly.
- Main companionPartial Differential Equations · Lawrence C. Evans
- Second voicePartial Differential Equations · Jürgen Jost
- ReferencePartial Differential Equations I: Basic Theory · Michael E. Taylor
- ReferencePartial Differential Equations II: Qualitative Studies of Linear Equations · Michael E. Taylor
Contents
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1
First-Order Equations and Characteristics
Transport, the method of characteristics for nonlinear first-order equations, and where classical solutions stop.
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2
Conservation Laws and Shocks
Burgers’ equation, shocks, the entropy condition, rarefactions, the Riemann problem and traffic flow.
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3
The Wave Equation
d’Alembert, spherical means, Huygens’ principle, Duhamel, energy and finite speed of propagation.
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4
Green’s Functions, Separation of Variables and Transforms
Green’s functions and images, Poisson kernels, separation of variables, Bessel functions and the Laplace transform.
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5
Power Series and the Cauchy–Kovalevskaya Theorem
Noncharacteristic surfaces, majorants, the Cauchy–Kovalevskaya theorem, and Lewy’s unsolvable equation.
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6
Perron’s Method and Barriers
Subharmonic functions, Perron’s solution of the Dirichlet problem, barriers, regular points and Lebesgue’s spine.
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7
Schauder Theory and the Continuity Method
Hölder estimates for the Newtonian potential, interior and global Schauder estimates, and existence by continuity.
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8
Lᵖ Theory and Strong Solutions
Calderón–Zygmund estimates, strong solutions, and the Alexandrov–Bakelman–Pucci maximum principle.
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9
De Giorgi–Nash–Moser
Moser iteration, the Harnack inequality and Hölder continuity for equations with rough coefficients.
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10
Harmonic Coordinates and the Regularity of Metrics
Metrics as matrix-valued functions, the coordinates in which the Ricci tensor is elliptic, the DeTurck–Kazdan regularity theorem, harmonic radius, and Einstein's equations in wave gauge.
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11
Linear Parabolic Equations in Full
Galerkin existence, energy estimates, regularity in space and time, and parabolic Harnack inequalities.
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12
Unique Continuation and Backward Uniqueness
Why a solution that vanishes on an open set vanishes everywhere: Holmgren, log-convexity, frequency functions and Carleman estimates, with backward uniqueness for the heat equation and the Ricci flow.
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13
Semigroups and Evolution Equations
Contraction semigroups, the Hille–Yosida theorem, analytic semigroups and Duhamel’s formula.
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14
Hyperbolic Equations and Systems
Weak solutions of second-order hyperbolic equations, symmetric hyperbolic systems and energy estimates.
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15
Spectral Theory and Heat Trace Asymptotics
Self-adjoint operators, the heat trace, Weyl’s law, and what the spectrum hears.
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16
Eigenvalue Problems and Perelman’s λ
Rayleigh quotients and min–max, Faber–Krahn and Cheeger’s inequality, ground states of Schrödinger operators, spectral clustering, and the first eigenvalue of −4Δ + R that begins Perelman’s first preprint.
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