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Book 6C
Nonlinear PDE: Variational and Geometric Methods
How nonlinear equations are solved
Regularity of minimisers, variational inequalities, the mountain pass, Noether’s theorem and moving planes, the Yamabe problem and critical exponents, fixed points and monotone operators, minimal surfaces, harmonic maps and their heat flow, monotonicity and ε-regularity, viscosity solutions, Monge–Ampère, optimal transport and Ricci curvature, and quasilinear parabolic systems. Written in full detail, to learn from directly.
- Main companionPartial Differential Equations · Lawrence C. Evans
- Main companionPartial Differential Equations III: Nonlinear Equations · Michael E. Taylor
- Second voicePartial Differential Equations · Jürgen Jost
Contents
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1
Regularity of Minimisers and Constraints
Second-derivative estimates for minimisers, nonlinear eigenvalue problems, and the obstacle problem.
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2
Critical Points and the Mountain Pass
Palais–Smale, the deformation lemma, the mountain pass theorem and Pohozaev’s identity.
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3
Symmetry, Noether’s Theorem and Moving Planes
Invariance gives conservation laws, and maximum principles give symmetry.
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4
The Yamabe Problem and Critical Exponents
Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.
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5
Fixed Points, Monotonicity and Sub- and Supersolutions
Schauder and Schaefer, Leray–Schauder a priori estimates, Minty–Browder and monotone iteration.
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6
Quasilinear Elliptic Equations and Minimal Surfaces
Gradient bounds, the Dirichlet problem for quasilinear equations, and the minimal surface equation.
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7
Harmonic Maps and Their Heat Flow
The Bochner formula for maps, Eells–Sampson, and bubbling in two dimensions.
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8
Monotonicity, ε-Regularity and Tangent Cones
Monotonicity formulas for minimal surfaces and harmonic maps, small energy implies smoothness, bubble trees, tangent cones and dimension reduction, with soap films, liquid crystals and pseudolocality.
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9
Hamilton–Jacobi Equations and Viscosity Solutions
Hopf–Lax, viscosity solutions, comparison by doubling variables, and optimal control.
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10
Fully Nonlinear Equations and Monge–Ampère
Concavity, Krylov–Safonov, Evans–Krylov, and the Monge–Ampère equation from optimal transport to Kähler geometry.
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11
Optimal Transport and Ricci Curvature
Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.
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12
Quasilinear Parabolic Systems
Short-time existence for quasilinear parabolic systems in full, and why the Ricci flow needs DeTurck.
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