Section E.1 List of exercises
1 Maximum principles for ordinary differential equations
Exercise 1.1.1 (PS1) Some basic counterexamples
Exercise 1.1.2 (PS1) A concrete application
Exercise 1.1.3 (PS1) Basic lemma for \(h \le 0\)
Exercise 1.1.4 Theorem 1.4
Exercise 1.1.5 Theorem 1.5
Exercise 1.1.6 (PS1) Alternative proof of Theorem 1.2
Exercise 1.2.1 Non-uniqueness
Exercise 1.2.2 (PS1) A ‘generalised’ maximum principle
2 Preliminaries
Exercise 2.1.1 (PS2) Summation convention
Exercise 2.2.1 (PS2) Derivatives of linear and quadratic functions
Exercise 2.2.2 (PS2) Derivative of the modulus
Exercise 2.3.1 (PS2) Derivatives after shifting and scaling
Exercise 2.3.2 (PS2) Derivatives of symmetric functions
Exercise 2.3.3 The Kelvin transform
Exercise 2.4.1 Series of \(C^k\) functions
Exercise 2.5.1 (PS3) A domain with a corner
Exercise 2.5.2 (PS3) Locally constant functions
Exercise 2.6.1 Lemma 2.31 for positive definite matrices
Exercise 2.6.2 (PS3) Invariance of the Laplacian
Exercise 2.6.3 The Laplacian and positive definiteness
Exercise 2.7.1 (PS3) Minimising a quadratic polynomial
Exercise 2.7.2 A \(C^2\) domain
Exercise 2.8.1 (PS3) Suprema, maxima, and closures
Exercise 2.8.2 (PS4) Sufficient condition for equicontinuity
Exercise 2.8.3 Non-existence of maxima and minima
Exercise 2.8.4 More diagonalisation
Exercise 2.8.5 Sequences of functions on \(\R\)
Exercise 2.8.6 Exhaustion by compact sets
Exercise 2.9.1 Divergence theorem and the Laplacian
Exercise 2.9.2 Dirichlet principle
Exercise 2.9.3 Non-constant coefficients
Exercise 2.9.4 Gradients and spheres
Exercise 2.10.1 Compositions and commutators
Exercise 2.10.2 Linearisation
3 Maximum principles for elliptic equations
Exercise 3.1.1 (PS4) Checking uniform ellipticity
Exercise 3.1.2 (PS4) Ellipticity after shifting and scaling
Exercise 3.1.3 Divergence form operators
Exercise 3.1.4 The symmetry assumption
Exercise 3.2.1 Non-uniqueness
Exercise 3.2.2 Subsolutions and supersolutions
Exercise 3.2.3 (PS4) A two-sided estimate
Exercise 3.2.4 (PS5) Domains contained in a slab
Exercise 3.3.1 (PS5) Normal derivatives of radial functions
Exercise 3.3.2 Minimum principles
Exercise 3.3.3 A strong comparison principle
Exercise 3.3.4 (PS5) Uniqueness for other boundary conditions
Exercise 3.3.5 Connectedness and the strong maximum principle
Exercise 3.3.6 (PS5) Maximum principles in the reverse order
Exercise 3.4.1 Weak maximum principle
Exercise 3.4.2 Uniqueness
Exercise 3.4.3 Strong maximum principle
Exercise 3.4.4 Hopf lemma
Exercise 3.4.5 Comparison
Exercise 3.4.6 Symmetry
4 The Dirichlet problem
Exercise 4.1.1 (PS6) A maximum principle for the gradient
Exercise 4.1.2 (PS6) Interior estimate for second derivatives
Exercise 4.1.3 Interior estimate for higher derivatives
Exercise 4.2.1 (PS6) Explicitly solving a Dirichlet problem
Exercise 4.2.2 (PS6) Harmonic functions are \(C^\infty\)
Exercise 4.2.3 Smooth but unbounded harmonic functions
Exercise 4.3.1 (PS7) Reflecting harmonic functions
Exercise 4.4.1 (PS7) Directly checking \(\abs x\) is subharmonic
Exercise 4.4.2 (PS7) Pointwise maximum of subharmonic functions
Exercise 4.4.3 (PS7) Harmonic liftings
Exercise 4.6.1 Getting a harmonic function
Exercise 4.6.2 Existence
Exercise 4.6.3 Extension
Exercise 4.6.4 Explicit formula
Exercise 4.6.5 Generalising
5 Radial symmetry
Exercise 5.1.1 (PS8) Necessity of \(u\gt0\) in Theorem 5.1
Exercise 5.1.2 (PS8) Proving \(u\gt0\)
Exercise 5.1.3 (PS8) Very old exam question
Exercise 5.1.4 (PS8) Laplacian of radially symmetric functions
Exercise 5.2.1 (PS9) Reflections are idempotent
Exercise 5.3.1 (PS9) Necessity of an assumption in Lemma 5.4
Exercise 5.4.1 A calculation needed in Lemma 5.6
Exercise 5.4.2 (PS9) Monotonicity in an annulus
Exercise 5.4.3 (PS9) Gidas–Ni–Nirenberg for an ellipse
6 Maximum principles for parabolic equations
Exercise 6.2.1 (PS10) Weak maximum principle for \(c\le0\)
Exercise 6.2.2 Asymptotics for the heat equation
Exercise 6.2.3 (PS10) Comparison principle
Exercise 6.3.1 (PS10) Uniqueness for semilinear problems
Exercise 6.3.2 (PS10) Bounding a solution to a semilinear problem
