Section E.2 List of definitions and results
1 Maximum principles for ordinary differential equations
Lemma 1.1 Basic lemma for \(h\equiv 0\)
Theorem 1.2 One-dimensional strong maximum principle for \(h\equiv 0\)
Theorem 1.3 One-dimensional Hopf lemma for \(h \equiv 0\)
Theorem 1.4 One-dimensional strong maximum principle for \(h\le 0\)
Theorem 1.5 One-dimensional Hopf lemma for \(h\le 0\)
Theorem 1.6 Uniqueness for boundary value problems
Theorem 1.7 Sturm separation theorem
2 Preliminaries
Example 2.3 Dot products
Example 2.4 Matrix multiplication
Definition 2.5 Vector and matrix norms
Definition 2.7 Multiindices and polynomials
Theorem 2.16 Chain rule
Definition 2.18 Radial symmetry
Theorem 2.20 \(C^k(\overline\Omega)\) is Banach
Theorem 2.22 Weierstrass approximation theorem
Definition 2.23 Open balls
Definition 2.24 Boundary
Definition 2.25 Interior and exterior ball properties
Definition 2.26 Connected set
Definition 2.27 Orthogonality
Theorem 2.28 Spectral theorem for symmetric matrices
Corollary 2.29 Bounds on quadratic forms
Definition 2.30 Definite and semi-definite
Lemma 2.31 Inner product between semi-definite matrices
Definition 2.32 Maximum and minimum
Definition 2.33 Supremum and infimum
Proposition 2.34 First and second derivative tests
Proposition 2.35 Multivariate Taylor’s theorem
Example 2.38 Subscript nightmare
Example 2.39 Escaping the nightmare
Theorem 2.40 Existence of maxima and minima
Theorem 2.43 Arzelà–Ascoli
Lemma 2.44 Exhaustion by compact sets
Theorem 2.49 Divergence theorem for balls
Theorem 2.51 Integrating in spherical polar coordinates
3 Maximum principles for elliptic equations
Definition 3.1 Ellipticity
Theorem 3.2 Weak maximum principle
Proposition 3.4 Comparison principle for \(c \le 0\)
Corollary 3.5 Uniqueness of solutions for \(c\le 0\)
Definition 3.6 Sub- and supersolutions
Lemma 3.7 Hopf lemma for balls
Theorem 3.8 Strong maximum principle
Theorem 3.9 Hopf lemma
4 The Dirichlet problem
Theorem 4.10 Mean value property
Theorem 4.11 Converse to mean value property
Lemma 4.16 Operations on subharmonic functions
Theorem 4.17 Existence and uniqueness for the Dirichlet problem
5 Radial symmetry
Theorem 5.1 Gidas–Ni–Nirenberg
6 Maximum principles for parabolic equations
Theorem 6.3 Weak maximum principle
Corollary 6.4 Comparison principle
Corollary 6.5 Uniqueness
Theorem 6.7 Comparison principle
