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Index Index

Arzelà–Ascoli, Theorem

average integrals, Notation

boundary, Definition

bounded, Definition

co-area formula, Theorem

compact subset, Definition

of \(\R^N\), Theorem

comparison principle

elliptic with \(c \le 0\), Proposition

parabolic

linear, Corollary

semilinear, Theorem

sub/superharmonic, Proposition

connected set, Definition

differential operator, Notation

Dirichlet problem, Chapter

existence and uniqueness, Theorem

divergence, Notation

divergence theorem, Theorem

elliptic equation

motivation, Paragraphs

ellipticity, Definition

ellipticity constant, Definition

equicontinuous, Definition

exhaustion by compact sets, Lemma

exterior ball property. See interior ball property

first derivative test, Proposition

Fréchet derivative, Notation

fundamental solution, Exercise

Gidas–Ni–Nirenberg, Theorem

gradient, Notation

harmonic, Definition

harmonic lifting, Definition

uniform convergence, Lemma Theorem

Hessian matrix, Notation

Hopf lemma, Theorem

one-dimensional

\(h=0\), Theorem

\(h\le 0\), Theorem

preliminary version, Lemma

hyperplane, Definition

infimum, Definition

interior ball property, Definition

Jacobian, Notation

Kronecker delta, Notation

Laplace operator, Notation

Laplacian, Notation

locally constant, Exercise

matrix, Notation

maximum, Definition

existence of, Theorem

global, Definition

local, Definition

maximum principle

one-dimensional

\(h\equiv0\), Theorem

\(h\le0\), Theorem

parabolic, Theorem

strong, Theorem

weak, Theorem

mean value property, Theorem

converse, Theorem

method of moving planes, Chapter

minimum. See maximum

minimum principles, Remark

one-dimensional, Paragraph

multiindex, Definition

partial derivative, Notation

norm

matrix, Definition

vector, Definition

normal derivative, Definition

normal vector, Definition

parabolic boundary, Paragraph

parabolic equation

linear, Paragraph

motivation, Paragraphs

semilinear, Paragraph

parabolic interior, Paragraph

parabolic operator, Paragraph

partial derivative, Notation

\(x\) versus \(t\), Notation

multiindex notation, Notation

named variables, Notation

second partials, Notation

partial differential equation (PDE)

definition, Definition

linear, Definition

homogeneous, Definition

linearisation, Exercise

order, Definition

semilinear, Definition

system of, Definition

polynomial, Definition

radial symmetry

definition, Definition

reflection, Definition

second derivative test, Proposition

subharmonic

definition for \(C^0\) functions, Definition

definition for \(C^2\) functions, Definition

subsolution, Definition

superharmonic. See subharmonic

supersolution, Definition

supremum, Definition

Taylor’s theorem, Proposition

unbounded domain, Exercise

uniformly bounded, Definition

uniformly elliptic, Definition

uniformly parabolic, Paragraph

uniqueness

elliptic equations, Corollary

parabolic equations, Corollary

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