Index of notations
General notations
d refers to the surface dimension (and thus, d = 1 or 2).
|α| =
∑d
j=1
αj for all multi-index α
Nd.
Cst is a generic notation for a constant whose exact value is of no impor-
tance.
A B means A Cst B.
s (s R) denotes the smallest integer larger or equal to s.
A + B
sr
is equal to A if s r and equal to A + B if s r.
A B stands for max{A, B}.
c.c. means “complex conjugate”.
Ω denotes the fluid domain.
S is the flat strip S = {(X, z)
Rd × R, −1 z 0}.
Γ is the boundary of Ω. We sometimes write Γ = (ζ, b), where ζ and b are
the surface and bottom parametrizations.
Γ Ht0+2(Rd)2 is the set of all ζ and b such that infRd (1+ εζ βb) 0.
ψ† is an explicit smoothing extension of ψ on S (see Notation 2.28).
ψh is the “harmonic extension” of ψ (see Notation 2.33).
Matrices and vectors
ez denotes the unit (upward) vector in the vertical direction.
n always denotes the unit upward normal vector evaluated at the bound-
ary of the fluid domain.
I always stands for the identity matrix (its size is usually obvious from
the context).
Iμ,γ is the (d + 1) × (d + 1) diagonal matrix with diagonal (

μ, γ

μ, 1)
when d = 2 and (

μ, 1) when d = 1.
Pμ,γ(Σ): see Equation (2.13).
|v|, where v is a vector of
Rd+1
stands for the euclidean norm of v.
|M|, where M is a (d + 1) × (d + 1) matrix is the matricial norm subor-
dinated to the euclidean norm.
Variables and standard operators
X
Rd
always refers to the horizontal variables; we sometimes write
X = (x, y) or X = (x1,x2) when d = 2 and X = x or X = x1 when d = 1.
z always refers to the vertical variable (also sometimes denoted xd+1)
=
(∂x,∂y)T
when d = 2 and = ∂x when d = 1.
Δ = ∂x
2
+ ∂y
2
when d = 2 and Δ = ∂x
2
when d = 1.
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