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. xvii

Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2013 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.