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
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