By Abdelhamid Meziani

This paper offers with elliptic equations within the aircraft with degeneracies. The equations are generated via a fancy vector box that's elliptic all over the place other than alongside an easy closed curve. Kernels for those equations are developed. homes of options, in an area of the degeneracy curve, are got via quintessential and sequence representations. An software to a moment order elliptic equation with a punctual singularity is given

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1 Now we estimate the functions ψ2 and φ2 that are defined in the previous lemmas. 6. 37) ψ2 (t) = φ1 i(1 + eix ) j with ψ4 satisfying |ψ4 (t)| ≤ K for every t ∈ R. 3. 1 35 Proof. 14) and obtain φ2 (t) + c1 (t)ψ2 (t) j (0) ψ2 (t) = −iEj ψ2 (t) + c1 (t)(jφ1 + φ2 (t)). 39) (l) (0) ψ2 = φ1 j (c1 φ2 )(l) , c1 (l) + i(l + Ej ) i(l + Ej ) l ∈ Z. 40) c1 (t) . 41) (l) + eix )j − (l + Ej ) (l) (c1 φ2 )(l) , c1 + i(l + Ej )(1 + eix ) i(l + Ej ) (0) (1 ψ3 = φ1 l ∈ Z. 42) (l) |ψ3 | ≤ K (0) |φ1 lc1 (l) | + |(c1 φ2 )(l) | .

Functions f (r, t) and g(ρ, θ) will be denoted by f (z) and g(ζ), where z = r λ eit and ζ = ρλ eiθ . 3) Ω2 (z, ζ) = ⎧ 1 ∗± ⎪ wj± (z)w−j (ζ) ⎪ ⎪ ⎪ ⎨ 2 Re(σ± )≥0 if r < ρ 1 ⎪ ⎪ − ⎪ ⎪ ⎩ 2 if r > ρ. 1) and where log denotes the principal branch of the logarithm in C\R− . In the next theorem, we will use the notation Δ1 = {(r, t, ρ, θ); 0 < r ≤ ρ}, Δ2 = {(r, t, ρ, θ); 0 < ρ ≤ r} and Int(Δ1 ), Int(Δ2 ) will denote their interiors. 37 38 5. 1. 5) ζ + iK(t, θ)L(z, ζ) ζ −z λν λν c(θ) r L(z, ζ) − L(z, ζ) 2a ρ are in C 1 (Δ1 ) ∪ C 1 (Δ2 ), meaning that the restrictions of C1,2 to Int(Δ1 ) (or to Int(Δ2 )) extend as C 1 functions to Δ1 (or Δ2 ).

8) = eij(t−θ) 1 + i K(t, θ) + O(j −2 ) j c(t) ij(t−θ) e + O(j −2 ) 2aj c(θ) ij(t−θ) e = −i + O(j −2 ) 2aj = O(j −2 ). 8). 1. 3. 2), consider the function fj (t) = tσj − tλ(j+ν) , 0 < t < 1. Then there are J0 > 0 and C > 0 such that |fj (t)| ≤ C , j2 ∀t ∈ (0, 1), j ≥ J0 . Proof. 2) and λ = a + ib (a > 0), we write σj = a(j + ν) + α 2β + i b(j + ν) + 2 j j where α > 0 and β ∈ R, depend on j, but are bounded. Hence, fj (t) = ta(j+ν)+(α/j) ti[b(j+ν)+2(β/j 2 )] − ta(j+ν) tib(j+ν) . We decompose fj as fj = gj + hj with 2 )) tα/j − 1 gj (t) = ta(j+ν) ti(b(j+ν)+2(β/j hj (t) = ta(j+ν) tib(j+ν) t2iβ/j − 1 .

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