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Extra resources for A behavior of generalized solutions of the Dirichlet problem for quasilienar elliptic divergence equations of second order near a conical point

Example text

For y ∈ Ve , we let y ∗ denote its reflection through ∂Ve . Then it is easy to check that ad1 (y) + bd2 (y) ≤ bd1 (y) + ad2 (y) = bd2 (y ∗ ) + ad1 (y ∗ ). By symmetry and the above inequality it follows that Rn |f1 | y + |z|e 2 |f2 | y − |z|e 2 e−(ad1 (y)+bd2 (y)) dy. dy ≤ 2 Ve We also define the three sets (which are respectively two strips and a half-space) Ve,1 = y ∈ Rn : − |z| ≤ y, e ≤ 0 ; 2 Ve,2 = y ∈ Rn : −|z| ≤ y, e ≤ − Ve,3 = {y ∈ Rn : y, e ≤ −|z|} . 46 |z| 2 ; Still by symmetry we have e−(ad1 (y)+bd2 (y)) dy ≤ Ve ,2 e−(ad1 (y)+bd2 (y)) , Ve ,1 and moreover one also finds e−(ad1 (y)+bd2 (y)) ≤ Ce−( 2 b+ 2 a)|z| dy, 3 1 Ve ,2 where the constant C depends on a and b.

A similar bound can be derived for A2K,m , . . 4: the term A6K,m indeed vanishes identically by oddness. 6 we obtain the assertion. 2 Derivatives of the coefficients αI,j for a general configuration In this subsection (modifying the arguments in the previous one) we consider the derivatives of the αI ’s for a general configuration of points satisfying (14) and (13). 7. We do not prove each of them since the methods are quite similar, but just list the necessary modifications. 8 Suppose (ya )a and (xI )I satisfy (13) and (14) respectively.

It then follows that (B k )IJ ≤ Cθ3k 0 |xI − xJ |k−1 + (k − 1)|xI − xJ |k−2 + (k − 1)(k − 2)|xI − xJ |k−3 + . . Lk−1 (k − 1)! ) e−|xI −xJ | . Now summing over k and rearranging the terms we obtain the expression for the Neumann series of B ∞ (B k )IJ = BIJ + (B 2 )IJ + (B 3 )IJ + (B 4 )IJ + . . L3 Cθ150 −|xI −xJ | e |xI − xJ |4 + 4|xI − xJ |3 + 12|xI − xJ |2 + 24|xI − xJ | + 24 + · · · . L4 50 Collecting all the terms with the same power of |xI − xJ | we can rewrite the series more conveniently as   2 3 ∞ 3 3 3 C C C θ0 θ0 (B k )IJ ≤ Cθ30 e−|xI −xJ | 1 + θ0 + + + ··· L L L k=1   2 3 3 3 3 3 C C C C |x − x | I J θ0 θ0 θ0 1 + θ0 + + Cθ30 e−|xI −xJ | + + ··· L L L L   2 2 3 3 3 3 3 C C C C |x − x | 1 I J θ0 θ0 θ0 1 + θ0 + + + ··· + Cθ30 e−|xI −xJ | 2 L L L L   3 2 3 3 3 3 3 C C |x − x | C C 1 I J θ0 θ0 θ0 1 + θ 0 + + Cθ30 e−|xI −xJ | + + ··· + ··· 3!