By Fernando Casas, Vicente Martínez (eds.)

The ebook incorporates a choice of contributions given on the 23th Congress on Differential Equations and functions (CEDYA) / thirteenth Congress of utilized arithmetic (CMA) that happened at Castellon, Spain, in 2013. CEDYA is popular because the congress of the Spanish Society of utilized arithmetic (SEMA) and constitutes the most discussion board and assembly aspect for utilized mathematicians in Spain. The papers incorporated during this booklet were chosen after a radical refereeing strategy and supply an outstanding precis of the hot task constructed by means of diverse teams operating quite often in Spain on purposes of arithmetic to a number of fields of technological know-how and expertise. the aim is to supply an invaluable reference of educational and commercial researchers operating within the region of numerical research and its applications.

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0; T / RN : Á 8 x x 2 ˆ A. D f in QT /@ /r . RN //: in RN (1) Our work consists in describing the asymptotic behavior of the solutions of this problem when " tends to zero. g. Ary ˚/ D j ˚ in RN ˚ periodic of period Y; (3) and Wj the corresponding eigenfunctions space, with dimension kj . Ar Ary ˚jk ˚jl / dy; 1 Ä l; k Ä kj ; j 2 Z n f0g; (4) where AN is the conjugated matrix of A. g. [5, 9, 11]). RN //: (5) 26 J. Casado-Díaz et al. We observe that in the previous theorem, the matrix Ah is the same matrix which appears in the homogenization of the elliptic problem div A.

7) 2 /D 1 U for all in a sector S0;2 for > 0 arbitrarily small. Furthermore, . 2 / D Œ0; 1/. 1 in [4] that D. RN /. To prove part (i), observe that, as in page 32–33 in [5], we can obtain an p expression for the operator . C I / 1 , provided Re. / > 0, as a convolution operator. The expression is uD. C / 1f D f; Re. x/ D p N 2 G2 . p x/; x 2 RN ; Re. p />0 where G2 is the Green’s function for . C I /. Now observe that if 2 S0; with > 0 then for D 2 C n . 1; 0 we p q / > 0. RN / can choose Re.

T; x/˚jk . QT /. Ary / z dydtdx D 0: Y From this equation, we conclude that the functions zkj satisfy (12). Y /=R/; which gives the initial conditions (13) for the functions zkj . We have then proved that the function z which appears in convergences (14), (15) agrees with the function z defined in the statement of Theorem 2. From these convergences and assuming and Á (and then z) smooth enough, we can now pass to the limit in the energy identity corresponding to Eq. (10) to prove (13). t u Acknowledgements The authors have been partially supported by the project MTM2011-24457 of the “Ministerio de Economía y Competitividad” of Spain.