Geodesic
$L^p$Regularity of Transmission Problems in Dihedral Domains
Bollettino della Unione matematica italiana, Série 8, 10B (2007) no. 3, pp. 633-660.

Voir la notice de l'article provenant de la source Biblioteca Digitale Italiana di Matematica

We consider the transmission problem for the Laplace operator in a straight cylinder with data in $L^p$. Applying the theory of the sums of operators in Banach spaces, we prove that the solution admits a decomposition into a regular part in $W^{2,p}$ and an explicit singular part.
Consideriamo il problema di trasmissione per l'operatore di Laplace per un cilindro retto con dati in $L^p$. Applicando la teoria delle somme di operatori negli spazi di Banach, dimostriamo che la soluzione ammette una decomposizione in una parte regolare in $W^{2,p}$ e una parte singolare esplicita. 
@article{BUMI_2007_8_10B_3_a11,
     author = {Aibeche, Aissa and Chikouche, Wided and Nicaise, Serge},
     title = {$L^p${Regularity} of {Transmission} {Problems} in {Dihedral} {Domains}},
     journal = {Bollettino della Unione matematica italiana},
     pages = {633--660},
     publisher = {mathdoc},
     volume = {Ser. 8, 10B},
     number = {3},
     year = {2007},
     zbl = {1178.47005},
     mrnumber = {2351535},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/BUMI_2007_8_10B_3_a11/}
}
TY  - JOUR
AU  - Aibeche, Aissa
AU  - Chikouche, Wided
AU  - Nicaise, Serge
TI  - $L^p$Regularity of Transmission Problems in Dihedral Domains
JO  - Bollettino della Unione matematica italiana
PY  - 2007
SP  - 633
EP  - 660
VL  - 10B
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/BUMI_2007_8_10B_3_a11/
LA  - en
ID  - BUMI_2007_8_10B_3_a11
ER  - 
%0 Journal Article
%A Aibeche, Aissa
%A Chikouche, Wided
%A Nicaise, Serge
%T $L^p$Regularity of Transmission Problems in Dihedral Domains
%J Bollettino della Unione matematica italiana
%D 2007
%P 633-660
%V 10B
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/BUMI_2007_8_10B_3_a11/
%G en
%F BUMI_2007_8_10B_3_a11
Aibeche, Aissa; Chikouche, Wided; Nicaise, Serge. $L^p$Regularity of Transmission Problems in Dihedral Domains. Bollettino della Unione matematica italiana, Série 8, 10B (2007) no. 3, pp. 633-660. http://geodesic.mathdoc.fr/item/BUMI_2007_8_10B_3_a11/

[1] J. O. Adeyeye, Generation of analytic semi-groups in $L_p(\Omega)$ by the Laplace operator, Bollettino UMI, Analisi Funzionale e Applicazioni, Serie VI, Vol. IV-C, no. 1 (1985), 113-128. | MR | Zbl

[2] S. Agmon, On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems, Comm. Pure Appl. Math., 15 (1962), 119-147. | DOI | MR | Zbl

[3] F. E. Browder, On the spectral theory of elliptic differential operators, I, Math. Ann., 142 (1960/61), 22-130. | fulltext EuDML | DOI | MR

[4] D. L. Burkholder, A geometrical characterisation of Banach spaces in which martingale difference sequences are unconditional, Ann. Probab., 9, (1981), 997-1011. | MR | Zbl

[5] P. Clément - P. Grisvard, Somme d'opérateurs et régularité $L^p$ dans les problémes aux limites, C. R. Acad. Sci., Paris, 314 (1992), 821-824. | MR | Zbl

[6] R. R. Coifman - G. Weiss, Transference methods in analysis, C.B.M.S. regional conference series in Math., 31, A.M.S., Providence, Rhode Island, 1977. | MR | Zbl

[7] S. Companato, Generation of analytic semi-groups by elliptic operators of second order in Hölder spaces, Ann. Scuola Nor. Sup. Pisa, 8 (1981), 495-512. | fulltext EuDML | MR

[8] G. Da Prato - P. Grisvard, Somme d'opérateurs linéaires et équations différentielles opérationnelles, J. Math. Pures Appl. IX Ser., 54 (1975). | MR | Zbl

[9] G. Dore - A. Venni, On the closedness of the sum of two closed operators, Mathematische Zeitschrift, 196 (1987), 270-286. | fulltext EuDML | DOI | MR | Zbl

[10] A. Favini - R. Labbas - K. Lemrabet - S. Maingot, Study of the limit of transmission problems in a thin layer by the sum theory of linear operators, Rev. Mat. Complut., 18 (2005), 143-176. | fulltext EuDML | MR | Zbl

[11] P. Grisvard, Equations différentielles abstraites, Annales de L'ENS, série 4, tome 2 (1969), 311-395. | fulltext EuDML | MR | Zbl

[12] P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics 24, Pitman, Boston, 1985. | MR | Zbl

[13] P. Grisvard, Edge behaviour of the solution of an elliptic problem, Math. Nachr., 132 (1987), 281-299. | DOI | MR | Zbl

[14] P. Grisvard, Problèmes aux limites dans les polygones, Mode d'emploi. EDF, Bulletin de la direction des études et des recherches, Série C, Mathématiques informatique, 1 (1989), 21-59. | MR

[15] P. Grisvard, Singularities in boundary value problems, Research notes in appl. Maths, 22, Masson Springer-Verlag, 1992. | MR | Zbl

[16] P. Grisvard, Singular behaviour of elliptic problems in non Hilbertian Sobolev spaces, J. Math. Pures Appl., 74 (1995), 3-33. | MR | Zbl

[17] B. Heinrich - S. Nicaise - B. Weber, Elliptic interface problems in axisymmetric domains. Part I: Singular functions of non-tensorial type, Math. Nachr, 186 (1997), 147-165. | DOI | MR | Zbl

[18] R. Labbas - M. Moussaoui - M. Najmi, Singular behavior of the Dirichlet problem in Hölder spaces of the solutions to the Dirichlet problem in a cone, Rend. Ist. Mat. Univ. Trieste, 30 (1998), 155-179. | MR | Zbl

[19] C. Martinez Carracedo - M. Sanz Alix, The Theory of Fractional Powers of Operators, Saul Lubkin, North-Holland, 2001. | MR | Zbl

[20] M. Moussaoui, On the Resolution of the Heat Equation with Discontinuous Coefficients, Semigroup Forum, 60 (2000), 187-201. | DOI | MR | Zbl

[21] S. Nicaise, Polygonal interface problems, Peter Lang, Berlin, 1993. | MR

[22] A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Math. Sciences, 44, Springer-Verlag, New York, 1983. | DOI | MR | Zbl

[23] P. E. Sobolevskii, Equation of Parabolic type in Banach space, Amer. Math. Soc. Transl., 2 (1965), 1-62.

[24] H. B. Stewart, Generation of analytic semi-groups by strongly elliptic operators, Amer. Math. Soc. Transl., 199 (1974), 141-162. | DOI | MR | Zbl