Elliptic Problems with Robin Boundary Coefficient-Operator Conditions in General L_p Sobolev Spaces and Applications

В статье доказаны некоторые новые результаты о полных операторно-дифференциальных уравнениях эллиптического типа второго порядка с граничными операторно-коэффициентными условиями Робина в пространстве L^{p}(0,1;X) в случае, когда p in1,+infty), а X - банахово UMD-пространство. Доказано существование, единственность и оптимальная регулярность классического решения. Статья дополняет и завершает предыдущие исследования авторов по данной проблематике.

Ключевые слова

абстрактные эллиптические дифференциальные уравнения второго порядка; граничные условия Робина; аналитическая полугруппа.

Литература

1. Bourgain J. Some Remarks on Banach Spaces in which Martingale Difference Sequences are Unconditional. Ark. Mat., 1983, vol. 21, pp. 163-168. DOI: 10.1007/BF02384306
2. Burkholder D.L. A Geometrical Characterisation of Banach Spaces in which Martingale Difference Sequences are Unconditional. Ann. Probab., 1981, vol. 9, pp. 997-1011. DOI: 10.1214/aop/1176994270
3. Favini A., Labbas R., Maingot S., Tanabe H., Yagi A. Complete Abstract Differential Equations of Elliptic Type in UMD Spaces. Funkcialaj Ekvacioj, 2006, vol. 49, pp. 193-214. DOI: 10.1619/fesi.49.193
4. Favini A., Labbas R., Maingot S., Tanabe H., Yagi A. A Simplified Approach in the Study of Elliptic Differential Equations in UMD Spaces and New Applications. Funkcialaj Ekvacioj, 2008, vol. 51, pp. 165-187. DOI: 10.1619/fesi.51.165
5. El Haial A., Labbas R. On the Ellipticity and Solvability of Abstract Second-Order Differential Equation. Electronic Journal of Differential Equations, 2001, vol. 57, pp. 1-18.
6. Favini A., Labbas R., Tanabe H., Yagi A. On the Solvability of Complete Abstract Differential Equations of Elliptic Type. Funkcialaj Ekvacioj, 2004, vol. 47, pp. 205-224.
7. Favini A., Labbas R., Maingot S., Tanabe H., Yagi A. On the Solvability and the Maximal Regularity of Complete Abstract Differential. Equations of Elliptic Type. Funkcialaj Ekvacioj, 2004, vol. 47, pp. 423-452. DOI: 10.1619/fesi.47.423
8. Favini A., Labbas R., Maingot S., Tanabe H., Yagi A. Etude Unifiee de Probl`emes Elliptiques dans le Cadre Holderien. C.R. Acad. Sci. Paris, Ser. I, 2005, vol. 341, pp. 485-490.
9. Favini A., Labbas R., Maingot S., Tanabe H., Yagi A. Necessary and Sufficient Conditions in the Study of Maximal Regularity of Elliptic Differential Equations in Holder Spaces. Discrete and Continuous Dynamical Systems, 2008, vol. 22, pp. 973-987. DOI: 10.3934/dcds.2008.22.973
10. Cheggag M., Favini A., Labbas R., Maingot S., Medeghri A. Abstract Differential Equations of Elliptic Type with General Robin Boundary Conditions in Holder Spaces. Applicable Analysis, 2012, vol. 91, no. 8, pp. 1453-1475. DOI: 10.1080/00036811.2011.635653
11. Cheggag M., Favini A., Labbas R., Maingot S., Medeghri A. Sturm - Liouville Problems for an Abstract Differential Equation of Elliptic Type in UMD Spaces. Differential and Integral Equations, 2008, vol. 21, no. 9-10, pp. 981-1000.
12. Cheggag M., Favini A., Labbas R., Maingot S., Medeghri A. Complete Abstract Differential Equations of Elliptic Type with General Robin Boundary Conditions in UMD Spaces. Discrete and Continuous Dynamical Systems - Series S, 2011, vol. 4, no. 3, pp. 1-16.
13. Dore G., Venni A. On the Closedness of the Sum of Two Closed Operators. Mathematicsche zeitschrift, 1987, vol. 196, pp. 124-136. DOI: 10.1007/BF01163654
14. Pruss J., Sohr H. On Operators with Bounded Imaginary Powers in Banach Spaces. Mathematicsche zeitschrift, 1990, vol. 203, pp. 429-452. DOI: 10.1007/BF02570748
15. Lions J.L., Peetre J. Sur Une Classe D'Espaces D'Interpolation. Publications Mathematiques de l'Institut des Hautes Etudes Scientifiques, 1964, vol. 19, no. 1, pp. 5-68. DOI: 10.1007/BF02684796
16. Triebel H. Interpolation Theory, Function Spaces, Differential Operators. Amsterdam, North Holland, 1978.
17. Favini A., Labbas R., Maingot S., Meisner M. Boundary Value Problem for Elliptic Differential Equations in Noncommutative Cases. Discrete and Continuous Dynamical Systems, 2013, vol. 33, no. 11-12, pp. 4967-4990.
18. Haase M. The Functional Calculus for Sectorial Operators. Operator Theory: Advances and Applications, 2006, vol. 169, pp. 19-60.
19. Lunardi A. Analytic Semigroups and Optimal Regularity in Parabolic Problems. Birkhauser, Basel, 1995.
20. Pazy A. Semigroups of Linear Operators and Applications to Partial Differential Equations. Berlin, Heidelberg, Tokyo, Springer-Verlag, 1983.
21. Engel K., Nagel R. One-Parameter Semigroups for Linear Evolution Equtions, New York, Springer Verlag, 2000.
22. Balakrishnan A.V. Fractional Powers of Closed Operators and the Semigroups Generated by Them. Pacific J. Math., 1960, vol. 10, pp. 419-437. DOI: 10.2140/pjm.1960.10.419
23. Pruss J., Sohr H. Imaginary Powers of Elliptic Second Order Differential Operators in L^{p}-Spaces. Hiroshima Math. J., 1993, vol. 23, pp. 161-192.
24. Cheggag M., Favini A., Labbas R., Maingot S., Medeghri A. Spectral Parameter Problems with Robin Boundary Coefficient-Operator Conditions in UMD Spaces and Applications. (To appear)