Free boundary value problems for abstract elliptic equations and applications

Abstract

The free boundary value problems for elliptic differential-operator equations are studied. Several conditions for the uniform maximal regularity with respect to boundary parameters and the Fredholmness in abstract L _p-spaces are given. In application, the nonlocal free boundary problems for finite or infinite systems of elliptic and anisotropic type equations are studied.

Keywords

Free boundary value problems / Sobolev-Lions type spaces / Differentialoperator equations / Maximal L _p regularity / Banach spaces / Operatorvalued multipliers / Interpolation of Banach spaces

References

[1] Adams R. A.. Sobolev Spaces, 1975, New York, San Francisco, London: Academic Press

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

[3] Amann H.. Linear and Quasi-linear Equations, 1, 1995, Basel: Birkhauser

[4] Aubin J. P.. Abstract boundary-value operators and their adjoint. Rend. Sem. Mat. Univ. Padova, 1970, 43: 1-33

[5] Besov O. V., Ilin V. P., Nikolskii S. M.. Integral Representations of Functions and Embedding Theorems, 1975, Moscow: Nauka

[6] Bourgain J.. Some remarks on Banach spaces in which martingale differences are unconditional. Arkiv Math., 1983, 21: 163-168

[7] Burkholder D. L.. A geometrical conditions that implies the existence certain singular integral of Banach space-valued functions. Proc. Conf. Harmonic Analysis in Honor of Antonu Zigmund, Chicago, 1981, Belmont: Wads Worth 270-286

[8] Clement P., de Pagter B., Sukochev F. A. Schauder decomposition and multiplier theorems. Studia Math., 2000, 138: 135-163

[9] Denk R., Hieber M. and Prüss, J., R-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Amer. Math. Soc., 166(788), 2003, viii+114.

[10] Dore C., Yakubov S.. Semigroup estimates and non-coercive boundary value problems. Semigroup Forum, 2000, 60: 93-121

[11] Escher J., Simonett G.. Maximal regularity for a free boundary problems. Nonlinear Diff. Eqs. Appl., 1995, 2: 463-510

[12] Favini A.. Su un problema ai limiti per certa equazini astratte del secondo ordine. Rend. Sem. Mat. Univ. Padova, 1975, 53: 211-230

[13] Grisvard P.. Elliptic Problems in Nonsmooth Domains, 1985, Boston: Pitman

[14] Guidotti P.. A 2-D free boundary problem with onset of a phase and singular elliptic boundary value problems. J. Evol. Eqs., 2002, 2: 395-424

[15] Karakas H. I., Shakhmurov V. B., Yakubov S.. Degenerate elliptic boundary value problems. Appl. Anal., 1996, 60: 155-174

[16] Krein S. G.. Linear Differential Equations in Banach Space, 1971, Providence, RI: A. M. S.

[17] Krein S. G.. Linear Equations in Banach Space, 1982, Basel: Birkhauser

[18] Lions J. L., Peetre J.. Sur une classe d’espaces d’interpolation. Inst. Hautes Etudes Sci. Publ. Math., 1964, 19: 5-68

[19] Lizorkin P. I.. (L _p, L _q)-Multiplicators of Fourier integrals. Dokl. Akad. Nauk SSSR, 1963, 152(4): 808-811

[20] Lunardi A.. Analytic Semigroups and Optimal Regularity in Parabolic Problems, 2003, Basel: Birkhauser

[21] Marius B.. Linear elliptic boundary value proplems in varying domains. Math. Nachr., 2003, 250: 17-24

[22] Nazarov S. A., Plammenevskii B. A.. Elliptic Problems in Domains with Piecewise Smooth Boundaries, 1994, New York: Walter de Gruyter

[23] Shakhmurov V. B.. Theorems about of compact embedding and applications. Doklady Akademii Nauk SSSR, 1978, 241(6): 1285-1288

[24] Shakhmurov V. B.. Theorems on compactness of embedding in weighted anisotropic spaces, and their applications. Doklady Akademii Nauk SSSR, 1986, 291(6): 612-616

[25] Shakhmurov V. B.. Imbedding theorems and their applications to degenerate equations. Diff. Eqs., 1988, 24(4): 475-482

[26] Shakhmurov V. B.. Coercive boundary value problems for regular degenerate differential-operator equations. J. Math. Anal. Appl., 2004, 292(2): 605-620

[27] Shakhmurov V. B.. Embedding theorems and maximal regular differential operator equations in Banachvalued function spaces. J. Inequalities and Appl., 2005, 2(4): 329-345

[28] Shakhmurov V. B.. Embedding and maximal regular differential operators in Banach-valued weighted spaces. Acta Math. Sin., 2006, 22(5): 1493-1508

[29] Shklyar A. Y.. Complete Second Order Linear Differential Equations in Hilbert Spaces, 1997, Basel: Birkhauser Verlak

[30] Sobolevskii P. E.. Coerciveness inequalities for abstract parabolic equations. Doklady Akademii Nauk SSSR, 1964, 57(1): 27-40

[31] Triebel H.. Interpolation Theory, Function Spaces, Differential Operators, 1978, Amsterdam: North-Holland

[32] Weis L.. Operator-valued Fourier multiplier theorems and maximal L _p regularity. Math. Ann., 2001, 319: 735-758

[33] Xiao T. J., Liang J.. The Cauchy Problem for Higher-Order Abstract Differential Equations, 1998, Berlin: Springer-Verlag

[34] Xiao T. J., Liang J.. Second order parabolic equations in Banach spaces with dynamic boundary conditions. Trans. Amer. Math. Soc., 2004, 356: 4787-4809

[35] Xiao T. J., Liang J.. Complete second order differential equations in Banach spaces with dynamic boundary conditions. J. Diff. Eqs., 2004, 200(1): 105-136

[36] Xiao T. J., Liang J.. Second order differential operators with Feller-Wentzell type boundary conditions. J. Funct. Anal., 2007, 254(6): 1467-1486

[37] Yakubov S.. Completeness of Foot Functions of Regular Differential Operators, 1994, New York: Longman, Scientific and Technical

[38] Yakubov S.. A nonlocal boundary value problem for elliptic differential-operator equations and applications. Integr. Equ. Oper. Theory, 1999, 35: 485-506

[39] Yakubov S., Yakubov Ya.. Differential-operator Equations, Ordinary and Partial Differential Equations, 2000, Boca Raton: Chapman and Hall/CRC