REFERENCES
M. S. Agranovich, “Nonself-adjoint elliptic operators in nonsmooth domains, ” Russian J. Math. Phys., 2, 139–148 (1994).
A. B. Antonevich, Linear Functional Equations. Operator Approach, Birkhäuser, Basel-Boston-Berlin (1995).
M. S. Birman and M. Z. Solomyak, “Spectral asymptotics of nonsmooth elliptic operators, ” Tr. Mosk. Mat. Obshch., 27, 3–52 (1972); English translation: Trans. Moscow Math. Soc., 27 (1972).
A. V. Bitsadze, “On the theory of nonlocal boundary-value problems, ” Dokl. Akad. Nauk SSSR, 277, 17–19 (1984); English translation: Sov. Math. Dokl., 30 (1984).
A. V. Bitsadze, “On some class of conditionally solvable nonlocal boundary-value problems for harmonicfunc tions, ” Dokl. Akad. Nauk SSSR 280, 521–524 (1985); English translation: Sov. Math. Dokl., 31 (1985).
A. V. Bitsadze and A. A. Samarskii, “On some simple generalizations of linear ellipticb oundary-value problems, ” Dokl. Akad. Nauk SSSR, 185, 739–740 (1969); English translation: Sov. Math. Dokl., 10 (1969).
J.-M. Bony, P. Courrege, and P. Priouret, “Semi-groups de Feller sur une variété à bord compacte et problèmes aux limites intégro-di. érentiels du second ordre donnant lieu au principe du maximum, ” Ann. Inst. Fourier, Grenoble, 18, 369–521 (1968).
C. Cancelier, “Problémes aux limites pseudo-di. érentiels donnant lieu au principe du maximum, ” Comm. Part. Differ. Equat., 11, 1677–1726 (1986).
N. Dunford and J. T. Schwartz, Linear Operators. Part 2: Spectral Theory, Interscience Publishers, New York-London (1963).
S. D. Eidel'man and N. V. Zhitarashu, “Nonlocal boundary-value problems for elliptic equations, ” Mat. Issled., 6, No. 2 (20), 63–73 (1971).
W. Feller, “The parabolic differential equations and the associated semi-groups of transformations, ” Ann. Math., 55, 468–519 (1952).
W. Feller, “Diffusion processes in one dimension, ” Trans. Amer. Math. Soc., 77, 1–30 (1954).
E. I. Galakhov, “On sufficientt conditions for the existence of Feller semigroups, ” Mat. Zametki, 60, 442–444 (1996); English translation: Math. Notes, 60 (1996).
E. I. Galakhov and A. L. Skubachevskii, “On nonnegative contractive semigroups with nonlocal conditions, ” Mat. Sb., 189, 45–78 (1998); English translation: Russ. Acad. Sci. Sb. Math., 189 (1998).
I. C. Gohberg and M. G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators in Hilbert Spaces [in Russian], Nauka, Moscow (1965); English translation: Amer. Math. Soc. (1969).
O. V. Guseva, “On the boundary-value problems for strongly ellipticsystems, ” Dokl. Akad. Nauk SSSR, 102, 1069–1072 (1955).
A. K. Gushchin and V. P. Mikhailov, “On solvability of nonlocal problems for a second-order elliptic equation, ” Mat. Sb., 185, 121–160 (1994); English translation: Russ. Acad. Sci. Sb. Math., 81 (1995).
V. A. Il'in and E. I. Moiseev, “Nonlocal boundary-value problems of the second kind for a Sturm-Liouville operator, ” Differents. Uravn., 23, 1422–1431 (1987); English translation: Di.erential Equations, 23 (1988).
V. A. Il'in and E. I. Moiseev, “An a priori bound for a solution of the problem conjugate to a nonlocal boundary-value problem of the first kind, ” Differents. Uravn., 24, 795–804 (1988); English translation: Di.erential Equations, 24 (1988).
Y. Ishikawa, “A remark on the existence of a diffusion process with non-local boundary conditions, ” J. Math. Soc. Jpn., 42, 171–184 (1990).
G. A. Kamenskii, A. D. Myshkis, and A. L. Skubachevskii, “Minimum value of a quadraticfunc tional and linear ellipticb oundary-value problems with deviating arguments, ” Differents. Uravn., 16, 1469–1473 (1980); English translation: Di.erential Equations, 16 (1981).
T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin-Heidelberg-New York (1966).
V. A. Kondrat'ev, “Boundary-value problems for elliptic equations in domains with conical or angular points, ” Tr. Mosk. Mat. Obshch., 16, 209–292 (1967); English translation: Trans. Moscow Math. Soc., 16 (1967).
A. M. Krall, “The development of general differential and general differential-boundary systems, ” Rocky Mountain J. Math., 5, 493–542 (1975).
O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations [in Russian], Nauka, Moscow (1964); English translation: Academic Press, New York-London (1968).
V. B. Lidskii, “On summability of series in main vectors of non-self-adjoint operators, ” Tr. Mosk. Mat. Obshch., 11, 3–35 (1962); English translation: Amer. Math. Soc. Transl., 40, 193–228 (1964).
E. I. Moiseev, “Spectral characteristics of a nonlocal boundary-value problem, ” Differents. Uravn., 30, 864–872 (1994); English translation: Differential Equations, 30 (1994).
J. NeČcas, Les Méthodes Directes en Théorie des Equations Elliptiques, Academia Tchecoslovaque des Sciences, Prague, 1967.
O. A. Oleinik and E. V. RadkeviČ, “Second order equations with nonnegative characteristic form, ” in: Progress in Science and Technology [in Russian], All-Union Institute for Scientific and Technical Information (VINITI), Akad. Nauk SSSR, Moscow (1971); English translation: Amer. Math. Soc., Providence, Rhode Island and Plenum Press, New York (1973).
G. G. Onanov and A. L. Skubachevskii, “Differential equations with displaced arguments in stationary problems in the mechanics of a deformed body, ” Prikl. Mekh., 15, 39–47 (1979); English translation: Sov. Appl. Mech., 15 (1979).
G. G. Onanov and E. L. Tsvetkov, “On the minimum of the energy functional with respect to functions with deviating argument in a stationary problem of elasticity theory, ” Russ. J. Math. Phys., 3, 491–500 (1996).
B. P. Paneyakh, “Certain nonlocal boundary-value problems for linear di.erential operators, ” Mat. Zametki, 35, 425–434 (1984); English translation: Math. Notes, 35 (1984).
V. V. Pod'yapolskii and A. L. Skubachevskii, “On completeness and the basis property of root functions for strongly ellipticfunc tional differential operators, ” Usp. Mat. Nauk, 51, 219–220 (1996); English translation: Russ. Math. Surveys, 51 (1997).
V. V. Pod'yapolskii and A. L. Skubachevskii, “On the main term of spectral asymptotics for nonlocal ellipticproblems, ” in: Abstracts of International Conference on Functional Spaces and Differential Operators Dedicated to the 75th Anniversary of the Birth of L. D. Kudryavtsev, Moscow (1998), p. 157.
V. V. Pod'yapolskii and A. L. Skubachevskii, “Asymptotics of eigenvalues for elliptic differential difference operators in bounded domains, ” in: Abstracts of International Conference Dedicated to the 90th Anniversary of the Birth of L. S. Pontryagin, Steklov Mathematical Institute, Moscow State University, Moscow (1998), pp. 86–88.
D. Przeworska-Rolewicz, Equations with Transformed Argument, PWN, Warszawa (1973).
V. S. Rabinovich, “On the solvability of differential-di.erence equations on Rn and in a half-space, ” Dokl. Akad. Nauk SSSR, 243, 1498–1502 (1978); English translation: Sov. Math. Dokl., 19 (1978).
A. V. Razgulin, “On autooscillations in nonlinear parabolic problems with transformed argument, ” Zh. Vychisl. Mat. Mat. Fiz., 33, 69–80 (1993); English translation: Comp. Math. Math. Phys., Pergamon Press, Great Britain (1993).
A. V. Razgulin, “Rotational multi-petal waves in optical systems with 2–D feedback, ” in: Chaos in Optics, Rajarshi Roy, Ed., Proc. SPIE, 2039, 342–352 (1993).
Ya. A. Roitberg, and Z. G. Sheftel', “On a class of general nonlocal elliptic problems, ” Dokl. Akad. Nauk SSSR, 192, 511–513 (1970); English translation: Sov. Math. Dokl., 11 (1970).
Ya. A. Roitberg and Z. G. Sheftel', “Nonlocal problems for elliptic equations and systems, ” Sib. Mat. Zh., 13, 165–181 (1972); English translation: Sib. Math. J., 13 (1972).
L. E. Rossovskii, “Coerciveness of functional differential equations, ” Mat. Zametki, 59, 103–113 (1996); English translation: Math. Notes, 59 (1996).
L. E. Rossovskii, “Fredholm property of boundary value problems for a class of functional differential equations, ” in: Abstracts of International Conference Dedicated to the 90th Anniversary of the Birth of L. S. Pontryagin, Steklov Mathematical Institute, Moscow State University, Moscow (1998), pp. 92–93.
L. E. Rossovskii, “The first boundary-value problem for some ellipticfunc tional-differential equations, ” Usp. Mat. Nauk, 53, 167 (1998); English translation: Russ. Math. Surveys.
W. Rudin, Functional Analysis, McGraw-Hill, New York (1973).
A. A. Samarskii, “Some problems of the theory of differential equations, ” Differents. Uravn., 16, 1925–1935 (1980); English translation: Di.erential Equations, 16 (1980).
K. Sato and T. Ueno, “Multi-dimensional diffusion and the Markov process on the boundary, ” J. Math. Kyoto Univ., 4, 529–605 (1965).
A. L. Skubachevskii, “On the spectrum of some nonlocal elliptic boundary-value problems, ” Mat. Sb., 117, 159, 548–558 (1982); English translation: Math. USSR Sb., 45 (1983).
A. L. Skubachevskii, “Some nonlocal elliptic boundary-value problems, ” Differents. Uravn., 18, 1590–1599 (1982); English translation: Di.erential Equations, 18 (1983).
A. L. Skubachevskii, “Nonlocal elliptic boundary-value problems with degeneration, ” Differents. Uravn., 19, 457–470 (1983); English translation: Di.erential Equations, 19 (1983).
A. L. Skubachevskii, “Nonlocal elliptic problems with a parameter, ” Mat. Sb., 121, 201–210 (1983); English translation: Math. USSR Sb., 49 (1984).
A. L. Skubachevskii, “Smoothness of generalized solutions of the first boundary-value problem for an ellipticdi.eren tial-di.erence equation, ” Mat. Zametki, 34, 105–112 (1983); English translation: Math. Notes, 34 (1984).
A. L. Skubachevskii, “The elliptic problems of A. V. Bitsadze and A. A. Samarskii, ” Dokl. Akad. Nauk SSSR, 278, 813–816 (1984); English translation: Sov. Math. Dokl., 30 (1984).
A. L. Skubachevskii, “Solvability of elliptic problems with Bitsadze-Samarskii boundary conditions, ” Differents. Uravn., 21, 701–706 (1985); English translation: Di.erential Equations, 21 (1985).
A. L. Skubachevskii, “Nonlocal boundary-value problems with a shift, ” Mat. Zametki, 38, 587–598 (1985); English translation: Math. Notes, 38 (1986).
A. L. Skubachevskii, “Elliptic problems with nonlocal conditions near the boundary, ” Mat. Sb., 129, 279–302 (1986); English translation: Math. USSR Sb., 57 (1987).
A. L. Skubachevskii, “The first boundary value problem for strongly elliptic differential-difference equations, ” J. Differ. Equat., 63, 332–361 (1986).
A. L. Skubachevskii, “Solvability of elliptic problems with nonlocal boundary conditions, ” Dokl. Akad. Nauk SSSR, 291, 551–555 (1986); English translation: Sov. Math. Dokl., 34 (1987).
A. L. Skubachevskii, “Eigenvalues and eigenfunctions of some nonlocal boundary-value problems, ” Differents. Uravn., 25, 127–136 (1989); English translation: Differential Equations, 25 (1989).
A. L. Skubachevskii, “On some problems for multidimensional diffusion processes, ” Dokl. Akad. Nauk SSSR, 307, 287–292 (1989); English translation: in Sov. Math. Dokl., 40 (1990).
A. L. Skubachevskii, “Model nonlocal problems for elliptic equations in dihedral angles, ” Differents. Uravn., 26, 119–131 (1990); English translation: Differential Equations, 26 (1990).
A. L. Skubachevskii, “Truncation-function method in the theory of nonlocal problems, ” Differents. Uravn., 27, 128–139 (1991); English translation: Di.erential Equations, 27 (1991).
A. L. Skubachevskii, “On the stability of the index of nonlocal elliptic problems, ” J. Math. Anal. Appl., 160, 323–341 (1991).
A. L. Skubachevskii, “On Feller semigroups for multidimensional diffusion processes, ” Dokl. Akad. Nauk, 341, 173–176 (1995); English translation: Russ. Acad. Sci. Dokl. Math.
A. L. Skubachevskii, “Nonlocal elliptic problems and multidimensional diffusion processes, ” Russ. J. Math. Phys., 3, 327–360 (1995).
A. L. Skubachevskii, “On some properties of elliptic and parabolic functional differential equations, ” Usp. Mat. Nauk, 51, 169–170 (1996); English translation: Russ. Math. Surveys, 51 (1996).
A. L. Skubachevskii, Elliptic Functional Differential Equations and Applications, Birkhéuser, Basel-Boston-Berlin (1997).
A. L. Skubachevskii, “On the normality of some elliptic functional differential operators, ” Funkts. Anal. Prilozh., 31, 60–65 (1997); English translation: Functional Anal. Appl., 31 (1997).
A. L. Skubachevskii, “Bifurcation of periodicsolutions for nonlinear parabolicfunc tional differential equations arising in optoelectronics, ” Nonlinear Anal., 32, 261–278 (1998).
A. L. Skubachevskii, “Elliptic differential-difference equations with degeneration, ” Tr. Mosk. Mat. Obshch., 59, 240–285 (1998); English translation: Trans. Mascow Math. Soc., 59 (1998).
A. L. Skubachevskii and E. L. Tsvetkov, “Second boundary-value problem for elliptic differentialdi fference equations, ” Differents. Uravn., 25, 1766–1776 (1989); English translation: Differential Equations, 25 (1990).
A. L. Skubachevskii and E. L. Tsvetkov, “General boundary-value problems for elliptic di.erentialdifference equations, ” Tr. S.-Peterburg. Mat. Obshch., 5, 223–288 (1998); English translation: Trans. S.-PetersburgM ath. Soc., 5 (1998).
K. Taira, Diffusion Processes and Partial Differential Equations, Academic Press, New York-London (1988).
K. Taira, “On the existence of Feller semigroups with boundary conditions, ” Mem. Amer. Math. Soc., 99, 1–65 (1992).
K. Taira, “Sur l'existence de processus de diffusion, ” Ann. Inst. Fourier, Grenoble, 29, 99–126 (1979).
E. L. Tsvetkov, “Solvability and spectrum of the third boundary-value problem for elliptic differentialdi fference equations, ” Mat. Zametki, 51, 107–114 (1992); English translation: Math. Notes, 51 (1992).
E. L. Tsvetkov, “On smoothness of generalized solutions of the third boundary-value problem for ellipticdifferen tial-difference equation, ” Ukr. Mat. Zh., 45, 1140–1150 (1993); English translation: Ukr. Math. J., 45 (1994).
A. D. Ventsel', “On boundary conditions for multidimensional diffusion processes, ” Teor. Veroyatn. Primen., 4, 172–185 (1959); English translation: Theory Prob. Appl., 4 (1959).
S. Watanabe, “Construction of diffusion processes with Wentzell's boundary conditions by means of Poisson point processes of Brownian excursions, ” in: Probability Theory, Banach Center Publications, Vol. 5, PWN-Polish Scientific Publishers, Warsaw (1979), pp. 255–271.
M. A. Vorontsov, V. Yu. Ivanov, and V. I. Shmalhausen, “Rotatory instability of the spatial structure of light fields in nonlinear media with two-dimensional feedback, ” in: Laser Optics in Condensed Matter, Plenum Press, New York-London, (1988), pp. 507–517.
M. A. Vorontsov, N. G. Iroshnikov, and R. L. Abernathy, “Difractive patterns in nonlinear optical two-dimensional feedback system with field rotation, ” Chaos, Solitons, and Fractals, 4, 1701–1716 (1994).
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Rossovskii, L.E., Skubachevskii, A.L. Solvability and Regularity of Solutions for Some Classes of Elliptic Functional-Differential Equations. Journal of Mathematical Sciences 104, 1008–1059 (2001). https://doi.org/10.1023/A:1009583608672
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DOI: https://doi.org/10.1023/A:1009583608672