On non-negative contractive semigroups with non-local conditions

E. I. Galakhov ; A. L. Skubachevskii

Sbornik. Mathematics, Tome 189 (1998) no. 1, pp. 43-74

Résumé

A second-order elliptic operator with non-local conditions in a bounded domain $Q\subset \mathbb R^n$ with boundary $\partial Q\in C^\infty$ is considered. The so-called 'non-transversal' case is investigated, that is, the case when the value of a function at each point $x\in \partial Q$ is related to the integral of this function over $\overline Q$ with respect to some Borel measure $\mu (x,dy)$. Sufficient conditions for the existence of a Feller semigroup whose infinitesimal generator is the closure in $C(\overline Q)$ of the elliptic operator under consideration are obtained.

Bibliographie

[1] Feller W., “The parabolic differential equations and the associated semi-groups of transformations”, Ann. of Math., 55:3 (1952), 468–519 | DOI | MR | Zbl

[2] Feller W., “Diffusion processes in one dimension”, Trans. Amer. Math. Soc., 77:1 (1954), 1–31 | DOI | MR | Zbl

[3] Venttsel A. D., “O granichnykh usloviyakh dlya mnogomernykh diffuzionnykh protsessov”, Teoriya veroyatn. i ee primen., 4:2 (1959), 172–185 | Zbl

[4] Sato K., Ueno T., “Multi-dimensional diffusion and the Markov process on the boundary”, J. Math. Kyoto Univ., 4:3 (1965), 529–605 | MR | Zbl

[5] Bony J. M., Courrege P., Priouret P., “Semigroupes de Feller sur une variété à bord compacte et problèmes aux limites intégro-différentiels du second ordre donnant lieu au principe du maximum”, Ann. Inst. Fourier (Grenoble), 18:2 (1968), 369–521 | MR | Zbl

[6] Cancelier C., “Problèmes aux limites pseudo-différentiels donnant lieu au principe du maximum”, Comm. Partial Differential Equations, 11 (15) (1986), 1677–1726 | DOI | MR | Zbl

[7] Taira K., Diffusion Processes and Partial Differential Equations, Academic Press, London, 1988 | MR | Zbl

[8] Taira K., “On the existence of Feller semigroups with boundary conditions”, Mem. Amer. Math. Soc., 99:475 (1992), 1–65 | MR

[9] Ishikawa Y., “A remark on the existence of a diffusion process with non-local boundary conditions”, J. Math. Soc. Japan, 42:1 (1990), 171–184 | DOI | MR | Zbl

[10] Skubachevskii A. L., “O nekotorykh zadachakh dlya mnogomernykh diffuzionnykh protsessov”, Dokl. AN SSSR, 307:2 (1989), 287–291

[11] Skubachevskii A. L., “O polugruppakh Fellera dlya mnogomernykh diffuzionnykh protsessov”, Dokl. AN, 341:2 (1995), 173–176 | MR

[12] Skubachevskiǐ A. L., “Nonlocal elliptic problems and multidimensional diffusion processes”, Russian Journal of Mathematical Physics, 3:3 (1995), 327–360 | MR | Zbl

[13] Galakhov E. I., “O dostatochnykh usloviyakh suschestvovaniya polugrupp Fellera”, Matem. zametki, 60:3 (1996), 442–444 | MR | Zbl

[14] Skubachevskii A. L., “Ellipticheskie zadachi s nelokalnymi usloviyami vblizi granitsy”, Matem. sb., 129 (171):2 (1986), 279–302 | MR

[15] Skubachevskiǐ A. L., “On the stability of index of nonlocal elliptic problems”, J. Math. Anal. Appl., 160:2 (1991), 323–341 | DOI | MR | Zbl

[16] Bensoussan A., Lions J. L., Impulse Control and Quasi-variational Inequalities, Gauthier-Villars, Paris, 1984 | MR

[17] Miranda K., Uravneniya s chastnymi proizvodnymi ellipticheskogo tipa, IL, M., 1957

[18] Guschin A. K., Mikhailov V. P., “O razreshimosti nelokalnykh zadach dlya ellipticheskogo uravneniya vtorogo poryadka”, Matem. sb., 185:1 (1994), 121–160 | MR | Zbl

[19] Tribel Kh., Teoriya interpolyatsii, funktsionalnye prostranstva, differentsialnye operatory, Mir, M., 1980 | MR

[20] Krein S. G., Lineinye uravneniya v banakhovykh prostranstvakh, Nauka, M., 1971 | MR