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Keywords:
semilinear elliptic problems; spaces with detached asymptotics; asymptotic behaviour near conical points
Summary:
We consider a class of semilinear elliptic problems in two- and three-dimensional domains with conical points. We introduce Sobolev spaces with detached asymptotics generated by the asymptotical behaviour of solutions of corresponding linearized problems near conical boundary points. We show that the corresponding nonlinear operator acting between these spaces is Frechet differentiable. Applying the local invertibility theorem we prove that the solution of the semilinear problem has the same asymptotic behaviour near the conical points as the solution of the linearized problem if the norms of the given right hand sides are small enough. Estimates for the difference between the solution of the semilinear and of the linearized problem are derived.
References:
[1] A. Azzam: Behaviour of solutions of Dirichlet problem for elliptic equations at a corner. Indian J. Pure Appl. Math. 10 (1979), 1453-1459. MR 0551783 | Zbl 0443.35021
[2] H. Blum R. Rannacher: On the boundary value problem of the biharmonic operator on domains with angular corners. Math. Methods Appl. Sci. 2 (1980), 556-581. DOI 10.1002/mma.1670020416 | MR 0595625
[3] M. Borsuk D. Portnyagin: Barriers on cones for degenerate quasilinear elliptic operators. Electron. J. Differential Equations 11 (1998), 1-8. MR 1613596
[4] Ph. Ciarlet: Mathematical Elasticity I. North-Holland, Amsterdam, 1988. MR 0936420
[5] M. Dobrowolski: On quasilinear elliptic equations in domains with conical boundary points. J. Reine Angew. Math. 394 (1989), 186-195. MR 0977441 | Zbl 0655.35022
[6] G. Dziuk: Das Verhalten von Lösungen semilinearer elliptischer Systeme an Ecken eines Gebietes. Math. Z. 159 (1978), 89-100. DOI 10.1007/BF01174570 | MR 0481479 | Zbl 0375.35025
[7] M. Feistauer: Mathematical Methods in Fluid Dynamics. Longman, New York, 1993. MR 1266627 | Zbl 0819.76001
[8] A. Friedman: Mathematics in Industrial Problems 2. Springer-Verlag, New York, 1989. MR 0968664
[9] A. Friedman: Mathematics in Industrial Problems 3. Springer-Verlag, New York, 1990. MR 0968664
[10] P. Grisvard: Elliptic Problems in Nonsmooth Domains. Pitman Publishing Inc., Boston, 1985. MR 0775683 | Zbl 0695.35060
[11] V. A. Kondrat'ev: Boundary problems for elliptic equations in domains with conical or angular points. Trans. Moscow Math. Soc. 16 (1967), 209-292. MR 0226187 | Zbl 0194.13405
[12] V. A. Kozlov V. G. Maz'ya: On the spectrum of an operator pencils generated by the Dirichlet problem in a cone. Mat. Sb. 73 (1992), 27-48. DOI 10.1070/SM1992v073n01ABEH002533 | MR 1124101
[13] V. A. Kozlov V. G. Maz'ya: On the spectrum of an operator pencils generated by the Neumann problem in a cone. St. Petersburg Math. J. 3 (1992), 333-353. MR 1137524
[14] V. A. Kozlov J. Rossmann: Singularities of solutions of elliptic boundary value problems near conical points. Math. Nachr. 170 (1994), 161-181. MR 1302373
[15] A. W. Leung: Systems of Nonlinear Partial Differential Equations. Kluwer, Dordrecht, 1989. MR 1621827 | Zbl 0691.35002
[16] M. Marcus V. J. Mizel: Complete characterization of functions which act, via superposition, on Sobolev spaces. Trans. Amer. Math. Soc. 251 (1979), 187-218. DOI 10.1090/S0002-9947-1979-0531975-1 | MR 0531975
[17] V. G. Maz'ya B. A. Plamenevsky: On the coefficients in the asymptotics of solutions of elliptic boundary value problems in domains with conical points. Math. Nachr. 76 (1977), 29-60. MR 0601608
[18] V. G. Maz'ya B. A. Plamenevsky: Weighted spaces with nonhomogeneous norms and boundary value problems in domains with conical points. Amer. Math. Soc. Transl. 123 (1984), 89-107.
[19] E. Miersemann: Asymptotic expansion of solutions of the Dirichlet problem for quasilinear elliptic equations of second order near a conical point. Math. Nachr. 135 (1988), 239-274. DOI 10.1002/mana.19881350120 | MR 0944231
[20] S. A. Nazarov: On the two-dimensional aperture problem for Navier-Stokes equations. C. R. Acad. Sci Paris, Sér. I Math. 323 (1996), 699-703. MR 1411068 | Zbl 0860.35096
[21] S. A. Nazarov K. I. Piletskas: Asymptotics of the solution of the nonlinear Dirichlet problem having a strong singularity near a corner point. Math. USSR Izvestiya 25 (1985), 531-550. DOI 10.1070/IM1985v025n03ABEH001305
[22] S. A. Nazarov B. A. Plamenevsky: Elliptic Problems in Domains with Piecewise Smooth Boundaries. Walter de Gruyter, Berlin, 1994. MR 1283387
[23] M. Orlt A.-M. Sändig: Regularity of viscous Navier-Stokes Flows in nonsmooth domains. Boundary Value Problems and Integral Equations in Nonsmooth Domains (M.Costabel, M.Dauge, S.Nicaise, eds.). Marcel Dekker Inc., 1995. MR 1301336
[24] L. Recke: Applications of the implicit function theorem to quasilinear elliptic boundary value problems with non-smooth data. Comm. Partial Differential Equations 20 (1995), 1457-1479. DOI 10.1080/03605309508821140 | MR 1349220 | Zbl 0838.35044
[25] P. Tolksdorf: On the Dirichlet problem for quasilinear equations in domains with conical boundary points. Comm. Partial Differential Equations 8 (1983), 773-817. DOI 10.1080/03605308308820285 | MR 0700735 | Zbl 0515.35024
[26] T. Valent: Boundary Value Problems of Finite Elasticity. Springer-Verlag, New York Inc., 1988. MR 0917733 | Zbl 0648.73019
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