Abstract:
This paper deals with solutions of the divergence for domains with external cusps. It is known that the classic results in standard Sobolev spaces, which are basic in the variational analysis of the Stokes equations, are not valid for this class of domains. For some bounded domains Ω ⊂ Rn presenting power type cusps of integer dimension m ≤ n - 2, we prove the existence of solutions of the equation div u = f in weighted Sobolev spaces, where the weights are powers of the distance to the cusp. The results obtained are optimal in the sense that the powers cannot be improved. As an application, we prove existence and uniqueness of solutions of the Stokes equations in appropriate spaces for cuspidal domains. Also, we obtain weighted Korn type inequalities for this class of domains.
Referencias:
- Acosta, G., Durán, R.G., Lombardi, A.L., Weighted Poincaré and Korn inequalities for Hölder α domains (2006) Math. Methods Appl. Sci., 29 (4), pp. 387-400
- Acosta, G., Durán, R.G., López, F., GARCÍ A, Work in progress; Acosta, G., Durán, R.G., Muschietti, M.A., Solutions of the divergence operator on John domains (2006) Adv. Math., 206 (2), pp. 373-401
- Arnold, D.N., Scott, L.R., Vogelius, M., Regular inversion of the divergence operator with Dirichlet boundary conditions on a polygon (1988) Ann. Sc. Norm. Super. Pisa Cl. Sci., 15 (4), pp. 169-192
- Babuska, I., Aziz, A.K., Survey lectures on the mathematical foundation of the finite element method (1972) The mathematical foundations of the finite element method with applications to partial differential equations, pp. 5-539. , edited by A.K. Aziz, Academic Press, New York
- Boffi, D., Brezzi, F., Demkowicz, L.F., Durán, R.G., Falk, R.S., Fortin, M., Finite elements, compatibility conditions, and applications (1939) Lecture Notes in Math, , Springer-Verlag, Berlin
- Bogovskii, M.E., Solution of the first boundary value problem for the equation of continuity of an incompressible medium (1979) Soviet Math. Dokl., 20, pp. 1094-1098
- Brenner, S.C., Scott, L.R., (1994) The mathematical theory of finite element methods, , Springer-Verlag, Berlin
- Brezzi, F., On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers (1974) Rev. Française Automat. Informat. Recherche Opéra-tionnelle Sér. Rouge, 8, pp. 129-151
- Brezzi, F., Fortin, M., Mixed and hybrid finite element methods (1991) Springer Ser. Comput. Math, 15. , Springer-Verlag, New York
- Buckley, S.M., Koskela, P., New Poincaré inequalities from old (1998) Ann. Acad. Sci. Fenn. Math., 23, pp. 251-260
- Calderón, A.P., Zygmund, A., On singular integrals (1956) Amer. J. Math., 78, pp. 289-309
- Ciarlet, P.G., Introduction to linear shell theory (1998) Ser. Appl. Math., , P. G. Ciarlet and P. L. Lions, eds., Gauthier-Villars
- Diening, L., Růžička, M., Schumacher, K., A decomposition technique for John domains (2010) Ann. Acad. Sci. Fenn. Math., 35, pp. 87-114
- Duoandikoetxea, J., Fourier analysis (2001) Grad. Stud. Math, 29. , Amer. Math. Soc
- Durán, R.G., (2005) The inf-sup condition for the Stokes equations: A constructive approach in general domains, pp. 270-272. , Mathematisches Forschungsinstitut Oberwolfach, Workshop on Gemischte und nicht-standard Finite-Elemente-Methoden mit Anwendungen, Extended abstract, Report No. 5
- Durán, R.G., López, F., GARCÍA: Solutions of the divergence and analysis of the Stokes equations in planar Hölder-α domains (2010) Math. Models Methods Appl. Sci., 20 (1), pp. 95-120
- Durán, R.G., Muschietti, M.A., An explicit right inverse of the divergence operator which is continuous in weighted norms (2001) Studia Math., 148 (3), pp. 207-219
- Friedrichs, K.O., On the boundary-value problems of the theory of elasticity and Korn's inequality (1947) Ann. of Math, 48 (2), pp. 441-471
- Galdi, G.P., (1994) An introduction to the mathematical theory of the Navier-Stokes equations, 1. , Linearized steady problems. - Springer Tracts Nat. Philos. 38, Springer-Verlag, New York
- Geymonat, G., Gilardi, G., (1998) Contre-exemples á l'inégalité de Korn et au Lemme de Lions dans des domaines irréguliers, pp. 541-548. , Équations aux Dérivées Partielles et Applications, Gauthiers-Villars, Éd. Sci. Méd. Elsevier, Paris
- Girault, V., Raviart, P.A., (1986) Finite element methods for Navier-Stokes equations, , Springer-Verlag, Berlin
- Gol'dshtein, V., Ukhlov, A., Weighted Sobolev spaces and embedding theorems (2009) Trans. Amer. Math. Soc., 361, pp. 3829-3850
- Kilpeläinen, T., Weighted Sobolev spaces and capacity (1994) Ann. Acad. Sci. Fenn. Ser. A I Math., 19, pp. 95-113
- Kufner, A., (1985) Weighted Sobolev spaces, , John Wiley & Sons, New York
- Ladyzhenskaya, O.A., (1969) The mathematical theory of viscous incompressible flow, , Gordon and Breach, New York
- Nicolás, F., (2008) Inverse of the divergence in Hardy spaces on star-shaped domains, , Preprint
- Stein, E.M., Note on singular integrals (1957) Proc. Amer. Math. Soc., 8 (2), pp. 250-254
- Stein, E.M., Harmonic analysis: Real-variable methods, orthogonality, and oscillatory integrals (1993) Princeton Math. Ser., 43. , Monographs in Harmonic Analysis III, Princeton Univ. Press, Princeton, NY
- Weck, N., Local compactness for linear elasticity in irregular domains (1994) Math. Meth. Appl. Sci., 17, pp. 107-113
Citas:
---------- APA ----------
Durán, R.G. & García, F.L.
(2010)
. Solutions of the divergence and Korn inequalities on domains with an external cusp. Annales Academiae Scientiarum Fennicae Mathematica, 35(1), 421-438.
http://dx.doi.org/10.5186/aasfm.2010.3527---------- CHICAGO ----------
Durán, R.G., García, F.L.
"Solutions of the divergence and Korn inequalities on domains with an external cusp"
. Annales Academiae Scientiarum Fennicae Mathematica 35, no. 1
(2010) : 421-438.
http://dx.doi.org/10.5186/aasfm.2010.3527---------- MLA ----------
Durán, R.G., García, F.L.
"Solutions of the divergence and Korn inequalities on domains with an external cusp"
. Annales Academiae Scientiarum Fennicae Mathematica, vol. 35, no. 1, 2010, pp. 421-438.
http://dx.doi.org/10.5186/aasfm.2010.3527---------- VANCOUVER ----------
Durán, R.G., García, F.L. Solutions of the divergence and Korn inequalities on domains with an external cusp. Ann. Acad. Sci. Fenn. Math. 2010;35(1):421-438.
http://dx.doi.org/10.5186/aasfm.2010.3527