Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-06T01:21:03.315Z Has data issue: false hasContentIssue false
  • English
  • Français

Weak Uniqueness for Elliptic Operators in ℝ3 with Time-Independent Coefficients

Published online by Cambridge University Press:  17 April 2009

Cristina Giannotti
Cristina Giannotti
Affiliation:
Dipartimento di Matematica e InformaticaVia Madonna delle CarceriI- 62032 Camerino (Macerata)Italy e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The author gives a proof with analytic means of weak uniqueness for the Dirichlet problem associated to a second order uniformly elliptic operator in ℝ3 with coefficients independent of the coordinate x3 and continuous in ℝ2 {0}.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 74 , Issue 1 , August 2006 , pp. 91 - 100
Copyright
Copyright © Australian Mathematical Society 2006

References

[1]Arena, O. and Manselli, P., ‘A class of elliptic operators in ℝ3 in non divergence form with measurable coefficients’, Matematiche (Catania) 48 (1993), 161–180.Google Scholar
[2]Caffarelli, L.A. and Cabrè, X., Fully nonlinear elliptic equations, American Mathematical Society Colloquium Publications 43 (Amer. Math. Society, Providence, R.I., 1995).Google Scholar
[3]Caffarelli, L.A., Crandall, M.G., Kocan, M. and Swiech, A., ‘On viscosity solutions of fully nonlinear equations with measurable ingredients’, Comm. Pure Appl. Math. 49 (1996), 365–397.3.0.CO;2-A>CrossRefGoogle Scholar
[4]Cerutti, M.C., Escauriaza, L. and Fabes, E.B., ‘Uniqueness in the Dirichlet problem for some elliptic operators with discontinuous coefficients’, Ann. Mat. Pura Appl. 4 163 (1993), 161–180.CrossRefGoogle Scholar
[5]Cerutti, M.C., Fabes, E.B. and Manselli, P., ‘Uniqueness for elliptic equations with time-independent coefficients’, in Progress in Elliptic and Parabolic PDEs, Pitman Research Notes in Math. 350 (Longman, Harlow, 1996), pp. 112–135.Google Scholar
[6]Crandall, M.G., Kocan, M., Sovavia, P. and Swiech, A., ‘On the equivalence of various notions of solutions of elliptic PDE's, with measurable ingredients’, in Progress in Elliptic and Parabolic PDEs, Pitman Research Notes in Math. 350 (Longman, Harlow, 1996), pp. 136–162.Google Scholar
[7]Escauriaza, L., ‘Uniqueness for the Dirichlet problem for time independent elliptic operators’, in Partial differential equations with minimal smoothness and applications, IMA Vol. Math. Appl. 42 (Springer-Verlag, New York, 1992), pp. 115–122.CrossRefGoogle Scholar
[8]Jensen, R., ‘Uniformly elliptic PDE's with bounded, measurable coefficients’, J. Fourier Anal. Appl. 2 (1996), 237–259.CrossRefGoogle Scholar
[9]Jensen, R., Kocan, M. and Swiech, A., ‘Good and viscosity solutions of fully nonlinear elliptic equations’, Proc. Amer. Math. Soc. 130 (2002), 533–542.CrossRefGoogle Scholar
[10]Krylov, N.V., ‘On one point weak uniqueness for elliptic equations’, Comm. Partial Differential Equations 17 (1992), 1759–1784.CrossRefGoogle Scholar
[11]Krylov, N.V., ‘On weak uniqueness for some diffusions with discontinuous coefficients’, Stochastic Processes. Appl. 113 (2004), 37–64.CrossRefGoogle Scholar
[12]Nadirashvili, N.S., ‘Non uniqueness in the martingale problem and the Dirichlet problem for uniformly elliptic operators’, Ann. Scuola Norm Sup. Pisa Cl. Sci. 4 24 (1997), 537–549.Google Scholar
[13]Safonov, M.V., ‘On a weak uniqueness for some elliptic equations’, Comm. Partial Differential Equations 19 (1994), 943–957.CrossRefGoogle Scholar
[14]Safonov, M.V., ‘Nonuniqueness for second order elliptic equations with measurable coefficients’, SIAM J. Math. Anal. 30 (1999), 879–895.CrossRefGoogle Scholar
[15]Talenti, G., ‘Equazioni lineari ellittiche in due variabili’, Matematiche (Catania) 21 (1966), 339–376.Google Scholar