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1998 | 128 | 2 | 171-198
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Schauder theorems for linear elliptic and parabolic problems with unbounded coefficients in $ℝ^{n}$

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We study existence, uniqueness, and smoothing properties of the solutions to a class of linear second order elliptic and parabolic differential equations with unbounded coefficients in $ℝ^n$. The main results are global Schauder estimates, which hold in spite of the unboundedness of the coefficients.
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Bibliografia
  • [1] D. G. Aronson and P. Besala, Parabolic equations with unbounded coefficients, J. Differential Equations 3 (1967), 1-14.
  • [2] J. S. Baras, G. O. Blankenship and W. E. Hopkins, Existence, uniqueness and asymptotic behavior of solutions to a class of Zakai equations with unbounded coefficients, IEEE Trans. Automat. Control 28 (1983), 203-214.
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  • [8] S. Cerrai, Some results for second order elliptic operators having unbounded coefficients, preprint, Scuola Norm. Sup. Pisa, 1996.
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  • [15] A. Lunardi, An interpolation method to characterize domains of generators of semigroups, Semigroup Forum 53 (1996), 321-329.
  • [16] A. Lunardi, On the Ornstein-Uhlenbeck operator in $L^2$ spaces with respect to invariant measures, Trans. Amer. Math. Soc. 349 (1997), 155-169.
  • [17] A. Lunardi and V. Vespri, Optimal $L^∞$ and Schauder estimates for elliptic and parabolic operators with unbounded coefficients, in: Reaction-Diffusion Systems, Proc., G. Caristi and E. Mitidieri (eds.), Lecture Notes in Pure and Appl. Math. 194, M. Dekker, 1997, 217-239.
  • [18] A. Lunardi and V. Vespri, Generation of strongly continuous semigroups by elliptic operators with unbounded coefficients in $L^p(ℝ^n)$, Rend. Mat., volume in honour of P. Grisvard, to appear.
  • [19] S. J. Sheu, Solution of certain parabolic equations with unbounded coefficients and its application to nonlinear filtering, Stochastics 10 (1983), 31-46.
  • [20] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, 1978.
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bwmeta1.element.bwnjournal-article-smv128i2p171bwm
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