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## Studia Mathematica

1998 | 128 | 2 | 171-198
Tytuł artykułu

### Schauder theorems for linear elliptic and parabolic problems with unbounded coefficients in $ℝ^{n}$

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Słowa kluczowe
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
171-198
Opis fizyczny
Daty
wydano
1998
otrzymano
1997-05-26
Twórcy
autor
• Dipartimento di Matematica, Università di Parma, Via D'Azeglio 85/A, 43100 Parma, Italy
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.
• [3] A. Bensoussan and J.-L. Lions, Applications of Variational Inequalities in Stochastic Control, North-Holland, Amsterdam, 1982.
• [4] P. Besala, On the existence of a fundamental solution for a parabolic differential equation with unbounded coefficients, Ann. Polon. Math. 29 (1975), 403-409.
• [5] W. Bodanko, Sur le problème de Cauchy et les problèmes de Fourier pour les équations paraboliques dans un domaine non borné, ibid. 28 (1966), 79-94.
• [6] P. Cannarsa and V. Vespri, Generation of analytic semigroups by elliptic operators with unbounded coefficients, SIAM J. Math. Anal. 18 (1987), 857-872.
• [7] S. Cerrai, Elliptic and parabolic equations in $ℝ^n$ with coefficients having polynomial growth, Comm. Partial Differential Equations 21 (1996), 281-317.
• [8] S. Cerrai, Some results for second order elliptic operators having unbounded coefficients, preprint, Scuola Norm. Sup. Pisa, 1996.
• [9] E. B. Davies, Heat Kernels and Spectral Theory, Cambridge Univ. Press, 1990.
• [10] M. H. Davis and S. I. Markus, An Introduction to Nonlinear Filtering, NATO Adv. Study Inst. Ser., Reidel, Dordrecht, 1980.
• [11] G. Da Prato and A. Lunardi, On the Ornstein-Uhlenbeck operator in spaces of continuous functions, J. Funct. Anal. 131 (1995), 94-114.
• [12] W. H. Fleming and S. K. Mitter, Optimal control and nonlinear filtering for nondegenerate diffusion processes, Stochastics 8 (1982), 63-77.
• [13] S. Ito, Fundamental solutions of parabolic differential equations and boundary value problems, Japan. J. Math. 27 (1957), 5-102.
• [14] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, Basel, 1995.
• [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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Bibliografia
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