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Gradient estimates for Dirichlet parabolic problems in unbounded domains. (English) Zbl 1061.35022
The authors consider the following Dirichlet problem: \[ \begin{cases} u_t(t,x)-Au(t,x)=0, & t\in(0,T),\;x\in \Omega,\\ u(t,\xi)=0, & t\in(0,T),\;\xi\in \partial\Omega,\\ u(0,x)=f(x), & x\in \Omega, \end{cases} \] where \(f\) is continuous and bounded in an unbounded smooth connected open set \(\Omega\subset {\mathbb R}^N.\) The operator \(A\) is a second-order elliptic one, with (possibly) unbounded regular coefficients. The authors determine new conditions on the coefficients of \(A\) yielding global estimates for the bounded classical solution.

35B45 A priori estimates in context of PDEs
35K65 Degenerate parabolic equations
35K20 Initial-boundary value problems for second-order parabolic equations
Full Text: DOI
[1] M. Bertoldi, S. Fornaro, Gradient estimates in parabolic problems with unbounded coefficients, Preprint del Dipartimento di Matematica, Universitá di Pharma, no. 316, 2003, Studia Math, to appear. · Zbl 1065.35076
[2] Cerrai, S., Second-order PDE’s in finite and infinite dimension, lecture notes in mathematics, vol. 1762, (2001), Springer Berlin
[3] Cannarsa, P.; Vespri, V., Generation of analytic semigroups by elliptic operators with unbounded coefficients, SIAM J. math. anal., 18, 857-872, (1987) · Zbl 0623.47039
[4] Cannarsa, P.; Vespri, V., Generation of analytic semigroups in the \(L^p\)-topology by elliptic operators in \(\mathbf{R}^N\), Israel J. math., 61, 235-255, (1988) · Zbl 0669.35026
[5] Da Prato, G.; Goldys, B.; Zabczyk, J., Ornstein – uhlenbeck semigroups in open sets of Hilbert spaces, C. R. acad. sci. Paris, serie I, 325, 433-438, (1997) · Zbl 0895.60083
[6] Da Prato, G.; Lunardi, A., On the ornstein – uhlenbeck operator in spaces of continuous functions, J. funct. anal., 131, 94-114, (1995) · Zbl 0846.47004
[7] Gilbarg, D.; Trudinger, N.S., Elliptic partial differential equations of second order, (1983), Springer Berlin · Zbl 0691.35001
[8] Has’minskii, R.Z., Stochastic stability of differential equations, (1980), Sijthoff and Noordhoff Alphen aan den Rijn-Germantown · Zbl 0276.60059
[9] Krylov, N.V., Lectures on elliptic and parabolic equations in Hölder spaces, (1996), American Mathematical Society Providence, RI · Zbl 0865.35001
[10] Ladyzhenskaja, O.A.; Solonnikov, V.A.; Ural’ceva, N.N., Linear and quasilinear equations of parabolic type, (1967), Nauka Moskow, English transl.: American Mathematical Society, Providence, RI, 1968 (in Russian)
[11] Ledoux, M., A simple analytic proof of an inequality by P. buser, Proc. amer. math. soc.,, 121, 3, 951-959, (1994) · Zbl 0812.58093
[12] Lunardi, A., Analytic semigroups and optimal regularity in parabolic problems, (1995), Birkhäuser Basel · Zbl 0816.35001
[13] Lunardi, A., Schauder theorems for linear elliptic and parabolic problems with unbounded coefficients in \(\mathbf{R}^N\), Studia math., 128, 2, 171-198, (1998) · Zbl 0899.35014
[14] G. Metafune, J. Prüss, A. Rhandi, R. Schnaubelt, \(L^p\)-regularity for a elliptic operators with unbounded coefficients, Preprint of the Institute of Analysis, Martin-Luther University, Halle-Wittenberg, no. 21, 2002.
[15] Pazy, A., Semigroups of linear operators and applications to partial differential equations, (1983), Springer Berlin · Zbl 0516.47023
[16] Priola, E., A counterexample to Schauder estimates for elliptic operators with unbounded coefficients, Atti dell’accademia nazionale dei lincei di roma, classe di scienze fisiche, matematiche e naturali, 12, 9, 15-25, (2001) · Zbl 1072.35521
[17] Priola, E., Dirichlet problems in a half space of a Hilbert space, Infinite dimensional anal. quantum probab. related topics, 2, 5, 257-291, (2002) · Zbl 1064.35207
[18] Priola, E., On a Dirichlet problem involving an ornstein – uhlenbeck operator, Potential anal., 18, 251-287, (2003) · Zbl 1218.35082
[19] Qian, Z., A gradient estimate on a manifold with convex boundary, Proc. roy. soc. Edinburgh sect. A, 127, 171-179, (1997) · Zbl 0885.58086
[20] Talarczyk, A., Dirichlet problem for parabolic equations on Hilbert spaces, Studia math., 141, 2, 109-142, (2000) · Zbl 0977.35057
[21] Thalmaier, A.; Wang, F.Y., Gradient estimates for harmonic functions on regular domains in Riemannian manifolds, J. funct. anal., 155, 109-124, (1998) · Zbl 0914.58042
[22] Varadhan, S.R.S., Lectures on diffusion problems and partial differential equations, (1980), Tata Institute of Fundamental Research Bombay · Zbl 0489.35002
[23] Wang, F.Y., On estimation of the logarithmic Sobolev constant and gradient estimates of heat semigroups, Probab. theory related fields, 108, 87-101, (1997) · Zbl 0874.58092
[24] F.Y. Wang, Gradient estimates of Dirichlet Heat Semigroups and Application to Isoperimetric Inequalities, Ann. Probab. 32 (2004) 424-440. · Zbl 1087.58017
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