Cerrai, Sandra Elliptic and parabolic equations in \(\mathbb{R} ^ n\) with coefficients having polynomial growth. (English) Zbl 0851.35049 Commun. Partial Differ. Equations 21, No. 1-2, 281-317 (1996). Let \(A_0 u(x)= {1\over 2} \sum^n_{i, j= 1} q_{ij} D_{ij} u(x)+ \sum^n_{i= 1} F_i(x) D_i u(x)\), where \(\{q_{ij}\}^n_{i, j= 1}\) denotes a symmetric positive definite matrix and the function \(F= (F_1,\dots, F_n)\) belongs to \(C^3(\mathbb{R}^n; \mathbb{R}^n)\) and may have polynomial growth. Using the probabilistic representation of the solution, the author proves existence and uniqueness theorems and Schauder type estimates for the solution of the problems \[ \lambda\varphi(x)- A_0 \varphi(x)= f(x),\quad x\in \mathbb{R}^n,\quad \lambda> 0; \] and \[ u_t(t, x)= A_0 u(t, x)+ f(t, x),\;0< t< T,\;x\in \mathbb{R}^n,\;u(0, x)= \varphi(x),\;x\in \mathbb{R}^n. \] Reviewer: A.D.Borisenko (Kiev) Cited in 8 Documents MSC: 35K15 Initial value problems for second-order parabolic equations 35J15 Second-order elliptic equations 60H30 Applications of stochastic analysis (to PDEs, etc.) Keywords:polynomial growth PDFBibTeX XMLCite \textit{S. Cerrai}, Commun. Partial Differ. Equations 21, No. 1--2, 281--317 (1996; Zbl 0851.35049) Full Text: DOI References: [1] DOI: 10.1016/0022-0396(67)90002-2 · Zbl 0149.06804 · doi:10.1016/0022-0396(67)90002-2 [2] Besala P., Ann. Polon. Math. 29 pp 403– (1975) [3] Cannarsa P., Preprint Scuola Normale Superiore [4] Cannarsa P., Preprint Scoula Normale Superiore [5] DOI: 10.1137/0518063 · Zbl 0623.47039 · doi:10.1137/0518063 [6] DOI: 10.1007/BF02573496 · Zbl 0817.47048 · doi:10.1007/BF02573496 [7] Da Prato G., j. funct. anal. [8] Da Prato G., Encyclopedia of mathematics and its applications [9] Elworthy K.D, J.Funct.Anal. 45 (1993) [10] Freidlin M.I., Random perturbations of dynamical systems (1983) [11] Lunardi A., Analytic semigroups and optimal regularity in parabolic problems (1995) · Zbl 0816.35001 · doi:10.1007/978-3-0348-9234-6 [12] Lunardi A., Preprint Scuola Normale Superiore [13] Lunardi A., Preprint Scuola Normale Superiore (1995) [14] Lunardi A., Preprint Scuola Normale Superiore (1995) [15] Triebel H., Interpolation theory, function spaces, differential operators (1986) · Zbl 0611.46036 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.