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Elliptic and parabolic equations in \(\mathbb{R} ^ n\) with coefficients having polynomial growth. (English) Zbl 0851.35049

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 \[ u_t(t, x)= A_0 u(t, x)+ f(t, x),\;0< t< T,\;x\in \mathbb{R}^n,\;u(0, x)= \varphi(x),\;x\in \mathbb{R}^n. \]

MSC:

35K15 Initial value problems for second-order parabolic equations
35J15 Second-order elliptic equations
60H30 Applications of stochastic analysis (to PDEs, etc.)
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