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Parabolic equations with VMO coefficients in Sobolev spaces with mixed norms. (English) Zbl 1133.35052
The author studies the Cauchy problem for second-order divergence and non-divergence type equations:
\[ \begin{aligned} L u(t,x)&= u_t(t,x) +a^{ij}(t,x)u_{x^ix^j}(t,x)+ b^i(t,x)u_{x^i}(t,x)+c(t,x) u(t,x),\\ {\mathcal L } u(t,x)&= u_t(t,x) +\big(a^{ij}(t,x)u_{x^i}(t,x)+ \bar b^j(t,x)u(t,x)\big)u_{x^j} + b^i(t,x)u_{x^i}(t,x)+c(t,x) u(t,x) \end{aligned} \] in the stripe \((0,T)\times{\mathbb R}^d,\) \(d\geq 1.\) The main coefficients supposed to be in \(\text{VMO}_x\), i.e. measurable in time and with vanishing mean oscillation in \(x.\) The unique solvability in Sobolev spaces with mixed \(L_q(L_p)\)-norms for \(L\) is proved if \(q\geq p\) and for \({\mathcal L}\) without this restriction. The technique is based on a priori pointwise estimates of the sharp functions of the second-order spatial derivatives of solutions.

MSC:
35K15 Initial value problems for second-order parabolic equations
35B45 A priori estimates in context of PDEs
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