## Differential and Integral Equations

### On $L_p$ estimates of optimal type for the parabolic oblique derivative problem with {VMO} coefficients

Peter Weidemaier

#### Abstract

We generalize recent results by L.G. Softova concerning $W_p^{2,1} (\Omega_T)$ estimates ($1 < p < \infty$) for second order parabolic operators with VMO coefficients and the boundary condition $$\sum_{i=1}^n b_i(\xi,t) \partial_i u(\xi,t) = g(\xi,t) \ \ \text{ on  \partial \Omega_T }$$ in the nondegenerate case (see Remark 2.2 i). While Softova assumed $[(\xi,t) \mapsto b_i(\xi,t)] \in Lip(\partial \Omega_T)$, we weaken this assumption to $b_i \in C^{ \alpha, \alpha/2 } (\partial \Omega_T)$ for some $\alpha > 1 -1/p$.

#### Article information

Source
Differential Integral Equations, Volume 18, Number 8 (2005), 935-946.

Dates
First available in Project Euclid: 21 December 2012

Weidemaier, Peter. On $L_p$ estimates of optimal type for the parabolic oblique derivative problem with {VMO} coefficients. Differential Integral Equations 18 (2005), no. 8, 935--946. https://projecteuclid.org/euclid.die/1356060151