## Differential and Integral Equations

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

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

#### 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

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356060151

**Mathematical Reviews number (MathSciNet)**

MR2150446

**Zentralblatt MATH identifier**

1212.35194

**Subjects**

Primary: 35K20: Initial-boundary value problems for second-order parabolic equations

Secondary: 35B45: A priori estimates 35R05: Partial differential equations with discontinuous coefficients or data

#### Citation

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