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In a wireless mesh network (WMN) with a number of stationary wireless routers, the aggregate capacity can be increased when each router is equipped with multiple network interface cards (NICs) and each NIC is assigned to a distinct orthogonal frequency channel. In this paper, given the logical topology of the network, we mathematically formulate a cross- layer fair bandwidth sharing problem as a non-linear mixed- integer network utility maximization problem. An optimal joint design, based on exact binary linearization techniques, is proposed which leads to a global maximum. A near-optimal joint design, based on approximate dual decomposition techniques, is also proposed which is practical for deployment. Performance is assessed through several numerical examples in terms of network utility, aggregate network throughput, and fairness index. Results show that our proposed designs can lead to multi-channel WMNs which are more efficient and fair compared to their single- channel counterparts. The performance gain on both efficiency and fairness increase as the number of available NICs per router or the number of available frequency channels increases.