The scarcity of high-quality annotations in many application scenarios has recently led to an increasing interest in devising learning techniques that combine unlabeled data with labeled data in a network. In this work, we focus on the label propagation problem in multilayer networks. Our approach is inspired by the heat diffusion model, which shows usefulness in machine learning problems such as classification and dimensionality reduction. We propose a novel boundary-based heat diffusion algorithm that guarantees a closed-form solution with an efficient implementation. We experimentally validated our method on synthetic networks and five real-world multilayer network datasets representing scientific coauthorship, spreading drug adoption among physicians, two bibliographic networks, and a movie network. The results demonstrate the benefits of the proposed algorithm, where our boundary-based heat diffusion dominates the performance of the state-of-the-art methods.