This paper focuses on the Dirichlet problem for Hamilton-Jacobi equations, specifically addressing the existence, uniqueness, and continuous dependence on boundary data of Lipschitz continuous maximal viscosity solutions. The author, Sandro Zagatti, considers a continuous function \( F : \Omega \times \mathbf{R} \times \mathbf{R}^n \to \mathbf{R} \) and a boundary datum \( \varphi \in W^{1,\infty}(\Omega) \). The problem is formulated as finding a function \( u \) such that \( F(x, u(x), Du(x)) = 0 \) in \( \Omega \) and \( u = \varphi \) on \( \partial \Omega \). The main results include: 1. **Existence**: The existence of a maximal viscosity solution \( \overline{u} \) is proven, which is unique and depends continuously on the boundary data. 2. **Uniqueness**: The viscosity solution \( \overline{u} \) is shown to be unique. 3. **Continuous Dependence**: If \( \varphi_k \) converges uniformly to \( \varphi_0 \) on \( \partial \Omega \), then the corresponding maximal solutions \( \overline{u}_k \) converge weakly* to \( \overline{u}_0 \) in \( W^{1,\infty}(\Omega) \). The paper also discusses the hypotheses required for the existence result and provides examples to illustrate the conditions. Additionally, it explores the homogeneous case where \( F \) depends only on \( Du \) and \( \varphi \equiv 0 \), and shows that under certain conditions, the solution is unique and continuous with respect to the boundary data.This paper focuses on the Dirichlet problem for Hamilton-Jacobi equations, specifically addressing the existence, uniqueness, and continuous dependence on boundary data of Lipschitz continuous maximal viscosity solutions. The author, Sandro Zagatti, considers a continuous function \( F : \Omega \times \mathbf{R} \times \mathbf{R}^n \to \mathbf{R} \) and a boundary datum \( \varphi \in W^{1,\infty}(\Omega) \). The problem is formulated as finding a function \( u \) such that \( F(x, u(x), Du(x)) = 0 \) in \( \Omega \) and \( u = \varphi \) on \( \partial \Omega \). The main results include: 1. **Existence**: The existence of a maximal viscosity solution \( \overline{u} \) is proven, which is unique and depends continuously on the boundary data. 2. **Uniqueness**: The viscosity solution \( \overline{u} \) is shown to be unique. 3. **Continuous Dependence**: If \( \varphi_k \) converges uniformly to \( \varphi_0 \) on \( \partial \Omega \), then the corresponding maximal solutions \( \overline{u}_k \) converge weakly* to \( \overline{u}_0 \) in \( W^{1,\infty}(\Omega) \). The paper also discusses the hypotheses required for the existence result and provides examples to illustrate the conditions. Additionally, it explores the homogeneous case where \( F \) depends only on \( Du \) and \( \varphi \equiv 0 \), and shows that under certain conditions, the solution is unique and continuous with respect to the boundary data.