This paper studies the Dirichlet problem for Hamilton-Jacobi equations and proves the existence, uniqueness, and continuous dependence on boundary data of Lipschitz continuous maximal viscosity solutions. The authors consider the equation $$ \mathcal{P}_{\varphi}:\quad\begin{cases}F(x,u,Du)=0&in\Omega\\ u=\varphi&in\partial\Omega,\end{cases} $$ where $ \Omega $ is an open bounded subset of $ \mathbb{R}^n $, $ F $ is a continuous function, and $ \varphi $ is a boundary datum in $ W^{1,\infty}(\Omega) $. The paper shows that under certain conditions, the maximal solution $ \overline{u} $ of the set $$ S_{\varphi}=\left\{u\in\varphi+W_{0}^{1,\infty}(\Omega):F^{**}(x,u,D u)\leq0\quad\mathrm{a.e.i n}\Omega\right\} $$ is a viscosity solution of both $ P_{\varphi} $ and $ P_{\varphi}^{**} $, where $ F^{**} $ is the lower convex envelope of $ F $ with respect to the last variable. The authors also prove that this solution depends continuously on the boundary data in the weak $ ^{*} $ topology of $ W^{1,\infty}(\Omega) $. The paper includes examples showing that the hypotheses cannot be weakened and discusses the implications of the results in the homogeneous case.This paper studies the Dirichlet problem for Hamilton-Jacobi equations and proves the existence, uniqueness, and continuous dependence on boundary data of Lipschitz continuous maximal viscosity solutions. The authors consider the equation $$ \mathcal{P}_{\varphi}:\quad\begin{cases}F(x,u,Du)=0&in\Omega\\ u=\varphi&in\partial\Omega,\end{cases} $$ where $ \Omega $ is an open bounded subset of $ \mathbb{R}^n $, $ F $ is a continuous function, and $ \varphi $ is a boundary datum in $ W^{1,\infty}(\Omega) $. The paper shows that under certain conditions, the maximal solution $ \overline{u} $ of the set $$ S_{\varphi}=\left\{u\in\varphi+W_{0}^{1,\infty}(\Omega):F^{**}(x,u,D u)\leq0\quad\mathrm{a.e.i n}\Omega\right\} $$ is a viscosity solution of both $ P_{\varphi} $ and $ P_{\varphi}^{**} $, where $ F^{**} $ is the lower convex envelope of $ F $ with respect to the last variable. The authors also prove that this solution depends continuously on the boundary data in the weak $ ^{*} $ topology of $ W^{1,\infty}(\Omega) $. The paper includes examples showing that the hypotheses cannot be weakened and discusses the implications of the results in the homogeneous case.