This paper presents a novel approach to solving the inviscid Burgers equation and its Hamilton-Jacobi form by treating them as degenerate elliptic problems. The authors introduce a duality-based method that involves formulating the primal PDE as constraints and using a convex auxiliary potential to optimize. This approach leads to a dual variational principle, where the Euler-Lagrange equations of the dual problem are equivalent to the primal PDE system. The method is applied to approximate weak solutions of the Burgers equation, which are non-unique and can be recovered through specific constructive approaches. The paper is structured into several sections: the introduction, the dual formulation of the Burgers equation in conservation and Hamilton-Jacobi forms, the algorithm for solving the problem, and a detailed discussion of five selected examples to evaluate the method's performance. The authors also provide appendices supporting various sections of the main narrative. The dual formulation of the Burgers equation is derived by multiplying the primal equation with a dual field and integrating by parts, resulting in a dual functional. The dual-to-primal mapping is then used to bridge the primal and dual variables. The degenerate ellipticity of the dual formulation is analyzed, and the weak formulation of the dual equations is constructed. The algorithm for solving the problem involves solving a series of space-time subdomains, with each stage solving a distinct dual boundary value problem. The Newton-Raphson method is used to approximate the solutions, and the quality of the results is evaluated using exact entropy solutions. The paper demonstrates the effectiveness of the dual scheme through five examples, including expansion fans, shocks, double shocks, half N-waves, and standing shocks. The results show that the dual scheme can accurately capture the behavior of these complex solutions, even in the presence of non-unique weak solutions.This paper presents a novel approach to solving the inviscid Burgers equation and its Hamilton-Jacobi form by treating them as degenerate elliptic problems. The authors introduce a duality-based method that involves formulating the primal PDE as constraints and using a convex auxiliary potential to optimize. This approach leads to a dual variational principle, where the Euler-Lagrange equations of the dual problem are equivalent to the primal PDE system. The method is applied to approximate weak solutions of the Burgers equation, which are non-unique and can be recovered through specific constructive approaches. The paper is structured into several sections: the introduction, the dual formulation of the Burgers equation in conservation and Hamilton-Jacobi forms, the algorithm for solving the problem, and a detailed discussion of five selected examples to evaluate the method's performance. The authors also provide appendices supporting various sections of the main narrative. The dual formulation of the Burgers equation is derived by multiplying the primal equation with a dual field and integrating by parts, resulting in a dual functional. The dual-to-primal mapping is then used to bridge the primal and dual variables. The degenerate ellipticity of the dual formulation is analyzed, and the weak formulation of the dual equations is constructed. The algorithm for solving the problem involves solving a series of space-time subdomains, with each stage solving a distinct dual boundary value problem. The Newton-Raphson method is used to approximate the solutions, and the quality of the results is evaluated using exact entropy solutions. The paper demonstrates the effectiveness of the dual scheme through five examples, including expansion fans, shocks, double shocks, half N-waves, and standing shocks. The results show that the dual scheme can accurately capture the behavior of these complex solutions, even in the presence of non-unique weak solutions.