This paper presents a novel approach to solving the inviscid Burgers equation in both conservation and Hamilton-Jacobi forms as degenerate elliptic problems. The method involves formulating the equation as a dual variational problem, where a strictly convex auxiliary potential is optimized to recover weak solutions. The dual variational principle is designed using a dual-to-primal (DtP) mapping, which allows for the transformation of the primal problem into a dual problem that can be solved using a Galerkin finite element method. The approach is shown to recover non-unique weak solutions and entropy solutions, which are essential for capturing the correct behavior of the inviscid Burgers equation. The dual formulation is analyzed in both conservation and Hamilton-Jacobi forms, with the latter involving a Hamiltonian function. The dual equations are shown to be degenerate elliptic, which has implications for the existence and uniqueness of solutions. The method is applied to various test cases, including expansion fans, shocks, double shocks, half N-waves, and N-waves, demonstrating its effectiveness in capturing the correct behavior of the inviscid Burgers equation. The results show that the dual scheme can accurately recover entropy solutions and handle discontinuities in the solution, even in the presence of shocks. The method is also shown to be robust in capturing the evolution of complex wave patterns, including the formation and dissipation of standing shocks. The approach is computationally efficient and provides a framework for solving a wide range of nonlinear hyperbolic problems.This paper presents a novel approach to solving the inviscid Burgers equation in both conservation and Hamilton-Jacobi forms as degenerate elliptic problems. The method involves formulating the equation as a dual variational problem, where a strictly convex auxiliary potential is optimized to recover weak solutions. The dual variational principle is designed using a dual-to-primal (DtP) mapping, which allows for the transformation of the primal problem into a dual problem that can be solved using a Galerkin finite element method. The approach is shown to recover non-unique weak solutions and entropy solutions, which are essential for capturing the correct behavior of the inviscid Burgers equation. The dual formulation is analyzed in both conservation and Hamilton-Jacobi forms, with the latter involving a Hamiltonian function. The dual equations are shown to be degenerate elliptic, which has implications for the existence and uniqueness of solutions. The method is applied to various test cases, including expansion fans, shocks, double shocks, half N-waves, and N-waves, demonstrating its effectiveness in capturing the correct behavior of the inviscid Burgers equation. The results show that the dual scheme can accurately recover entropy solutions and handle discontinuities in the solution, even in the presence of shocks. The method is also shown to be robust in capturing the evolution of complex wave patterns, including the formation and dissipation of standing shocks. The approach is computationally efficient and provides a framework for solving a wide range of nonlinear hyperbolic problems.