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A Lax-Oleinik type explicit formula for 1D scalar balance laws has beenrecently obtained for the pure initial value problem by Adimurthi et al. in[1]. In this article, by introducing a suitable boundary functional, weestablish a Lax-Oleinik type formula for the initial boundary value problem.For the pure initial value problem, the solution for the correspondingHamilton-Jacobi equation turns out to be the minimizer of a functional on theset of curves known as h-curves. In the present situation, part of the h-curvejoining any two points in the quarter plane may cross the boundary $x = 0.$This phenomenon breaks the simplicity of the minimization process through theboundary functional compared to the case of conservation laws. Moreover, thiscomplicates the verification of the boundary condition in the sense of Bardos,le Roux, and Nedelec [2]. To verify the boundary condition, the boundary pointsare classified into three types depending on the structure of the minimizers atthose points. Finally, by introducing characteristic triangles, we constructgeneralized characteristics and show that the explicit solution is entropyadmissible.

Initial boundary value problem for 1D scalar balance laws with strictly convex flux