Existence Solution for a Non-smooth System

‎A variational method for a mixed boundary value problem in mathematical physics is considered‎. ‎Using two-field Lagrange multipliers‎, ‎we would investigate a variational formulation containing a mixed variational problem which is equivalent with a problem of saddle point type‎. ‎With a keen focus on two-field Lagrange multipliers‎, ‎we introduce a variational formulation comprising a mixed variational problem that seamlessly aligns with a saddle point problem (Problem 3})‎. Consequently, the distinctive solvability of the weak formulation we propose is intricately governed by the principles of saddle point theory, marking a significant advancement in our understanding of mathematical physics under challenging conditions.