Emergent_Metrics_from_a_Minimal_Structural_Scalar

This manuscript develops a weak variational formulation of the Euler–Lagrange equations for partial differential equations associated with Lagrangians that may fail classical coercivity, convexity, or smoothness. Instead of focusing on the convergence of energies or minimizing sequences, the work establishes the stability and well-posedness of the Euler–Lagrange operator itself as a distribution under spatial regularization. Under minimal assumptions of local integrable growth and compatibility with mollification, it is proved that the Euler–Lagrange operators of regularized fields converge to a unique distributional limit. This yields a rigorous weak Euler–Lagrange identity for rough fields phi in L-infinity(0,T; H1_loc(Omega)) with dt phi in L2(0,T; H-1_loc(Omega)), even when standard coercivity and convexity assumptions are violated. The framework applies to subquadratic and borderline quadratic gradient growth and covers a broad class of degenerate and singular variational models, including porous-medium type energies, nonconvex phase-field theories, and p-Laplacian structures with 1 < p ≤ 2. By focusing on the convergence of the Euler–Lagrange operators rather than on minimizers, the paper provides a mathematically precise bridge between formal variational calculus and the analysis of degenerate and noncoercive partial differential equations.