Weighted Energy-Dissipation principle for gradient flows in metric spaces

الأصل · EN

This paper develops the so-called Weighted Energy-Dissipation (WED) variational approach for the analysis of gradient flows in metric spaces. This focuses on the minimization of the parameter-dependent global-in-time functional of trajectories Iε[u] = ∫₀∞ e⁻ᵗ/ε(12 |u'|²(t) + 1εϕ(u(t))) t, featuring the weighted sum of energetic and dissipative terms. As the parameter ε is sent to 0, the minimizers uε of such functionals converge, up to subsequences, to curves of maximal slope driven by the functional ϕ. This delivers a new and general variational approximation procedure, hence a new existence proof, for metric gradient flows. In addition, it provides a novel perspective towards relaxation.