Vectorial problems : sharp Lipschitz bounds and borderline regularity

• Main Author: De filippis, Cristiana
• Format: Dissertation
• Language: English
• Published: University of Oxford 2020
• Subjects: Partial Differential Equations; Regularity theory

This thesis is devoted to the proof of fine regularity properties of solutions to a broad class of variational problems including models from geometry, material science, continuum mechanics and particle physics. Our starting point is the analysis of the behavior of manifold-constrained minima to certain non-homogeneous functionals: under sharp assumptions, we prove that they are regular everywhere, except on a negligible, "singular" set of points. The presence of the singular set is in general unavoidable. Looking at minima as solutions to the associated Euler-Lagrange system does not help: it presents an additional component generated by the curvature of the manifold having critical growth in the gradient variable. For instance, sphere-valued harmonic maps satisfy in a suitably weak sense −∆u = |Du|2u. This turns out to be an insurmountable obstruction to regularity. It is then natural to consider general systems of type − div a(x, Du) = f (0.0.1) and study how the features of f and of the partial map x 7→ a(x, z) influence the regularity of solutions. In this respect, we are able to cover non-linear tensors with exponential type growth conditions as well as with unbalanced polynomial growth: we prove everywhere Lipschitz regularity for vector-valued solutions to (0.0.1) under optimal assumptions on forcing term and space-depending coefficients, [76]. When the system in (0.0.1) has the Double Phase structure: − div (|Du|p−2Du + a(x)|Du|q−2Du)= − div (|F|p−2F + a(x)|F| q−2F) 0 ≤ a(·) ∈ C0,α, 1 ≤ q/p ≤ 1 + α/n, we complete the Calderón-Zygmund theory started in [62] by dealing with the delicate borderline case q/p = 1 + α/n, which has been left open so far. Finally, we propose a new approach to the analysis of variational integrals with (p, q)-growth based on convex duality.