Improved Hölder regularity for strongly elliptic PDEs

Authors: Kari Astala, Albert Clop, Daniel Faraco (ICMAT-UAM), Jarmo Jaaskelainen and Aleksis Koski

Source: Journal de Mathematiques Pures Et Appliquees

Date of publication: August 2020



Hilbert XIX problem can be interpreted as trying to mathematically prove or disprove whether nature uses beauty as a selection mechanism. At least in solid mechanics, the principle of minimum total potential energy interprets this question as asking whether the minimizers of integral functionals are smooth if the lagrangian is smooth. This is the starting point of the theory of elliptic regularity. For convex scalar functionals the answer is yes. The solution to the HIlbert problem was obtained independently in the 1960 by Nash in Princeton and by Enrio De Giorgi in a small village in the south of Italy. The research on elliptic regularity has been enormous since then modifying the properties of the lagrangian in multiple ways. A branch of the theory has been to make quantitative versions of De Giorgi-Nash theory.

That is to say make precise the more convex and regular is the lagrangian the more regular is the minima. Recall, that the starting point of the Calculus of Variations as stablished by Euler and Lagrange in mid eighteen century is that extremal of variational problems satisfy the corresponding Euler Lagrange equation. Convexity of the functional translates into ellipticity of the equation and regularity of the lagrangian into regularity of the Euler Lagrange equation. Perhaps the easiest setting is the case of an autonomous functional (which is a natural assumption for the author). The classical theory yields the non linearity is C¹ the minimizers are for every 0 < α < 1.

The reviewed article is part of a program of understanding elliptic equations with complex analytic methods, particularly the theory of quasiconformal mappings. To the surprise of many we prove a striking new result in the autonomous case. If the nonlinear function in the Euler Lagrange equation is only Lipchitz, it follows that the regularity of the solution depend on the ellipticity constants. Being pessimistic one would then expect after linearization, that the solution behaves as the solution of a linear elliptic equation with measurable coefficients. Indeed, this was the strategy known back to Hilbert’s times. However, we prove that if the original equation is autonomous there is an automatic improve of the integrability. In summary we obtain two coefficients 1/K < α(K) < β(K) < 1 such that u ∈ but not in

The proof reduces the second order equation to a nonlinear elliptic system (autonomous) using the, by now, popular method of Bers and Niremberg. This formalism allows to detect an improvement for a Poincare inequality which is the ingredient that remain oblivious to approaches based on real variable methods. It remains as a beautiful challenge for the energetic reader to decide wether α(K) = β(K) or not.