Generalised solutions for fully nonlinear PDE systems and existence–uniqueness theorems

Abstract

We introduce a new theory of generalised solutions which applies to fully nonlinear PDE systems of any order and allows for merely measurable maps as solutions. This approach bypasses the standard problems arising by the application of Distributions to PDEs and is not based on either integration by parts or on the maximum principle. Instead, our starting point builds on the probabilistic representation of derivatives via limits of difference quotients in the Young measures over a toric compactification of the space of jets. After developing some basic theory, as a first application we consider the Dirichlet problem and we prove existence-uniqueness-partial regularity of solutions to fully nonlinear degenerate elliptic 2nd order systems and also existence of solutions to the ∞-Laplace system of vectorial Calculus of Variations in L ∞ .

On the numerical approximation of $$\infty $$ ∞ -harmonic mappings

“…In the vectorial case when n = 1, it is true in the sense of D-solutions of [27]. We conjecture this to also be true in the case of (1.1) when both n, N > 1, but this is not a consequence of the current results of [26] since the method of the existence proof was based on an ad-hoc method (an analytic counterpart of Gromov's "Convex Integration" for a differential inclusion) rather than on p-harmonic approximations. A complete proof of this conjecture, at least to date, eludes us, but recently we have made significant progress in this regard.…”

“…Motivated partly by the equations arising in L ∞ , a duality-free theory of generalised solutions which applies to general fully nonlinear systems of any order [26]. In particular, it allows to make sense of (1.1) in the appropriate regularity class of W 1,∞ (Ω, R N ) mappings.…”

“…In particular, it allows to make sense of (1.1) in the appropriate regularity class of W 1,∞ (Ω, R N ) mappings. Among other results, in [26] is the existence of a solution to the Dirichlet problem for (1.1) and in [28] variational characterisations of ∞-harmonic maps in terms of the functional (1.3) are proven. The idea behind this new notion of so-called D-solutions is briefly explained in Sects.…”

“…Up to the early 2010s, all considerations were restricted to the scalar case. The general vectorial case of (1.1) as well as the study of the associated PDE systems arising from more general first order functionals (1.5) has been initiated in a series of recent papers [20][21][22][23][24][25][26][27][28]. Besides the intrinsic mathematical interest of the field, L ∞ functionals can be applied in a variety of scenarios.…”

“…This happens because the range of the Jacobian matrix Du(x) ∈ R Nn may not be constant throughout the domain Ω. The appropriate duality-free notion of generalised solution for (1.1) has very recently been proposed in [26] and is based on the probabilistic representation of the Hessian. The latter may not exist classically and we define it "weakly" in a duality-free manner by considering the limits of difference quotients as Young measures.…”