Vectorial Calculus of Variations in L-infinity, generalised solutions for fully nonlinear PDE systems and applications to Data Assimilation

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Finding the extremal values of some physically meaningful, quantifiable entity is a ubiquitous problem of great importance in science. From antiquity, when the problem might have been to find the perimeter that enclosed the largest area of land, to the most sophisticated application nowadays, a complete solution to such problems always opens large horizons for applications and is intrinsically interesting to mathematicians, as it translates usually into hard and often technical questions. But the answers impact applications and everyday life, as in the example above.In particular, in classical Calculus of Variations one seeks to minimise a functional defined on a class of maps, typically such functionals are integrals and model some "energy". The extrema of these functionals satisfy a certain system of PDE (Partial Differential Equations) known as the Euler-Lagrange equations. In the early 1960s G. Aronsson initiated the study of functionals which are instead defined as a maximum. Except for the intrinsic mathematical interest connected to geometric problems, minimising the "max" of an energy provides more realistic models as opposed to the classical case of the "average" energy. "Calculus of Variations in L-infinity", as this area is known today, has undergone huge development since. However, until recently the theory was restricted exclusively to the scalar case and to first order variational problems (involving minimisation of the map and its first derivatives). In the early 2010s the PI pioneered the study of vectorial L-infinity problems for maps valued in higher-dimensional spaces and involving perhaps higher order derivatives. The vectorial case is of interest to a large number of real-world applications. The main reason that hindered the development of the vector case was the absence of the appropriate analytic framework: the new complicated equations possess singular "solutions" and a theory is needed in order to make rigorous sense and to be studied effectively. The problem is that standard PDE approaches based on either duality/integration-by-parts or on the maximum principle do not apply. In particular, the systems arising are non-divergence, highly nonlinear, degenerate and with discontinuous coefficients. The situation is analogous to that the mathematical community faced in the 1910s when attempting to understand and make rigorous sense of the "Dirac Delta" which arose in Quantum Theory. The development of the theory of "generalised functions" allowed the understanding of fundamental physical phenomena.Motivated by the newly discovered equations, the PI very recently proposed a novel theory of "generalised solutions" for fully nonlinear PDE systems of any order which allows for discontinuous solutions and coefficients. This approach is duality-free and relies on the probabilistic interpretation of those derivatives which do not exist classically. Our theory is a nonlinear alternative to distributions compatible with all existing approaches. In this setting, the PI has recently begun studying successfully certain cases of the L-infinity equations. The proposed research will continue the study of L-infinity variational problems and of their equations in the proper analytic framework. We are interested in developing new mathematical tools in order to study 1st and 2nd order variational problems and the associated PDE systems. A further particular focus will be to apply our results to models of variational Data Assimilation in Earth sciences and in weather forecasting. Mathematically, Data Assimilation faces problems which are not exactly solvable and instead one tries to minimise an "error" which describes the deviation of approximate solutions from being the exact solution we would like to have. By replacing the standard models currently used with their "max" counterparts, we could obtain better predictions: spikes of large errors are at the outset excluded when minimising the maximum.