Higher order L∞ variational problems and the ∞-Polylaplacian

Nikos Katzourakis (Reading, UK)

Calculus of Variations in L∞ is a relatively new field initiated by Aronsson in the 1960s which is under active research since. Minimising the supremum of a function of the gradient is very challenging because the equations arising as the analogues of the Euler-Lagrange equations are non-divergence and highly degenerate. However, it provides more realistic models than the classical average functionals (integrals). In this talk I will discuss a very recent advance made jointly with T. Pryer (Reading, UK), where we initiated the study of 2nd order variational problems in L∞ , seeking to minimise the L∞ norm of a function of the hessian. We also derived and studied the associated PDE. The latter is fully nonlinear and of 3rd order. Special cases arise when the function is the Euclidean length of either the full hessian or of the Laplacian, leading to the ∞-Polylaplacian and the ∞-Bilaplacian respectively. Our analysis relies heavily on the recently proposed by the speaker theory of D-solutions, a general duality-free notion of generalised solutions for fully nonlinear PDE systems which do not support integration-by-parts.

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