Existence, Uniqueness and Structure of Second Order Absolute Minimisers

Abstract

Let Ω ⊆ R n be a bounded open C 1,1 set. In this paper we prove the existence of a unique second order absolute minimiser u∞ of the functionalwith prescribed boundary conditions for u and Du on ∂Ω and under natural assumptions on F. We also show that u∞ is partially smooth and there exists a harmonic function f∞ ∈ L 1 (Ω) such that F(x, ∆u∞(x)) = e∞ sgn f∞(x)for all x ∈ {f∞ = 0}, where e∞ is the infimum of the global energy.

Explicit $$\infty$$ ∞ -harmonic functions in high dimensions

“…[9]). Since then, the field has been developed enormously by N. Katzourakis in the series of papers ( [11][12][13][14][15][16][17][18][19]) and also in collaboration with the author, Abugirda, Croce, Manfredi, Moser, Parini, Pisante and Pryer ( [1,2,6,[20][21][22][23][24][25]). A standard difficulty of (1.1) is that it is nondivergence form equation and since in general smooth solutions do not exist, the definition of generalised solutions is an issue.…”

On the numerical approximation of p‐biharmonic and ∞‐biharmonic functions

“…Theorem guarantees convergence to a candidate ∞‐harmonic function. The correct notion of weak solution to the limiting problem is that of ‐solutions . The solution is probabilistic in nature and interpreted in a weak sense.…”

“…, it has been shown that in one spatial dimension the problem does indeed have a unique absolutely minimizing ‐solution and in Ref. for higher spatial dimension.…”

A minimisation problem in L∞with PDE and unilateral constraints

“…In this paper, motivated by the problem of optical tomography and by the recent developments in Calculus of Variations in L ∞ appearing in the papers [38,39,40], we consider the problem of minimising over the class of all admissible parameters ξ a certain cost functional which measures the deviation of the solution v on the boundary ∂Ω from some predictionṽ of its values. Given the high complexity of the optical tomography problem, in this work which is the companion paper of [37] we will make the simplifying assumption that the diffusion coefficients A, B and the optical terms K, L do not depend explicitly on the dye distribution.…”