2018
DOI: 10.1007/s00205-018-1305-6
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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.

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Cited by 20 publications
(34 citation statements)
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References 39 publications
(49 reference statements)
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“…[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.…”
Section: Introductionmentioning
confidence: 99%
“…[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.…”
Section: Introductionmentioning
confidence: 99%
“…Theorem guarantees convergence to a candidate ‐harmonic function. The correct notion of weak solution to the limiting problem left()Δu3||normalD()Δu2=0,inΩ,u=g,on∂Ω,Du=Dg,on∂Ω, is that of D‐solutions . The solution is probabilistic in nature and interpreted in a weak sense.…”
Section: Approximation Via the P‐bilaplacianmentioning
confidence: 99%
“…, it has been shown that in one spatial dimension the problem does indeed have a unique absolutely minimizing D‐solution and in Ref. for higher spatial dimension.…”
Section: Introduction and The ∞‐Bilaplacianmentioning
confidence: 95%
“…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.…”
Section: Introductionmentioning
confidence: 99%