2018 # 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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“…[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.…”

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.…”

confidence: 99%

“…Theorem guarantees convergence to a candidate ∞ ‐harmonic function. The correct notion of weak solution to the limiting problem $$\left\{\begin{array}{c}left{\left(\Delta u\right)}^{3}{\left(\mathrm{normalD}\left(\Delta u\right)\right)}^{2}=0,\phantom{\rule{1em}{0ex}}\text{in}\phantom{\rule{0.5em}{0ex}}\Omega ,\\ u=g,\phantom{\rule{6.4em}{0ex}}\mathrm{on}\phantom{\rule{0.5em}{0ex}}\mathrm{\partial \Omega},\\ \mathrm{D}u=\mathrm{D}g,\phantom{\rule{5em}{0ex}}\mathrm{on}\phantom{\rule{0.5em}{0ex}}\mathrm{\partial \Omega},\end{array}\right.$$ is that of $\mathcal{D}$‐solutions . The solution is probabilistic in nature and interpreted in a weak sense.…”

confidence: 99%

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

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.…”

confidence: 99%