On a class of second order variational problems with constraints

1999 ◽  
Vol 111 (1) ◽  
pp. 1-28 ◽  
Author(s):  
Moshe Marcus ◽  
Alexander J. Zaslavski
Keyword(s):  
Variational Problems ◽  
Second Order

Second order duality for variational problems involving generalized convexity

OPSEARCH ◽  
2014 ◽  
Vol 52 (3) ◽  
pp. 582-596 ◽  
Author(s):  
Anurag Jayswal ◽  
I. M. Stancu-Minasian ◽  
Sarita Choudhury
Keyword(s):  
Generalized Convexity ◽  
Variational Problems ◽  
Second Order ◽  
Second Order Duality

scholarly journals Second-order L∞ variational problems and the ∞-polylaplacian

2020 ◽  
Vol 13 (2) ◽  
pp. 115-140 ◽  
Author(s):  
Nikos Katzourakis ◽  
Tristan Pryer
Keyword(s):  
Critical Point ◽  
Numerical Experiments ◽  
Lagrange Equation ◽  
Variational Problems ◽  
Second Order ◽  
Third Order ◽  
Fully Nonlinear ◽  
Euclidean Length ◽  
Special Cases

AbstractIn this paper we initiate the study of second-order variational problems in {L^{\infty}}, seeking to minimise the {L^{\infty}} norm of a function of the hessian. We also derive and study the respective PDE arising as the analogue of the Euler–Lagrange equation. Given {\mathrm{H}\in C^{1}(\mathbb{R}^{n\times n}_{s})}, for the functional\mathrm{E}_{\infty}(u,\mathcal{O})=\|\mathrm{H}(\mathrm{D}^{2}u)\|_{L^{\infty}% (\mathcal{O})},\quad u\in W^{2,\infty}(\Omega),\mathcal{O}\subseteq\Omega,{}the associated equation is the fully nonlinear third-order PDE\mathrm{A}^{2}_{\infty}u:=(\mathrm{H}_{X}(\mathrm{D}^{2}u))^{\otimes 3}:(% \mathrm{D}^{3}u)^{\otimes 2}=0.{}Special cases arise when {\mathrm{H}} is the Euclidean length of either the full hessian or of the Laplacian, leading to the {\infty}-polylaplacian and the {\infty}-bilaplacian respectively. We establish several results for (1) and (2), including existence of minimisers, of absolute minimisers and of “critical point” generalised solutions, proving also variational characterisations and uniqueness. We also construct explicit generalised solutions and perform numerical experiments.

An invariance theory for second‐order variational problems

1975 ◽  
Vol 16 (7) ◽  
pp. 1374-1379 ◽  
Author(s):  
J. D. Logan ◽  
J. S. Blakeslee
Keyword(s):  
Variational Problems ◽  
Second Order ◽  
Invariance Theory
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