TY - JOUR
T1 - Second-Order L ∞ Variational Problems and the ∞-Polylaplacian
AU - Katzourakis, Nikos
AU - Pryer, Tristan
N1 - Journal: Advances in Calculus of Variations, 31 pages, 13 figures
PY - 2020/4/1
Y1 - 2020/4/1
N2 - In this paper we initiate the study of second-order variational problems in L∞, seeking to minimise the L∞ 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 H∈C1(ℝn×ns), for the functionalE∞(u,풪)=∥∥H(D2u)∥∥L∞(풪),u∈W2,∞(Ω),풪⊆Ω,the associated equation is the fully nonlinear third-order PDEA2∞u:=(HX(D2u))⊗3:(D3u)⊗2=0.Special cases arise when H is the Euclidean length of either the full hessian or of the Laplacian, leading to the ∞-polylaplacian and the ∞-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.
AB - In this paper we initiate the study of second-order variational problems in L∞, seeking to minimise the L∞ 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 H∈C1(ℝn×ns), for the functionalE∞(u,풪)=∥∥H(D2u)∥∥L∞(풪),u∈W2,∞(Ω),풪⊆Ω,the associated equation is the fully nonlinear third-order PDEA2∞u:=(HX(D2u))⊗3:(D3u)⊗2=0.Special cases arise when H is the Euclidean length of either the full hessian or of the Laplacian, leading to the ∞-polylaplacian and the ∞-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.
KW - math.AP
KW - math.NA
U2 - 10.1515/acv-2016-0052
DO - 10.1515/acv-2016-0052
M3 - Article
VL - 13
JO - Advances in Calculus of Variations
JF - Advances in Calculus of Variations
SN - 1864-8258
IS - 2
ER -