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Existence, uniqueness and structure of second order absolute minimisers
OAI: oai:purehost.bath.ac.uk:openaire_cris_publications/3dbcd953-13ec-4971-b29f-4834250864ab • DOI: https://doi.org/10.1007/s00205-018-1305-6
Abstract
Let Ω⊆Rn be a bounded open C1,1 set. In this paper we prove the existence of a unique second order absolute minimiser u∞ of the functional
E∞(u,O):=∥F(⋅,Δu)∥L∞(O),O⊆Ωmeasurable,
with 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∞∈L1(Ω) such that
F(x,Δu∞(x))=e∞sgn(f∞(x))
for all x∈{f∞≠0} , where e∞ is the infimum of the global energy.
E∞(u,O):=∥F(⋅,Δu)∥L∞(O),O⊆Ωmeasurable,
with 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∞∈L1(Ω) such that
F(x,Δu∞(x))=e∞sgn(f∞(x))
for all x∈{f∞≠0} , where e∞ is the infimum of the global energy.