2014/08/22 by Nikos Katzourakis, Katzourakis, Nikos · 1 citation
Engineering · Mathematics · #32A50 #32W50 (Secondary) #35D30 #35J46 #35J47 #35J60 (Primary) #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Bounded function #Counterexample #Discrete mathematics #Energy (signal processing) #FOS: Mathematics #Fourier transform #Mathematical analysis #Mathematics #Nonlinear Partial Differential Equations #Nonlinear system #Order (exchange) #Perturbation (astronomy) #Physics #Pure mathematics #Quantum mechanics #Sobolev space #Stability and Controllability of Differential Equations #Uniqueness #math.AP #msc:32A50 #msc:32W50 #msc:35D30 #msc:35J46 #msc:35J47 #msc:35J60
paper · pdf · doi:10.48550/arxiv.1408.5423
Journal: Nonlinear Differential Equations and Applications, 27 pages, final version
openalex publication_date 2014/08/22 · arxiv created 2016/02/29 · arxiv updated 2016/03/01 · openalex created_date 2025/10/24 · openalex updated_date 2026/08/05
We consider the problem of existence and uniqueness of strong a.e. solutions\nu: \ℝn longrightarrow \ℝN to the fully nonlinear PDE\nsystem \
label1
tag1 F(
cdot,D2u )
,=
, f,
text a.e. on\n
mathbbRn, when f\∈ L2(\ℝn)N and F is a Carath 'eodory\nmap. eqref1 has not been considered before. The case of bounded domains has\nbeen studied by several authors, firstly by Campanato and under Campanato's\nellipticity condition on F. By introducing a new much weaker notion of\nellipticity, we prove solvability of eqref1 in a tailored Sobolev "energy"\nspace and a uniqueness estimate. The proof is based on the solvability of the\nlinearised problem by Fourier transform methods, together with a "perturbation\ndevice" which allows to use Campanato's near operators. We also discuss our\nhypothesis via counterexamples and give a stability theorem of strong global\nsolutions for systems of the form eqref1.\n