Existence and uniqueness of global solutions to fully nonlinear second order elliptic systems

We consider the problem of existence and uniqueness of strong a.e. solutions u:Rn⟶RNu:Rn⟶RN to the fully nonlinear PDE system F(⋅,D2u)=f, a.e. on Rn,(1) F(⋅,D2u)=f, a.e. on Rn,(1) when f∈L2(Rn)Nf∈L2(Rn)N and F is a Carathéodory map. (1) has not been considered before. The case of bounded domains has been studied by several authors, firstly by Campanato and under Campanato’s ellipticity condition on F. By introducing a new much weaker notion of ellipticity, we prove solvability of (1) in a tailored Sobolev “energy” space and a uniqueness estimate. The proof is based on the solvability of the linearised problem by Fourier transform methods, together with a “perturbation device” which allows to use Campanato’s near operators. We also discuss our hypothesis via counterexamples and give a stability theorem of strong global solutions for systems of the form (1)

Global regularity in Orlicz-Morrey spaces of solutions to nondivergence elliptic equations with VMO coefficients

We show continuity in generalized Orlicz-Morrey spaces MΦ,ϕ(ℝn) of sublinear integral operators generated by Calder´on-Zygmund operator and their commutators with BMO functions. The obtained estimates are used to study global regularity of the solution of the Dirichlet problem for linear uniformly elliptic operator (Formula presented.) with discontinuous coefficients. We show that Lu ∈ MΦ, φ implies the second-order derivatives belong to MΦ,φ. © 2018 Texas State Universit

Global regularity in Orlicz-Morrey spaces of solutions to nondivergence elliptic equations with VMO coefficients

We show continuity in generalized Orlicz-Morrey spaces M?,?(Rn) of sublinear integral operators generated by Calder´on-Zygmund operator and their commutators with BMO functions. The obtained estimates are used to study global regularity of the solution of the Dirichlet problem for linear uniformly elliptic operator (Formula presented.) with discontinuous coefficients. We show that Lu ? M?, ? implies the second-order derivatives belong to M?,?. © 2018 Texas State UniversityIndian National Science Academy --V. S. Guliyev and M. Omarova were partially supported by the 1st Azerbaijan-Russia Joint Grant Competition (Agreement number No. 18-51-06005), and by a grant from the Presidium of Azerbaijan National Academy of Science 2015. -

The Dirichlet problem in a class of generalized weighted Morrey spaces

We show continuity in generalized weighted Morrey spaces Mp,φ (w) of sub-linear integral operators generated by some classical integral operators and commutators. The obtained estimates are used to study global regularity of the solution of the Dirichlet problem for linear uniformly elliptic operators with discontinuous data