Parabolic systems with measurable coefficients in weighted Orlicz spaces

We consider a parabolic system in divergence form with measurable coefficients in a non-smooth bounded domain when the associated nonhomogeneous term belongs to a weighted Orlicz space. We generalize the Calder'{o}n-Zygmund theorem for the weak solution of such a system as an optimal estimate in weighted Orlicz spaces, by essentially proving that the spatial gradient is as integrable as the nonhomogeneous term under a possibly optimal assumption on the coefficients and a minimal geometric assumption on the boundary of the domain

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)

Elliptic systems in generalized Morrey spaces

We obtain local regularity in generalized Morrey spaces for the strong solutions to 2b-order linear elliptic systems with discontinuous coefficients