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Register a new userGlobal Lorentz estimates for non-uniformly nonlinear elliptic equations via fractional maximal operators
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This paper is a contribution to the study of regularity theory for nonlinear elliptic equations. The aim of this paper is to establish some global estimates for non-uniformly elliptic in divergence form as follows begin{align*} -mathrm{div}(| abla u|^{p-2} abla u + a(x)| abla u|^{q-2} abla u) = - mathrm{div}(|mathbf{F}|^{p-2}mathbf{F} + a(x)|mathbf{F}|^{q-2}mathbf{F}), end{align*} that arises from double phase functional problems. In particular, the main results provide the regularity estimates for the distributional solutions in terms of maximal and fractional maximal operators. This work extends that of cite{CoMin2016,Byun2017Cava} by dealing with the global estimates in Lorentz spaces. This work also extends our recent result in cite{PNJDE}, which is devoted to the new estimates of divergence elliptic equations using cut-off fractional maximal operators. For future research, the approach developed in this paper allows to attain global estimates of distributional solutions to non-uniformly nonlinear elliptic equations in the framework of other spaces.
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This paper continues the development of regularity results for quasilinear measure data problems begin{align*} begin{cases} -mathrm{div}(A(x, abla u)) &= mu quad text{in} Omega, quad quad qquad u &=0 quad text{on} partial Omega, end{cases} end{align*} in Lorentz and Lorentz-Morrey spaces, where $Omega subset mathbb{R}^n$ ($n ge 2$), $mu$ is a finite Radon measure on $Omega$, and $A$ is a monotone Caratheodory vector valued operator acting between $W^{1,p}_0(Omega)$ and its dual $W^{-1,p}(Omega)$. It emphasizes that this paper studies the `very singular case $1<p le frac{3n-2}{2n-1}$ and the problem is considered under the weak assumption, where the $p$-capacity uniform thickness condition is imposed on the complement of domain $Omega$. There are two main results obtained in our study pertaining to the global gradient estimates of solutions in Lorentz and Lorentz-Morrey spaces involving the use of maximal and fractional maximal operators. The idea for writing this working paper comes directly from the recent results by others in the same research topic, where global estimates for gradient of solutions for the `very singular case still remains a challenge, specifically related to Lorentz and Lorentz-Morrey spaces.
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