arXiv subjects
Publications and source records attributed to Mariia O. Savchenko.
We study the local regularity properties of weak solutions to a special class of anisotropic doubly nonlinear parabolic operators, whose prototype is the anisotropic Trudinger's equation. We prove a parabolic Harnack inequality, valid without any restrictions on the exponents $p_i$s. When the range of diffusion exponents is restricted, solutions are Hölder continuous.
We define a suitable class $\mathcal{PDG}$ of functions bearing unbalanced energy estimates, that are embodied by local weak subsolutions to doubly nonlinear, double-phase, Orlicz-type and fully anisotropic operators. Yet we prove that members of $\mathcal{PDG}$ are locally bounded, under critical, sub-critical and limit growth conditions typical of singular parabolic operators, with quantitative a priori estimates that follow the lines of the pioneering work of Ladyzhenskaya, Solonnikov and Uraltseva \cite{LadSolUra}. These local bounds are new in the sub-critical cases, even for the classic $p$-Laplacean equations, since no extra-integrability condition is needed.
We prove the weak Harnack inequality for the functions $u$ which belong to the corresponding De Giorgi classes $DG^{-}(Ω)$ under the additional assumption that $u\in L^{s}_{loc}(Ω)$ with some $s> 0$. In particular, our result covers new cases of functionals with a variable exponent or double-phase functionals under the non-logarithmic condition.