arXiv subjects
Publications and source records attributed to Mariia Savchenko.
We study the local H\"older continuity of nonnegative solutions to doubly nonlinear equations by introducing a new technique that allows us to treat the cases where the equation is both singular and degenerate, up to specific Barenblatt numbers. Our argument relies on a new integral $L^1$-$L^1$ Harnack estimate, of independent interest.
We prove expansion of positivity and reduction of the oscillation results to the local weak solutions to a doubly nonlinear anisotropic class of parabolic differential equations with bounded and measurable coefficients, whose prototype is \begin{equation*} u_t-\sum\limits_{i=1}^N \left( u^{(m_i-1)(p_i-1)} \ |u_{x_i}|^{p_i-2} \ u_{x_i} \right)_{x_i}=0 , \end{equation*} for a restricted range of $p_i$s and $m_i$s, that reflects their competition for the diffusion. The positivity expansion relies on an exponential shift and is presented separately for singular and degenerate cases. Finally we present a study of the local oscillation of the solution for some specific ranges of exponents, within the singular and degenerate cases.
We establish the continuity of bounded solutions to the anisotropic elliptic equation $$-\sum\limits_{i=1}^N\Big(|u_{x_i}|^{p_i-2} u_{x_i}\Big)_{x_i}=f(x),\quad x\in Ω,\quad f(x)\in L^1(Ω)$$ under the conditions $$\min\limits_{1\leqslant i\leqslant N} p_i >1,\quad \sum\limits_{i=1}^N \frac{1}{p_i}=1$$ and $$\lim\limits_{ρ\rightarrow 0}\,\sup\limits_{x\in Ω}\int\limits^ρ_0\Big(\int\limits_{B_r(x)}|f(y)|\,dy\Big)^{\frac{1}{N-1}}\frac{dr}{r}=0.$$ In the standard case $p_1=...=p_N=N$, these conditions recover the known results for the $N$-Laplacian.
In the case $q> p\dfrac{n+2}{n}$, we give a proof of the weak Harnack inequality for non-negative super-solutions of degenerate double-phase parabolic equations under the additional assumption that $u\in L^{s}_{loc}(Ω_{T})$ with some $s >p\dfrac{n+2}{n}$.
We prove Harnack's type inequalities for bounded non-negative solutions of degenerate parabolic equations with $(p,q)$ growth $$ u_{t}-{\rm div}\left(\mid \nabla u \mid^{p-2}\nabla u + a(x,t) \mid \nabla u \mid^{q-2}\nabla u \right)=0,\quad a(x,t) \geq 0 , $$ under the generalized non-logarithmic Zhikovs conditions $$ \mid a(x,t)-a(y,τ)\mid \leqslant Aμ(r) r^{q-p},\quad (x,t),(y,τ)\in Q_{r,r}(x_{0},t_{0}),$$ $$\lim\limits_{r\rightarrow 0}μ(r) r^{q-p}=0,\quad \lim\limits_{r\rightarrow 0}μ(r)=+\infty,\quad \int\limits_{0} μ^{-β}(r)\frac{dr}{r} =+\infty,$$ with some $β>0$.