arXiv subjects
Publications and source records attributed to Igor Skrypnik.
We prove a local weak Harnack inequality for nonnegative weak super-solutions to the anisotropic Trudinger equation. As an application, we show a proof of the Harnack inequality that bypasses any sort of Krylov-Safonov covering argument. We further develop an analysis of global sub-potential lower bounds which ultimately leads to the identification of a Barenblatt-type profile.
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.
We study the boundary behavior of solutions to parabolic double-phase equations through the celebrated Wiener's sufficiency criterion. The analysis is conducted for cylindrical domains and the regularity up to the lateral boundary is shown in terms of either its $p$ or $q$ capacity, depending on whether the phase vanishes at the boundary or not. Eventually we obtain a fine boundary estimate that, when considering uniform geometric conditions as density or fatness, leads us to the boundary Hölder continuity of solutions. In particular, the double-phase elicits new questions on the definition of an adapted capacity.
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 study the decay towards the extinction that pertains to local weak solutions to fully anisotropic equations whose prototype is \[ \partial_t u= \sum_{i=1}^N \partial_i (|\partial_i u|^{p_i-2} \partial_i u), \qquad 1<p_i<2. \] Their rates of extinction are evaluated by means of several integral Harnack-type inequalities which constitute the core of our analysis and that are obtained for anisotropic operators having full quasilinear structure. Different decays are obtained when considering different space geometries. The approach is motivated by the research of new methods for strongly nonlinear operators, hence dispensing with comparison principles, while exploiting an intrinsic geometry that affects all the variables of the solution.
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 $β>0$.
We prove continuity and Harnack's inequality for bounded solutions to the equation $$ {\rm div}\big(|\nabla u|^{p(x)-2}\,\nabla u \big)=0, \quad p(x)= p + L\frac{\log\log\frac{1}{|x-x_{0}|}}{\log\frac{1}{|x-x_{0}|}},\quad L > 0, $$ under the precise non-logarithmic condition on the function $p(x)$.