arXiv subjects
Publications and source records attributed to Mykhailo V. Voitovych.
We consider the Dirichlet problem for quasilinear elliptic equations with Musielak-Orlicz (p,q)-growth and non-logarithmic conditions on the coefficients. A sufficient Wiener-type condition for the regularity of a boundary point is established.
We prove the continuity of bounded solutions for a wide class of parabolic equations with $(p,q)$-growth $$ u_{t}-{\rm div}\left(g(x,t,|\nabla u|)\,\frac{\nabla u}{|\nabla u|}\right)=0, $$ under the generalized non-logarithmic Zhikov's condition $$ g(x,t,{\rm v}/r)\leqslant c(K)\,g(y,τ,{\rm v}/r), \quad (x,t), (y,τ)\in Q_{r,r}(x_{0},t_{0}), \quad 0<{\rm v}\leqslant Kλ(r), $$ $$ \quad \lim\limits_{r\rightarrow0}λ(r)=0, \quad \lim\limits_{r\rightarrow0} \frac{λ(r)}{r}=+\infty, \quad \int_{0} λ(r)\,\frac{dr}{r}=+\infty. $$ In particular, our results cover new cases of double-phase parabolic equations.
We prove continuity and Harnack's inequality for bounded solutions to elliptic equations of the type $$ \begin{aligned} {\rm div}\big(|\nabla u|^{p-2}\,\nabla u+a(x)|\nabla u|^{q-2}\,\nabla u\big)=0,& \quad a(x)\geqslant0, \\ |a(x)-a(y)|\leqslant A|x-y|^αμ(|x-y|),& \quad x\neq y, \\ {\rm div}\Big(|\nabla u|^{p-2}\,\nabla u \big[1+\ln(1+b(x)\, |\nabla u|) \big] \Big)=0,& \quad b(x)\geqslant0, \\ |b(x)-b(y)|\leqslant B|x-y|\,μ(|x-y|),& \quad x\neq y, \end{aligned} $$ $$ \begin{aligned} {\rm div}\Big(|\nabla u|^{p-2}\,\nabla u+ c(x)|\nabla u|^{q-2}\,\nabla u \big[1+\ln(1+|\nabla u|) \big]^β \Big)=0,& \quad c(x)\geqslant0, \, β\geqslant0,\phantom{=0=0} \\ |c(x)-c(y)|\leqslant C|x-y|^{q-p}\,μ(|x-y|),& \quad x\neq y, \end{aligned} $$ under the precise choice of $μ$.
We introduce elliptic and parabolic $\mathcal{B}_{1}$ classes that generalize the well-known $\mathfrak{B}_{p}$ classes of DeGiorgi, Ladyzhenskaya and Ural'tseva with $p>1$. New classes are applied to prove pointwise continuity of solutions of elliptic and parabolic equations with nonstandard growth conditions. Our considerations cover new cases of variable exponent and $(p, q)$-phase growth including the ,,singular-degenerate'' parabolic case $p<2<q$.