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Register a new user$mathcal{B}_{1}$ classes of DeGiorgi-Ladyzhenskaya-Uraltseva and their applications to elliptic and parabolic equations with generalized Orlicz growth conditions
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We introduce elliptic and parabolic $mathcal{B}_{1}$ classes that generalize the well-known $mathfrak{B}_{p}$ classes of DeGiorgi, Ladyzhenskaya and Uraltseva 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 $p<2<q$.
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We prove the continuity of bounded solutions for a wide class of parabolic equations with $(p,q)$-growth $$ u_{t}-{rm div}left(g(x,t,| abla u|),frac{ abla u}{| abla u|}right)=0, $$ under the generalized non-logarithmic Zhikovs condition $$ g(x,t,{rm v}/r)leqslant c(K),g(y,tau,{rm v}/r), quad (x,t), (y,tau)in Q_{r,r}(x_{0},t_{0}), quad 0<{rm v}leqslant Klambda(r), $$ $$ quad limlimits_{rrightarrow0}lambda(r)=0, quad limlimits_{rrightarrow0} frac{lambda(r)}{r}=+infty, quad int_{0} lambda(r),frac{dr}{r}=+infty. $$ In particular, our results cover new cases of double-phase parabolic equations.
We investigate existence and uniqueness of bounded solutions of parabolic equations with unbounded coefficients in $Mtimes mathbb R_+$, where $M$ is a complete noncompact Riemannian manifold. Under specific assumptions, we establish existence of solutions satisfying prescribed conditions at infinity, depending on the direction along which infinity is approached. Moreover, the large-time behavior of such solutions is studied. We consider also elliptic equations on $M$ with similar conditions at infinity.
The solvability in Sobolev spaces $W^{1,2}_p$ is proved for nondivergence form second order parabolic equations for $p>2$ close to 2. The leading coefficients are assumed to be measurable in the time variable and two coordinates of space variables, and almost VMO (vanishing mean oscillation) with respect to the other coordinates. This implies the $W^{2}_p$-solvability for the same $p$ of nondivergence form elliptic equations with leading coefficients measurable in two coordinates and VMO in the others. Under slightly different assumptions, we also obtain the solvability results when $p=2$.
We study both divergence and non-divergence form parabolic and elliptic equations in the half space ${x_d>0}$ whose coefficients are the product of $x_d^alpha$ and uniformly nondegenerate bounded measurable matrix-valued functions, where $alpha in (-1, infty)$. As such, the coefficients are singular or degenerate near the boundary of the half space. For equations with the conormal or Neumann boundary condition, we prove the existence, uniqueness, and regularity of solutions in weighted Sobolev spaces and mixed-norm weighted Sobolev spaces when the coefficients are only measurable in the $x_d$ direction and have small mean oscillation in the other directions in small cylinders. Our results are new even in the special case when the coefficients are constants, and they are reduced to the classical results when $alpha =0$
We consider both divergence and non-divergence parabolic equations on a half space in weighted Sobolev spaces. All the leading coefficients are assumed to be only measurable in the time and one spatial variable except one coefficient, which is assumed to be only measurable either in the time or the spatial variable. As functions of the other variables the coefficients have small bounded mean oscillation (BMO) semi-norms. The lower-order coefficients are allowed to blow up near the boundary with a certain optimal growth condition. As a corollary, we also obtain the corresponding results for elliptic equations.
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