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Register a new userInterior continuity, continuity up to the boundary and Harnacks inequality for double-phase elliptic equations with non-logarithmic conditions
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We prove continuity and Harnacks inequality for bounded solutions to elliptic equations of the type $$ begin{aligned} {rm div}big(| abla u|^{p-2}, abla u+a(x)| abla u|^{q-2}, abla ubig)=0,& quad a(x)geqslant0, |a(x)-a(y)|leqslant A|x-y|^{alpha}mu(|x-y|),& quad x eq y, {rm div}Big(| abla u|^{p-2}, abla u big[1+ln(1+b(x), | abla u|) big] Big)=0,& quad b(x)geqslant0, |b(x)-b(y)|leqslant B|x-y|,mu(|x-y|),& quad x eq y, end{aligned} $$ $$ begin{aligned} {rm div}Big(| abla u|^{p-2}, abla u+ c(x)| abla u|^{q-2}, abla u big[1+ln(1+| abla u|) big]^{beta} Big)=0,& quad c(x)geqslant0, , betageqslant0,phantom{=0=0} |c(x)-c(y)|leqslant C|x-y|^{q-p},mu(|x-y|),& quad x eq y, end{aligned} $$ under the precise choice of $mu$.
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We continue to study regularity results for weak solutions of the large class of second order degenerate quasilinear equations of the form begin{eqnarray} text{div}big(A(x,u, abla u)big) = B(x,u, abla u)text{ for }xinOmega onumber end{eqnarray} as considered in our previous paper giving local boundedness of weak solutions. Here we derive a version of Harnacks inequality as well as local Holder continuity for weak solutions. The possible degeneracy of an equation in the class is expressed in terms of a nonnegative definite quadratic form associated with its principal part. No smoothness is required of either the quadratic form or the coefficients of the equation. Our results extend ones obtained by J. Serrin and N. Trudinger for quasilinear equations, as well as ones for subelliptic linear equations obtained by Sawyer and Wheeden in their 2006 AMS memoir article.
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.
Let $Omega subset mathbb{R}^{n+1}$ be an open set whose boundary may be composed of pieces of different dimensions. Assume that $Omega$ satisfies the quantitative openness and connectedness, and there exist doubling measures $m$ on $Omega$ and $mu$ on $partial Omega$ with appropriate size conditions. Let $Lu=-mathrm{div}(A abla u)$ be a real (not necessarily symmetric) degenerate elliptic operator in $Omega$. Write $omega_L$ for the associated degenerate elliptic measure. We establish the equivalence between the following properties: (i) $omega_L in A_{infty}(mu)$, (ii) $L$ is $L^p(mu)$-solvable for some $p in (1, infty)$, (iii) every bounded null solution of $L$ satisfies a Carleson measure estimate with respect to $mu$, (iv) the conical square function is controlled by the non-tangential maximal function in $L^q(mu)$ for some (or for all) $q in (0, infty)$ for any null solution of $L$, (v) $L$ is $mathrm{BMO}(mu)$-solvable, and (vi) every bounded null solution of $L$ is $varepsilon$-approximable for any $varepsilon>0$. On the other hand, we obtain a qualitative analogy of the previous equivalence. Indeed, we characterize the absolute continuity of $omega_L$ with respect to $mu$ in terms of local $L^2(mu)$ estimates of the truncated conical square function for any bounded null solution of $L$. This is also equivalent to the finiteness $mu$-almost everywhere of the truncated conical square function for any bounded null solution of $L$.
We classify positive solutions to a class of quasilinear equations with Neumann or Robin boundary conditions in convex domains. Our main tool is an integral formula involving the trace of some relevant quantities for the problem. Under a suitable condition on the nonlinearity, a relevant consequence of our results is that we can extend to weak solutions a celebrated result obtained for stable solutions by Casten and Holland and by Matano.
For a general class of divergence type quasi-linear degenerate parabolic equations with measurable coeffcients and lower order terms from non-linear Kato-type classes, we prove local boundedness and continuity of solutions, and the intrinsic Harnack inequality for positive solutions.
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