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Register a new userHarnacks inequality and Holder continuity for weak solutions of degenerate quasilinear equations with rough coefficients
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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.
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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 $mu$.
In this paper we study existence and spectral properties for weak solutions of Neumann and Dirichlet problems associated to second order linear degenerate elliptic partial differential operators $X$, with rough coefficients of the form $$X=-text{div}(P abla )+{bf HR}+{bf S^prime G} +F$$ in a geometric homogeneous space setting where the $ntimes n$ matrix function $P=P(x)$ is allowed to degenerate. We give a maximum principle for weak solutions of $Xuleq 0$ and follow this with a result describing a relationship between compact projection of the degenerate Sobolev space $QH^{1,p}$ into $L^q$ and a Poincare inequality with gain adapted to $Q$.
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.
This article addresses the local boundedness and Holder continuity of weak solutions to kinetic Fokker-Planck equations with general transport operators and rough coefficients. These results are due to the mixing effect of diffusion and transport. Although the equation is parabolic only in the velocity variable, it has a hypoelliptic structure provided that the transport part $partial_t+b(v)cdot abla_x$ is nondegenerate in some sense. We achieve the results by revisiting the method, proposed by Golse, Imbert, Mouhot and Vasseur in the case $b(v)= v$, that combines the elliptic De Giorgi-Nash-Moser theory with velocity averaging lemmas.
This article gives an existence theory for weak solutions of second order non-elliptic linear Dirichlet problems of the form {eqnarray} ablaP(x) abla u +{bf HR}u+{bf SG}u +Fu &=& f+{bf Tg} textrm{in}Theta u&=&phitextrm{on}partial Theta.{eqnarray} The principal part $xiP(x)xi$ of the above equation is assumed to be comparable to a quadratic form ${cal Q}(x,xi) = xiQ(x)xi$ that may vanish for non-zero $xiinmathbb{R}^n$. This is achieved using techniques of functional analysis applied to the degenerate Sobolev spaces $QH^1(Theta)=W^{1,2}(Omega,Q)$ and $QH^1_0(Theta)=W^{1,2}_0(Theta,Q)$ as defined in recent work of E. Sawyer and R. L. Wheeden. The aforementioned authors in referenced work give a regularity theory for a subset of the class of equations dealt with here.
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