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 co
nsidered 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.
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 $Q$.
We give an elementary proof of a compact embedding theorem in abstract Sobolev spaces. The result is first presented in a general context and later specialized to the case of degenerate Sobolev spaces defined with respect to nonnegative quadratic for
ms. Although our primary interest concerns degenerate quadratic forms, our result also applies to nondegener- ate cases, and we consider several such applications, including the classical Rellich-Kondrachov compact embedding theorem and results for the class of s-John domains, the latter for weights equal to powers of the distance to the boundary. We also derive a compactness result for Lebesgue spaces on quasimetric spaces unrelated to Euclidean space and possibly without any notion of gradient.
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} Th
e 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.
