arXiv subjects
Publications and source records attributed to Scott Rodney.
We prove regularity estimates for weak solutions to the Dirichlet problem for a divergence form elliptic operator. We give $L^p$ estimates for the second derivative for $p<2$. Our work generalizes results due to Miranda [28].
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 $$X=-\text{div}(P\nabla )+{\bf HR}+{\bf S^\prime G} +F$$ in a geometric homogeneous space setting where the $n\times n$ matrix function $P=P(x)$ is allowed to degenerate. We give a maximum principle for weak solutions of $Xu\leq 0$ 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 Poincaré 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 forms. 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} \nabla'P(x)\nabla u +{\bf HR}u+{\bf S'G}u +Fu &=& f+{\bf T'g} \textrm{in}Θu&=&ϕ\textrm{on}\partial Θ.{eqnarray} The principal part $ξ'P(x)ξ$ of the above equation is assumed to be comparable to a quadratic form ${\cal Q}(x,ξ) = ξ'Q(x)ξ$ that may vanish for non-zero $ξ\in\mathbb{R}^n$. This is achieved using techniques of functional analysis applied to the degenerate Sobolev spaces $QH^1(Θ)=W^{1,2}(Ω,Q)$ and $QH^1_0(Θ)=W^{1,2}_0(Θ,Q)$ 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.
We derive local boundedness estimates for weak solutions of a large class of second order quasilinear equations. The structural assumptions imposed on an equation in the class allow vanishing of the quadratic form associated with its principal part and require no smoothness of its coefficients. The class includes second order linear elliptic equations as studied by D. Gilbarg and N. S. Trudinger [1998] and second order subelliptic linear equations as studied by E. Sawyer and R. L. Wheeden [2006 and 2010]. Our results also extend ones obtained by J. Serrin [1964] concerning local boundedness of weak solutions of quasilinear elliptic equations.