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Register a new userExistence and Spectral Theory for Weak Solutions of Neumann and Dirichlet Problems for Linear Degenerate Elliptic Operators with Rough Coefficients
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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 $Q$.
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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.
It is shown that the theory of real symmetric second-order elliptic operators in divergence form on $Ri^d$ can be formulated in terms of a regular strongly local Dirichlet form irregardless of the order of degeneracy. The behaviour of the corresponding evolution semigroup $S_t$ can be described in terms of a function $(A,B) mapsto d(A ;B)in[0,infty]$ over pairs of measurable subsets of $Ri^d$. Then [ |(phi_A,S_tphi_B)|leq e^{-d(A;B)^2(4t)^{-1}}|phi_A|_2|phi_B|_2 ] for all $t>0$ and all $phi_Ain L_2(A)$, $phi_Bin L_2(B)$. Moreover $S_tL_2(A)subseteq L_2(A)$ for all $t>0$ if and only if $d(A ;A^c)=infty$ where $A^c$ denotes the complement of $A$.
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 consider the Dirichlet and Neumann problems for second-order linear elliptic equations: $$-triangle u +operatorname{div}(umathbf{b}) =f quadtext{ and }quad -triangle v -mathbf{b} cdot abla v =g$$ in a bounded Lipschitz domain $Omega$ in $mathbb{R}^n$ $(ngeq 3)$, where $mathbf{b}:Omega rightarrow mathbb{R}^n$ is a given vector field. Under the assumption that $mathbf{b} in L^{n}(Omega)^n$, we first establish existence and uniqueness of solutions in $L_{alpha}^{p}(Omega)$ for the Dirichlet and Neumann problems. Here $L_{alpha}^{p}(Omega)$ denotes the Sobolev space (or Bessel potential space) with the pair $(alpha,p)$ satisfying certain conditions. These results extend the classical works of Jerison-Kenig (1995, JFA) and Fabes-Mendez-Mitrea (1998, JFA) for the Poisson equation. We also prove existence and uniqueness of solutions of the Dirichlet problem with boundary data in $L^{2}(partialOmega)$.
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.
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