oalib
Search Results: 1 - 10 of 100 matches for " "
All listed articles are free for downloading (OA Articles)
Page 1 /100
Display every page Item
Harnack inequality for a class of degenerate elliptic operators  [PDF]
J. D. Fernandes,J. Groisman,S. T. Melo
Mathematics , 2002,
Abstract: We prove a Harnack inequality for a class of two-weight degenerate elliptic operators. The metric distance is induced by continuous Grushin-type vector fields. It is not know whether there exist cutoffs fitting the metric balls. This obstacle is bypassed by means of a covering argument that allows the use of rectangles in the Moser iteration.
A New Proof of the Sharp Hardy Inequality to the Grushin Type Operators
Grushin型算子精确Hardy不等式的一个新证明(英文)

LIU Hai-Feng,NIU Peng-Cheng,
刘海峰
,钮鹏程

中国科学院研究生院学报 , 2006,
Abstract: A generalized sharp Hardy inequality to the degenerate elliptic operators with respect to Grushin type operators is proved by ingeniously choosing test functions and calculating the maximum value of a quadratic equation.Some interesting corollaries are also listed.
Analysis of degenerate elliptic operators of Gru?in type  [PDF]
Derek W. Robinson,Adam Sikora
Mathematics , 2013,
Abstract: We analyze degenerate, second-order, elliptic operators $H$ in divergence form on $L_2(\Ri^{n}\times\Ri^{m})$. We assume the coefficients are real symmetric and $a_1H_\delta\geq H\geq a_2H_\delta$ for some $a_1,a_2>0$ where \[ H_\delta=-{\nabla}_{x_1}\cdot(c_{\delta_1, \delta'_1}(x_1)\,\nabla_{x_1})-c_{\delta_2, \delta'_2}(x_1)\,\nabla_{x_2}^2 \;. \] Here $x_1\in\Ri^n$, $x_2\in\Ri^m$ and $c_{\delta_i, \delta'_i}$ are positive measurable functions such that $c_{\delta_i, \delta'_i}(x)$ behaves like $|x|^{\delta_i}$ as $x\to0$ and $|x|^{\delta_i'}$ as $x\to\infty$ with $\delta_1,\delta_1'\in[0,1\rangle$ and $\delta_2,\delta_2'\geq0$. Our principal results state that the submarkovian semigroup $S_t=e^{-tH}$ is conservative and its kernel $K_t$ satisfies bounds \[ 0\leq K_t(x\,;y)\leq a\,(|B(x\,;t^{1/2})|\,|B(y\,;t^{1/2})|)^{-1/2} \] where $|B(x\,;r)|$ denotes the volume of the ball $B(x\,;r)$ centred at $x$ with radius $r$ measured with respect to the Riemannian distance associated with $H$. The proofs depend on detailed subelliptic estimations on $H$, a precise characterization of the Riemannian distance and the corresponding volumes and wave equation techniques which exploit the finite speed of propagation. We discuss further implications of these bounds and give explicit examples that show the kernel is not necessarily strictly positive, nor continuous.
Hardy Type Inequalities Related to Degenerate Elliptic Differential Operators  [PDF]
Lorenzo D'Ambrosio
Mathematics , 2006,
Abstract: We prove some Hardy type inequalities related to quasilinear second order degenerate elliptic differential operators L_p(u):=-\nabla_L^*(\abs{\nabla_L u}^{p-2}\nabla_L u). If \phi is a positive weight such that -L_p\phi>= 0, then the Hardy type inequality c\int_\Omega \frac{\abs u^p}{\phi ^p}\abs{\nabla_L \phi}^p d\xi \le \int_\Omega\abs{\nabla_L u}^p d\xi holds. We find an explicit value of the constant involved, which, in most cases, results optimal. As particular case we derive Hardy inequalities for subelliptic operators on Carnot Groups.
Sharp spectral multipliers for a new class of Grushin type operators  [PDF]
Peng Chen,Adam Sikora
Mathematics , 2012,
Abstract: We describe weighted Plancherel estimates and sharp Hebisch-M\"uller-Stein type spectral multiplier result for a new class of Grushin type operators. We also discuss the optimal exponent for Bochner-Riesz summability in this setting.
Global Irregularity For Degenerate Elliptic Operators  [PDF]
Michael Christ
Mathematics , 1995,
Abstract: Examples are given of degenerate elliptic operators on smooth, compact manifolds that are not globally regular in $C^\infty$. These operators degenerate only in a rather mild fashion. Certain weak regularity results are proved, and an interpretation of global irregularity in terms of the associated heat semigroup is given.
The Harnack inequality for a class of degenerate elliptic operators  [PDF]
Francois Hamel,Andrej Zlatos
Mathematics , 2011,
Abstract: We prove a Harnack inequality for distributional solutions to a type of degenerate elliptic PDEs in $N$ dimensions. The differential operators in question are related to the Kolmogorov operator, made up of the Laplacian in the last $N-1$ variables, a first-order term corresponding to a shear flow in the direction of the first variable, and a bounded measurable potential term. The first-order coefficient is a smooth function of the last $N-1$ variables and its derivatives up to certain order do not vanish simultaneously at any point, making the operators in question hypoelliptic.
H lder classes relative to degenerate elliptic operators as interpolation spaces
Ugo Gianazza,Vincenzo Vespri
Le Matematiche , 1998,
Abstract: The well known characterization ofH lder classes as interpolation spaces is here extended under suitable hypotheses to a class of spaces wherethe H lder continuity is given in terms of an intrinsic distance relative to degenerate elliptic operators of H rmander type.
Inverse source problem and null controllability for multidimensional parabolic operators of Grushin type  [PDF]
Karine Beauchard,Piermarco Cannarsa,Masahiro Yamamoto
Mathematics , 2013, DOI: 10.1088/0266-5611/30/2/025006
Abstract: The approach to Lipschitz stability for uniformly parabolic equations introduced by Imanuvilov and Yamamoto in 1998, based on Carleman estimates, seems hard to apply to the case of Grushin-type operators of interest to this paper. Indeed, such estimates are still missing for parabolic operators degenerating in the interior of the space domain. Nevertheless, we are able to prove Lipschitz stability results for inverse source problems for such operators, with locally distributed measurements in arbitrary space dimension. For this purpose, we follow a mixed strategy which combines the appraoch due to Lebeau and Robbiano, relying on Fourier decomposition, with Carleman inequalities for heat equations with nonsmooth coefficients (solved by the Fourier modes). As a corollary, we obtain a direct proof of the observability of multidimensional Grushin-type parabolic equations, with locally distributed observations, which is equivalent to null controllability with locally distributed controls.
Inverse coefficient problem for Grushin-type parabolic operators  [PDF]
Karine Beauchard,Piermarco Cannarsa
Mathematics , 2013,
Abstract: The approach to Lipschitz stability for uniformly parabolic equations introduced by Imanuvilov and Yamamoto in 1998 based on Carleman estimates, seems hard to apply to the case of Grushin-type operators studied in this paper. Indeed, such estimates are still missing for parabolic operators degenerating in the interior of the space domain. Nevertheless, we are able to prove Lipschitz stability results for inverse coefficient problems for such operators, with locally distributed measurements in arbitrary space dimension. For this purpose, we follow a strategy that combines Fourier decomposition and Carleman inequalities for certain heat equations with nonsmooth coefficients (solved by the Fourier modes).
Page 1 /100
Display every page Item


Home
Copyright © 2008-2017 Open Access Library. All rights reserved.