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Grushin型退化椭圆算子的分析

Analysis of degenerate elliptic operators of Grushin type

Derek W. Robinson, Adam Sikora

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42 pages
发表信息：
Math. Zeit. {\bf260} (2008) 475-508
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摘要：

我们分析退化，二阶，椭圆算子$H$在$L_2({\bf R}^{n}\times{\bf R}^{m})$上的发散形式。我们假设系数是实对称的，并且对于某些$a_1,a_2>0$是$a_1H_δ\geq H\geq a_2H_δ$，其中\[ H_δ=-\nabla_{x_1} c_{δ_1, δ'_1}(x_1) \nabla_{x_1}-c_{δ_2, δ'_2}(x_1) \nabla_{x_2}^2 . \]这里$x_1\in{\bf R}^n$，$x_2\in{\bf R}^m$和$c_{δ_i, δ'_i}$是正可测函数，使得$c_{δ_i, δ'_i}(x)$的行为类似于$|x|^{δ_i}$作为$x\to0$，$|x|^{δ_i'}$作为$x\to\infty$，具有$δ_1,δ_1'\in[0,1>$和$δ_2,δ_2'\geq0$。我们的主要结果表明，子马尔可夫半群$S_t=e^{-tH}$是保守的，它的核$K_t$满足界限\[ 0\leq K_t(x ;y)\leq a (|B(x ;t^{1/2})| |B(y ;t^{1/2})|)^{-1/2} \]，其中$|B(x ;r)|$表示的体积球$B(x ;r)$中心在$x$与半径$r$测量相对于与$H$黎曼距离。证明依赖于详细的次椭圆估计$H$，精确表征的黎曼距离和相应的卷和波动方程技术，利用有限的传播速度。我们讨论了这些界限的进一步影响，并给出明确的例子，表明内核不一定是严格的积极的，也不连续。

英文摘要：

We analyze degenerate, second-order, elliptic operators $H$ in divergence form on $L_2({\bf R}^{n}\times{\bf R}^{m})$. We assume the coefficients are real symmetric and $a_1H_δ\geq H\geq a_2H_δ$ for some $a_1,a_2>0$ where \[ H_δ=-\nabla_{x_1} c_{δ_1, δ'_1}(x_1) \nabla_{x_1}-c_{δ_2, δ'_2}(x_1) \nabla_{x_2}^2 . \] Here $x_1\in{\bf R}^n$, $x_2\in{\bf R}^m$ and $c_{δ_i, δ'_i}$ are positive measurable functions such that $c_{δ_i, δ'_i}(x)$ behaves like $|x|^{δ_i}$ as $x\to0$ and $|x|^{δ_i'}$ as $x\to\infty$ with $δ_1,δ_1'\in[0,1>$ and $δ_2,δ_2'\geq0$. 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.

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