Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
173 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Degenerate elliptic operators in $L_p$-spaces with complex $W^{2,\infty}$-coefficients (1607.06881v1)

Published 23 Jul 2016 in math.AP

Abstract: Let $c_{kl} \in W{2,\infty}(\mathbb{R}d, \mathbb{C})$ for all $k,l \in {1, \ldots, d}$. We consider the divergence form operator $A = - \sum_{k,l=1}d \partial_l (c_{kl} \, \partial_k) $in $L_2(\mathbb{R}d)$ when the coefficient matrix satisfies $(C(x) \, \xi, \xi) \in \Sigma_\theta$ for all $x \in \mathbb{R}d$ and $\xi \in \mathbb{C}d$, where $\Sigma_\theta$ be the sector with vertex 0 and semi-angle $\theta$ in the complex plane. We show that for all $p$ in a suitable interval the contraction semigroup generated by $-A$ extends consistently to a contraction semigroup on $L_p(\mathbb{R}d)$. For those values of $p$ we present a condition on the coefficients such that the space $C_c\infty(\mathbb{R}d)$ of test functions is a core for the generator on $L_p(\mathbb{R}d)$. We also examine the operator $A$ separately in the more special Hilbert space $L_2(\mathbb{R}d)$ setting and provide more sufficient conditions such that $C_c\infty(\mathbb{R}d)$ is a core.

Summary

We haven't generated a summary for this paper yet.