AI 中文总结
研究非局部\(\rho -\)拉普拉斯算子，通过确定核\(\rho\)条件，建立其与积分 - 微分椭圆算子类的联系，并在更弱\(\rho\)条件下建立最大值和比较原理，涵盖更广泛算子族，且对核假设最小。
AI 中文摘要
我们研究非局部\(\rho -\)拉普拉斯算子，其定义为与一般径向核\(\rho\)相关的非局部散度和梯度算子的复合：\(\Delta _\rho u = \text{div}_\rho (D_\rho u)\)。我们的第一个主要贡献是在该算子与费尔南德斯 - 雷亚尔和罗斯 - 奥顿所研究的积分 - 微分椭圆算子类之间建立精确联系，确定核\(\rho\)上保证属于该类的明确条件。第二个主要贡献涉及\(\rho -\)拉普拉斯算子的最大值和比较原理。我们在比积分 - 微分类所需条件严格更弱的\(\rho\)条件下建立了强最大值原理和弱最大值原理，从而涵盖了更广泛的算子族。结果仅对核有最小假设，尤其不依赖任何分数型可比性条件。
英文摘要
We study the nonlocal $ρ$-Laplacian, defined as the composition of the nonlocal divergence and gradient operators associated with a general radial kernel $ρ$: $Δ_ρu=\mbox{div}_ρ\left(D_ρu\right)$. Our first main contribution is to establish a precise connection between this operator and the class of integro-differential elliptic operators studied by Fernández-Real and Ros-Oton (\cite{FernandezRos}), identifying explicit conditions on the kernel $ρ$ that guarantee membership in this class. Our second main contribution concerns maximum and comparison principles for the $ρ$-Laplacian. We establish both a strong and a weak maximum principle under conditions on $ρ$ that are strictly weaker than those required for membership in the integro-differential class, thereby covering a genuinely broader family of operators. The results require only minimal assumptions on the kernel, and in particular do not rely on any fractional-type comparability condition.
