Abstract In this article, we study the logarithmic Laplacian operator which is a singular integral operator with symbol We show that this operator has the integral representation with and where Γ is the Gamma function, is the Digamma function and is the Euler Mascheroni constant. This operator arises as formal derivative of fractional Laplacians at We develop the functional analytic framework for Dirichlet problems involving the logarithmic Laplacian on bounded domains and use it to characterize the asymptotics of principal Dirichlet eigenvalues and eigenfunctions of as As a byproduct, we then derive a Faber-Krahn type inequality for the principal Dirichlet eigenvalue of Using this inequality, we also establish conditions on domains giving rise to the maximum principle in weak and strong forms. This allows us to also derive regularity up to the boundary of solutions to corresponding Poisson problems.

中文翻译：

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对数拉普拉斯算子的狄利克雷问题

摘要 在本文中，我们研究了对数拉普拉斯算子，它是一个带符号的奇异积分算子。我们证明了这个算子具有积分表示，其中Γ 是伽玛函数，是二伽玛函数，是欧拉马斯切罗尼常数。这个算子作为分数拉普拉斯算子的形式导数出现在我们开发了涉及有界域上的对数拉普拉斯算子的狄利克雷问题的泛函分析框架，并用它来表征主狄利克雷特征值的渐近性和作为副产品，我们然后推导出 Faber -Krahn 型不等式的主要 Dirichlet 特征值 使用这个不等式，我们还建立了在弱和强形式中产生最大值原理的域上的条件。