The First and Second Monotone Integral Principles for Fundamental Solutions of Uniformly Elliptic Equations
نویسنده
چکیده
Abstract. Two optimal monotone integral principles (equivalently for the Laplacian, two sharp iso-weighted-volume inequalities) are established through extending the first and second integral bounds of H. Weinberger for the Green functions (i.e., fundamental solutions) of uniformly elliptic equations in terms of the layer-cake formula, a one-dimensional monotone integral principle, and the isoperimetric and Jenson’s inequalities with sharp constants. Surprisingly, a special setting of the first principle can be used to not only verify the lowdimensional Pólya conjecture for the principal eigenvalue of the Laplacian but also to characterize the geometry of the Nash inequality for a strong uniform elliptic equation.
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