Cross theorems with singularities are important results in mathematics that deal with the behavior of solutions to partial differential equations (PDEs) in the presence of singularities. These theorems provide insights into the regularity and decay properties of solutions near singular points, and they play a crucial role in the study of PDEs with non-smooth coefficients or irregular domains.
In the mathematical perspective, cross theorems with singularities establish quantitative estimates on the regularity of solutions to PDEs near singular points. These singularities can arise from various sources, such as the presence of corners, edges, or non-smooth coefficients in the equations. The theorems characterize the decay rates and the optimal regularity of solutions in the vicinity of these singular points.
Cross theorems are particularly relevant in the study of elliptic and parabolic PDEs. For elliptic equations, the theorems address the regularity of solutions in domains with non-smooth boundaries or coefficients. They provide estimates on the smoothness of solutions near the singular points and reveal the connection between the regularity of solutions and the geometric properties of the domain.
