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We study the lower semicontinuity for functionals defined on compact sets in R^2 with a finite number of connected components and finite length which depend on their normal vector. We apply the result to the study of quasi-static growth of brittle fractures in linearly elastic inhomogeneous and anisotropic bodies.
In this paper, we prove an extension theorem for spheres of square radii in $mathbb{F}_q^d$, which improves a result obtained by Iosevich and Koh (2010). Our main tool is a new point-hyperplane incidence bound which will be derived via a cone restric
We present a new data-driven paradigm for variational brittle fracture mechanics. The fracture-related material modeling assumptions are removed and the governing equations stemming from variational principles are combined with a set of discrete data
A new quantum action-based theory, Dynamic Quantized Fracture Mechanics (DQFM), is presented that modifies continuum-based dynamic fracture mechanics. The crack propagation is assumed as quantized in both space and time. The static limit case corresp
Let $X$ be a ball Banach function space on ${mathbb R}^n$. In this article, under the mild assumption that the Hardy--Littlewood maximal operator is bounded on the associated space $X$ of $X$, the authors prove that, for any $fin C_{mathrm{c}}^2({mat
Classical statistical average values are generally generalized to average values of quantum mechanics, it is discovered that quantum mechanics is direct generalization of classical statistical mechanics, and we generally deduce both a new general con