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Let me state and prove the following: Proposition. Let $E \subset \mathbb R^n$ be a set of finite perimeter. For $\mathcal L^1$-a.e. $\rho>0$ the following equality holds: $$ P(E \cap B_{\rho}) …

I would like to know if the answer to the following question is known. Let $d \ge 3$. What is the value of $$ \theta(d) := \max_{\mu} \int_{S^{d-1}} \int_{S^{d-1}} \cdots \int_{S^{d-1}} |x_1 \w …

It is well known that an integrable function $u \colon \mathbb R^d \to \mathbb R$ is said to be of bounded variation iff the distributional gradient $Du$ is (representable by) a finite Radon measure, …

The gradient/differential/exterior differential/divergence/curl are all strictly related first order differential operators. As far as I understood, they are the base of (co)homological theories in Eu …

Let $f \colon \mathbb R^d \to \mathbb R$ be a measurable function. Consider the following property: (P) there exist a negligible set $N \subset \mathbb R^d$ and function $T_f \in L^p(\mathbb R^d) …

I am reading this paper and I came across the following sentence: Throughout the paper we silently assume [...] that the density character (i.e. the minimum cardinality of a dense subset) of ev …

My background is essentially Geometric Measure Theory and its application to partial differential equations (e.g. linear and non-linear hyperbolic conservation laws). These are currently my research i …

1. Introduction. By-now classical results assert that minimal surfaces (in $\mathbb R^n$) are generically "smooth" out of a "small" set. Question. What are the known regularity results for anis …

The answer to your questions 1. & 2. can be found in Theorem 3.95 of the book Ambrosio, Fusco, Pallara Functions of Bounded Variation and Free Discontinuity Problems. Another very recent and excellen …