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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 …

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, …

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 …

Let $V \colon [0,1]\times \mathbb R^d \to \mathbb R^d$ be a Borel vector field which is globally bounded, $V \in L^\infty$. I am looking for a reference for the following result (which I suppose it …

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}) …

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 …

I am looking for a reference for the following kind of results. Let $\Gamma$ be the space of Lipschitz curves $\text{Lip}([0,1]; \mathbb R^d)$ equipped with the sup norm. Let $B$ be a Borel subset of …