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What is the relationship between the definitions of $s$-capacity (page 13 here) and $p$-capacity (here) of a set? Are they equivalent? If not, what inequalities hold? What is the difference (in terms …

Let us consider the Burgers equation $$u_t + (u^2)_x = 0$$ In Liu, Tai-Ping; Pierre, Michel, Source-solutions and asymptotic behavior in conservation laws, J. Differ. Equations 51, 419-441 (1984). Z …

Let us consider $$\mathcal X = \{(a,b)|a \in A, \ b = f(a)\}, $$ where $A \subset L^1(\mathbb R)$ has empty interior and $f:L^1 \to L^1$ is a bijective map. Does $\mathcal X$ also have empty interior? …

Let $\Omega$ be an open (non empty) set and $u:\Omega \subset \mathbb{R}^N \to \mathbb{R}^M$ be a function such that the Hausdorff dimension of its graph is $N$. Let $\tilde u = u$ a.e. Is it true th …

Consider the problem $$(\star) \quad \begin{cases} \frac{d}{dt} X(t,x) = \begin{cases} - 1 & \text{ if } X(t,x) >0, \\ 1 & \text{ if } X(t,x) < 0 \end{cases}, &t \in [0,T],\\ X(0,x) = x, &x \in \mat …

Let $u,\phi:\mathbb R \to \mathbb R$ be smooth functions and $\Omega_\epsilon$ be a bounded domain in $\mathbb R$ with diameter $\epsilon>0$ (consider for exaple the ball $B_{\epsilon/2}(0)$). Is it t …

Let $u:\mathbb R_+ \times \Omega \subset \mathbb R^N \to \mathbb R$ (sufficiently smooth). Are the following statements are equivalent? For every $\tau >0$ and level surface $S$ of $u(\tau,\cdot)$, i …

What is an example of vector field $c: \mathbb R_+ \times \mathbb R^N \to \mathbb R^N$ with $c \in L^1(\mathbb{R}_+, BV(\mathbb R^N))$ without bounded divergence $div_x c$ but such that there exists a …

How can we estimate the Kolmogorov $\epsilon$-entropy $$H_\epsilon (A,L^1(\mathbb R))$$ where $ A = \{f:\mathbb R \to [0,K] \text{ s.t. $f \in L^1$ and has total variation $TV(f) \le M$}\} $?

Consider the initial value problem $$\dot x(t) = F(t,x), \quad t \in (0,T)$$ with given initial datum $$x(0) = x_0 \in \mathbb R.$$ More precisely we consider the integral equation $$x(t)=x(0)+\int_0^ …

Let $u$ be a divergence-free vector field $u:\mathbb R^n \to \mathbb R^ n$. Does the following inequality hold? $$\Big( \int_{\mathbb R^n} |u|^2 dx\Big)^2 \le C\Big(\int_{\mathbb R^n} |u|^2|x|^2 dx \B …

If $f$ is Lipschitz, then the following holds for the Hausdorff dimension: $$\dim_H f(A) \le \dim_H A.$$ Is the same true for the box counting dimension?

Let $u: \Omega\subset \mathbb{R}^N \to \mathbb{R}^M$ be a $BV$ function. Is the box counting dimension of the graph of $u$ equal to $N$? How can we prove it? The analogous question for the Hau …

Let $u \in [BV(\mathbb R^N)]^N$. We have $$D^{jump} u(x) = a(x) \otimes b(x)|D^{jump}u|,$$ where $a,b \in \mathbb S^{N-1}$. What is the projection of $D^{jump}u$ in the direction $a$? And how can w …

Motivated by my previous question Alberti rank-one theorem and irregular jump discontinuities, I'd like to ask the following: Is there a function $u \in BV_{loc}(\mathbb{R}^2)$ such that $Du$ is a …