MathOverflow

I have recently developed a geometric proof of the Pythagorean Theorem using the Power of a Point Theorem. While I am aware that proofs using circle power exist, I found my specific construction using concentric circles to be somewhat unique, and I would...

For every $1 \leq j \leq N$, let $L_j(x) = a_{j,1} x_1 + ... + a_{j,R} x_R$ be linear forms with real coefficients. The first $R$ of them are $L_j(x) = a_{j,j} x_j$, $1 \leq j \leq R$.
and $N > R$. Let$$Z(c, \varrho) = \{ x \in [0,1]^R: | L_j (x) - c_j | < \varrho \text{ for each } 1 \leq j \leq N\}$$I am interested in obtaining an upper bound for the measure of $Z(c, \varrho)$. An easy upper bound is $\frac{\varrho^R}{|a_...

Let $p>1$, $\mu>0$, and let $u \in W^{1,p}(\mathbb{R}^n)\setminus\{0\}$.In a paper I am reading, the authors use the following inequality
$p \int_{\mathbb{R}^n} |u(x)|^p \log\!\left(\frac{|u(x)|}{\|u\|_{L^p(\mathbb{R}^n)}}\right)\,dx+ \frac{n}{p}\log\!\left(\frac{p\mu e}{n\mathcal{L}_p}\right)\int_{\mathbb{R}^n} |u(x)|^p\,dx\le \mu \int_{\mathbb{R}^n} |\nabla u(x)|^p\,dx,$...

Let us consider the following linear system with damping:$$\begin{cases}u_t - u_x = -\frac{1}{2} (u+v)\\v_t + v_x = -\frac{1}{2} (u+v)\end{cases}$$Let's write the solution as $w=(u,v)$ corresponding to initial data $w_0 := w(0,\cdot) \in L^1(\mathbb...