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It is well known that $\mathbb{R}^2\ncong \mathbb{R}$. It is also known that $\mathbb{Q}^2\cong \mathbb{Q}$. It is a corollary to Sierpiński's theorem which states that every countable metric space wi …

Let $f:\mathbb{R}^2 \to \mathbb{R}^2$ be a smooth map whose differential has fixed distinct singular values $0<\sigma_1<\sigma_2$ and an everywhere positive determinant (which is the product $\sigm …

Let $g:[0,\infty] \to [0,\infty]$ be a smooth strictly increasing function satisfying $g(0)=0$ and $g^{(k)}(0)=0$ for every natural $k$. Is $\sqrt g$ is infinitely (right) differentiable at $x=0$? …

This is a cross-post. Let $U \subseteq \mathbb R^n$ be an open subset, and let $f:U \to \mathbb R$ be smooth. Suppose that $x \in U$ is a strict local minimum point of $f$. Let $df^k(x):(\mathbb R^n)^ …

Let $V$ be a compactly-supported vector field on $\mathbb{R}^2$, whose zeros inside some open neighbourhood of the closed unit disk $\mathbb{D}^2$ are isolated. Does there exist a sequence of vec …

Let $\Omega \subseteq \mathbb{R}^n$ be an open bounded domain with a smooth boundary. Fix $1<p<n$. Let $A \in W^{1,p}(\Omega;\text{End}(\mathbb{R}^n)) \cap C(\Omega;\text{End}(\mathbb{R}^n))$ with $\d …

While analyzing a variational problem, I came to the following question: Let $\mathbb D^n \subseteq \mathbb{R}^n$ be the closed $n$-dimensional unit ball, and let $f: \mathbb D^n \to \mathbb{R}^n$ …

This is a cross-post. Let $\Omega_1,\Omega_2 \subseteq \mathbb R^2$ be open, connected, bounded, with non-empty $C^1$ boundaries. Let $f_n:\bar\Omega_1 \to \bar\Omega_2$ be $C^1$ be bijective maps wit …

While solving a problem in calculus of variations, I came to the following question: Let $A,B$ be two real $2 \times 2$ matrices with positive determinants, and suppose that $\operatorname{rank}(A-B)= …

Let $\mathbb{D}^2$ be the closed two-dimensional unit disk, and let $g:\mathbb{D}^2 \to \mathbb{R}$ be a non-constant harmonic function (smooth up to the boundary). Does there exist a sequence of …

I am not sure if this is exactly research-level, but I am struggling to find a proof for the following claim: Let $F:[0,\infty) \to [0,\infty)$ be a $C^2$ strictly convex function. Let $\lambda_n \in …

Let $M$ be a smooth closed manifold, and let $g_0$ be a Riemannian metric on $M$. Let $U$ be a neighbourhood of $p \in M$, and suppose that we are given a metric $g$ on $U$, which satisfies $\| g-g_ …

While working on a variational problem I have reached to the following question: Let $f:(0,\infty) \to [0,\infty)$ be a $C^1$ function satisfying $f(1)=0$. Suppose that $f(x)$ is strictly increasing …

Let $M,N$ be smooth, connected, compact, oriented, two-dimensional Riemannian manifolds, with $C^k$ boundaries. Let $f:M \to N$ be a Lipschitz continuous weakly harmonic map**, and assume that $f(\par …

Let $a>b>0$. Suppose we want to minimize $$ f(x)=(x-a)^2+(1/x-b)^2, $$ over $x>0$. Equating $f'(x)=0$ leads to the quartic equation $$ g(x)=x^4-ax^3+bx-1=0. \tag{1} $$ Question: Is the smallest positi …