All Questions

Let $\Omega$ be an open bounded subset of $\mathbb{R}^n$ with $C^1$ boundary. Let $$ \begin{matrix} F: \mathbb{R}^n \times \mathbb{R}^N \times \mathbb{R}^{nN} \to \mathbb{R},& \\ (x,z,p) \mapsto F(...

Think of a beach ball on an pool of water or sand. Let $\left(\mathcal{M}^2,g\right)$ be a surface homeomorphic to a sphere, endowed with a Riemannian metric $g$, and $\left(\mathcal{N}^2,h\right)$ a ...

Let's assume we have a continuous and finite scalar function $f(x,y)$ over the $xy$ plane ($\mathbb{R}^{2}$) and this function is to be integrated over a bounded area (surface) $A\subset\mathbb{R}^{2}...

I am having difficulty formulating a problem, which involves optimizing a contour shape, into a well-posed variational form that would give a reasonable answer. Within a bounded region on the $xy$ ...

Let $g$ be a Riemannian metric on the $d$-dimensional flat space $\mathbb R^d$, and consider the usual Lagrangian $$L(x, \dot x) = \tfrac 1 2 g_{ij}(x) \dot x^i \dot x^j.$$ Let $\hat g := \sqrt g$ ...