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Certain surfaces in mechanics are endowed with the fundamental forms \begin{align} \text{I} &= \mathrm{d}u^2+\mathrm{d}v^2+2\cos\gamma\: \mathrm{d}u\: \mathrm{d}v \\ \text{II} &= \alpha\left(\gamma \r …

Consider the following system of PDEs for the dependent variables $\tau=\tau(u,v)$ and $\gamma=\gamma(u,v)$, with $(u,v)\in [0,a]^2$. $$ \begin{cases} \tau_u&=F\left( \gamma,\gamma_u,\gamma_v,\gamma_{ …

(This question comes as a particular case with specific boundary conditions of the system shown in mathSE) Consider the PDE system $$ \begin{cases} \xi_u^2+\eta_u^2=\left(1+\dfrac{\xi^2+\eta^2}{4} \ri …

Let $\Omega_1$ and $\Omega_2$ be (simply connected) domains on $\mathbb{R}^2$, with coordinates $(x,y)$ and $(X,Y)$ respectively. Given a (smooth) function $Z(X,Y)$ such that $Z\left(\partial \Omega_2 …

Consider a flat, simply connected surface endowed with the Riemannian metric $g_0=e^{2\Omega(u,v)}\left(\mathbb{d}^2u +\mathbb{d}^2v \right)$, so that $\Omega(u,v)$ is an arbitrary harmonic function. …

This is a cross-post. In the context of two dimensional elasticity theory, when considering deformations of flat membranes into spherical caps, one encounters the following hyperbolic system \begin{al …

Consider the equation $$ \begin{equation} \frac{\partial^2f}{\partial x\partial y}=f \end{equation} $$ on a Jordan domain (i.e. the interior of a simple, closed curve on the plane). The equation is hy …