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Results tagged with ap.analysis-of-pdes
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user 171439
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
1
vote
0
answers
78
views
Nonlinear, 1st order system of PDEs with variables interchanged
(This question comes as a particular case with specific boundary conditions of the system shown in mathSE)
Consider the PDE system
$$
\begin{cases}
\xi_u^2+\eta_u^2=\left(1+\dfrac{\xi^2+\eta^2}{4} \ri …
- 134
1
vote
Asymptotics for repulsive aggregation(-diffusion) equation
(Too long to be a comment.) A hand-waving, physicist calculation of a specific example for $d=3$ goes as follows. It may be put in more rigours terms for higher but not for lower dimensions. Let's tak …
- 134
1
vote
0
answers
292
views
Solutions of a Gauss–Codazzi-like system of nonlinear PDEs
Consider the following system of PDEs for the dependent variables $\tau=\tau(u,v)$ and $\gamma=\gamma(u,v)$, with $(u,v)\in [0,a]^2$.
$$
\begin{cases}
\tau_u&=F\left( \gamma,\gamma_u,\gamma_v,\gamma_{ …
- 134
1
vote
1
answer
254
views
Beltrami equation with harmonic coefficient
I need to find solutions to the Beltrami equation
$$
\frac{\partial w}{\partial\overline{{z}}}=e^{i\varphi(z)}\frac{\partial w}{\partial z}
$$
for $w=w(z,\overline{z})$ and $\varphi(z)$ some given, r …
- 134
5
votes
2
answers
271
views
Linear hyperbolic PDE on compact two dimensional domain
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 …
- 134
12
votes
0
answers
395
views
A model of pillows
(The same system with slightly different questions has been asked in MSE.)
Let $\Omega\subset \mathbb{R}^2$ be some simply connected planar domain. We seek for a mapping $\mathbf{r}:\Omega\rightarrow …
- 134
6
votes
2
answers
602
views
Non-linear hyperbolic PDE
I have the following PDE in two dimensions
$$
2\partial_x\partial_y\sqrt{1-u^2}+\left(\partial^2_x-\partial^2_y \right)u=0,
$$
with $u=u(x,y)$ with values between $-1$ and $1$, or alternatively
$$
2\p …
- 134
6
votes
0
answers
158
views
Nonlinear-PDE arising from flat conformal Chebyshev nets
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. …
- 134
3
votes
0
answers
170
views
Non-linear, hyperbolic, 2nd order system of PDEs
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 …
- 134
2
votes
1
answer
155
views
Hyperbolic system of PDEs with elliptic-like boundary contions
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 …
- 134
1
vote
Accepted
Linear elliptic equation
The equation is happily linear, so depending on the domain you may find separable analytical solutions to the Dirichlet problem thanks to Sturm–Liouville.
For instance, let's take the domain to be a u …
- 134
5
votes
1
answer
365
views
Systems of (hyperbolic) 2nd order PDEs with lower order constraints
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 …
- 134