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Results tagged with elliptic-pde
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user 1441
Questions about partial differential equations of elliptic type. Often used in combination with the top-level tag ap.analysis-of-pdes.
37
votes
Accepted
Does a Riemannian manifold with bounded geometry admit an isometric proper embedding into Eu...
It does not hold for hyperbolic plane.
It follows since the volume growth of the hyperbolic plane is exponential, while volume growth of $\mathbb{R}^N$ is polynomial.
Postcript. Let us say that a Riem …
- 45k
2
votes
The minimal surface equation in a Riemannian metric
There are no simpler expressions for your case.
Essentially you ask for a minimal-surface equation assuming that your surface is a graph in local coordinates, but any smooth-regular surface is a graph …
- 45k
1
vote
Solutions to $\Delta u\ge u^2$
Let $(\theta,r)$ be the polar coordinates on the plane.
Consider the Riemannian metric of the form
$$g=\left(\begin{smallmatrix}1&0\\0&\ell^2(r)\end{smallmatrix}\right).$$
Let $f(\theta,r)=\phi(r)$.
T …
- 45k
0
votes
Accepted
Uniqueness affine curvature
Note that closed curves made by two arcs of ellipses of the given areas, say $a$ and $b$ may be not related to each other by an affine transformation.
They have constant affine curvatures on two arcs, …
- 45k