3
votes
2answers
183 views

A question on certain elliptic PDE

Consider the elliptic PDE "CR" $$\begin{cases} U_{xx}=V_{yy}\\U_{yy}=-V_{xx} \end{cases}$$ And its consequence "LAP" $$U_{xxxx}+U_{yyyy}=0$$. Somehow, these equations are similar to the Cauchi ...
2
votes
1answer
141 views

The distribution of roots of elliptic polynomial

If $p(x)$ is an $n$ variables polynomial of even degree with complex coefficients which satisfies the strong elliptic condition, that is, Re$p(x) \ge C|x|^{2m}$ for any $x \in \mathbb R^n$ where $2m$ ...
6
votes
2answers
128 views

Conditions on a unit vector field to be the Gauss map of some surface immersed in R^3?

Let $U$ be a bounded domain in $R^2$ and let $n : U \to S^2$. Which (necessary/sufficient) conditions must $n$ satisfy in order that there exist an immersion $f : U \to R^3$ such that $n(x)$ is the ...
1
vote
1answer
208 views

Heat equation of spatial complex variable

Suppose that $v(t, z)$ is analytic with respect to the complex variable $z$ and differentiable with respect to the real variable $t$ and satisfies the partial differential equation $$\frac{\partial ...
8
votes
1answer
687 views

Question about an estimate in Hörmander's proof of Cartan's Theorem B

I have been working through the proof of Cartan's Theorem B that Hörmander gives in his book 'Introduction to Complex Analysis in Several Variables'. When I began, I skipped over some of the initial ...
2
votes
1answer
374 views

Hölder estimates for the Complex Monge-Ampere equation

If on a bounded smooth, pseudoconvex domain in $\mathbb{C}^n$, $\mathrm{det} ( \mathrm{Hess}(u)) = f$ ($f>0$, $\mathrm{Hess}(u)>0$, $u=0$ on the boundary), if $f \in C^{k, \alpha}$, is $u \in ...
4
votes
3answers
454 views

Is there a PDE for this phenomenon?

At a point on a surface an incompressible fluid begins to up well at a constant rate and spread across the surface. Is there a physical law - like the heat equation - that describes the flow? Will ...
1
vote
0answers
236 views

Strong minimum principle for maximal plurisubharmonic functions

Suppose $u$ is a bounded maximal plurisubharmonic function in a bounded domain $D \in \Bbb C^n$. If $u$ is $C^2$ one can see that $u$ cannot have a local strict minimum inside $D$. Is there an analog ...
3
votes
3answers
640 views

Monge Ampere equations

I am a graduate student trying to understand complex Monge-Ampere equations(mostly on complex manifolds with or without boundary, but also in C^n), but I can't put my hand on any monograph/textbook ...
4
votes
0answers
312 views

Do there exist generalized conformal maps that preserve elliptic measure?

Let $D_1$ and $D_2$ be two bounded simply connected Jordan domains in $\mathbb{R}^2$. By Carathéodory's Theorem there exists a homeomorphism $f:\bar{D}_1 \to \bar{D}_2$ such that the restriction ...
7
votes
2answers
710 views

Characterize where the Dirichlet Problem for the Laplacian is always solvable

Conway's 1978 textbook Functions of One Complex Variable I gives an unsatisfying characterization of the regions for which the Dirichlet Problem can always be solved, and then comments no cleaner ...
1
vote
1answer
285 views

metrics compatible with conformal structures

I have three related questions: (1) How does one describe the possible Riemannian metrics that are compatible with a conformal structure on a two dimensional surface? (2) Can all conformable ...
8
votes
0answers
365 views

Parametrisations for Null Temperature Functions: nonuniqueness of solutions to the Heat Equation

Disclaimer I expect this is a highly open problem, but maybe I'm wrong and someone has come up with some answers besides those given here. In any case, all information appreciated, thanks! Definition ...