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A bit of General Relativity and Causality theory One feature of general relativity is that the space-time is modelled as a Lorentzian manifold. The Lorentzian metric on the manifold has signature (-+ …

There are no "rapid decaying harmonic 2-forms" in Minkowski space. Consider the expression $$ 0 = \mathrm{d} \star \mathrm{d} F = \partial^i \partial_{[i}F_{jk]} = \frac13 (\Box F_{jk} + \partial_ …

I answered about the incompleteness theorem in the other thread. Let's talk about some of his other contributions here. (This list is definitely incomplete*, but just some stuff off the top of my head …

Yes, there is one such example: $u \equiv 0$. The answer above is not facetious! That $u$ is in fact the only example (modulo measure zero modifications). By Titchmarsh's theorem, if $u\in L^2(\ …

Okay, after figuring out which paper you were trying to link to in the third link, I decided that it is better to just give an answer rather then a bunch of comments. So... there are several issues at …

Somewhat related to the ergodic hypothesis mentioned in another answer is the assumption that generic non-linearities leads to thermalization and equipartition of energy. To be more precise, start wit …

The statement It seems that the classical programme of the PDE community, i.e., (i) existence (ii) uniqueness (iii) regularity, heavily employing concepts from functional analysis, has not found prom …

If your goal is to get some understanding of the Clay Problem, you can't really go wrong with first reading the official problem statement and then reading the papers referred to in the document. …

One of the reasons that it may be difficult to model Minkowski space based on cellular automata is that there are no "non-trivial" finite sub-groups of $O(3,1)$, where non-trivial means that it doesn' …

Yes it follows from the previous sentence. Since $\tau > (n-2)/2 \geq 0$ by assumption you have that $|\varphi^i| \leq C \rho$ from the definition of the norm. This implies that $\tilde{\rho} \leq C \ …

How about the study of minimal surfaces (physical applications in soap films etc.)? In fact one might argue the Lagrangian formulation of minimal surfaces (the problem of Plateau) is one of the oldest …

The stationary / travelling wave / soliton regime of the KdV equation and its cousins give a lot of examples. For the original KdV, under the travelling wave ansatz we have the third order equation $$ …

I think in your question, as currently formulated, the whole rotating cosmic string is a red herring. If I interpret your notation correctly, $a$ and $\kappa$ are constants. And hence locally you can …

I am somewhat doubtful that the question as posed as any sort of reasonable answer. (Also, I don't really see how the Lorentzian metric even enter into the problem.) (a) ANY hyperbolic PDE in (1+3) d …

Write $z = e^{i2\theta}$ where $\theta$ is as in your second formulation, you have that the equation is equivalent to $$ -2i \partial^2_{xy} (z - \bar{z})+ (\partial^2_{xx} - \partial^2_{yy})(z + \bar …