Concerning to "we claim that u^\gamma is \alpha-Hölder continuous", this comes from the definition and the fact that the LHS of the inequality on the line right above is finite.
The key point here is to deal with f \circ g, like
u(x)^\alpha = f(u(x)) with f(t) =t^\alpha.
Oh thanks, I am reading. BTW, I have another approach: Consider the Lorentzian metric
d{s^2} = - dt \otimes dt + {g_{ij}}(x,t)d{x^i} \otimes d{x^j}.
Einstein equations in vacuum, i.e., G_{ij} = 0 become
\frac{{{\partial ^2}{g_{ij}}}}{{\partial {t^2}}} + 2{R_{ij}} +...
Thanks, what I am expecting is to show that studying Cauchy problems for Einstein equations make sense. My advisor suggests me to show that the vacuum Einstein equation is indeed a hyperbolic equation but I am still not able. Having this fact, it is reasonable to study Cauchy problem for the...
I learn from a book saying that the vacuum Einstein equation can be rewritten as
{\partial _t}{g_{\alpha \beta }} = - 2N{k_{\alpha \beta }} + {\mathbb{L}_X}{g_{\alpha \beta }}
and
{\partial _t}{k_{\alpha \beta }} = - {\nabla _\alpha }{\nabla _\beta }N + N\left( {{R_{\alpha \beta }} +...
Hello friends,
I am now interested in Einstein scalar field equations with a very little knowledge about Physics. I would like to ask why vacuum Einstein equations are hyperbolic equations? As far as I know that for 3+1-dimensional manifold V, we can convert the vacuum Einstein equations as a...
Einstein-scalar field action --> Einstein-scalar field equations
Dear friends,
Just a small question I do not know how to derive.
From the Einstein-scalar field action defined by
S\left( {g,\psi } \right) = \int_{} {\left( {R(g) - \frac{1}{2}\left| {\nabla \psi } \right|_g^2 - V\left(...
Having a Lorentzian 4-manifold, the Einstein vacuum equations of general relativity read
\overline R_{\alpha \beta} - \frac{1}{2}\overline g_{\alpha\beta}\overline R=0
where \overline R the scalar curvature, \overline g_{\alpha\beta} the metric tensor and \overline R_{\alpha\beta} the Ricci...