A question about notation in PDE

[BiGyElLoWhAt]

I'm reading through one of my profs papers, or starting. Actually it's 2 of my old profs, one I had for linear and one I had for diff eq. My question is in Section 1 of this paper.

"We begin with an analysis of a second order quasilinear partial dif-ferential inequality for real valued functions of n real variables,
$\Delta u - B|u|^{\epsilon} \geq 0$"
where B and epsilon are constants, and B> 0 ; 0<epsilon<1

What does the delta represent? Is it something to the effect of $\sum_n \partial_{x_i}$? Similar to Del? If not what does this mean?

 

[BiGyElLoWhAt]

-.-
Ok, another question, not too much farther down. What does $\mathbb{C}^{\alpha}$ where alpha is between 0 and 1 imply?
Starting to think this paper may be over my head, especially if we're dealing with partial complex dimensions.

 

[TeethWhitener]

What does the delta represent?

Δ is the Laplacian operator.

What does Cα\mathbb{C}^{\alpha} where alpha is between 0 and 1 imply?

Look up "Holder space" on Wikipedia.

And that's about all the help I can offer. This paper is way over my head.

 

[BiGyElLoWhAt]

Thanks