By Giovanni Leoni

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Extra resources for A First Course in Sobolev Spaces

Sample text

Let u : [a, b] -> R be an increasing function and let r > 0. x+ u (x) < V-x for every x E E, then Lo (u. (E)) < rGo (E) . Proof. Consider the function w (x) := -u (-x), x E [-b, -a]. , (-u (E)) = Gp (w (-E)) < rGo (-E) = r1 (E) . 28. Let u : [a, b] -p R be an increasing function. Prove that the set of right endpoints of intervals of constancy of u is at most countable. 29. Let u : [a, b] --+ R be an increasing function and let R > 0. If E C (a, b) is such that D+u (x) := lim sup u (y) - u (x) > R /X for every x E E, then RGo (E) < f-o (u (E)) .

For each n E N write In = (an, bn) and define the continuous increasing function un : R - [0, oo) as follows: un(x) 0 if x < an, x - a, if an bn. Note that 0 < un (x) < diam In for all x E R. Set u(x) := Eun(x), x E R. 16), 00 1 0 < u (x) - E un (x) < E diam In < 1, n=1 n=1+1 and so the series of functions is uniformly convergent. In particular, this implies that u is continuous. Since each un is nonnegative and increasing, it follows that u has the same properties. It remains to show that u is not differentiable in E.

The smallest vector space of functions u : I -+ R that contains all monotone functions (respectively, bounded monotone functions) is given by the space BPVOC (I) (respectively, BPV (I)). Moreover, every function in BPVOC (I) (respectively, BPV (I)) may be written as a difference of two increasing functions (respectively, two bounded increasing functions). Proof. Let u, v : I - R and let t e R. , u + VarJ v. 7) space. 1. 10 the space BPV°C (I) (respectively, BPV (I)) contains all monotone functions (respectively, bounded monotone functions).