Null controllability properties of some degenerate parabolic equations.
Null controllability properties of some degenerate parabolic equations.
Null controllability properties of some degenerate parabolic equations.
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Useful 1-D Hardy type inequalities<br />
The simplest one :<br />
Lemma<br />
Given L > 0 and α ∈ [0, 1), then the following inequality holds for<br />
all z ∈ D((0, L])<br />
∫ L<br />
0<br />
x α−2 z(x) 2 dx ≤<br />
∫<br />
4 L<br />
(1 − α) 2 x α z x (x) 2 dx.<br />
0<br />
Comments :<br />
◮ natural extension by density to absolutely continuous<br />
functions z such that<br />
z(x) → 0<br />
x→0 +<br />
and<br />
◮ the constant<br />
4<br />
(1−α) 2 is optimal ;<br />
◮ the result is false for α = 1 ;<br />
◮ similar version for α ∈ (1, 2).<br />
∫ L<br />
0<br />
x α z x (x) 2 dx < ∞;<br />
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