Verify of sub-additivity conditions
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Picture below is from Lions, Pierre-Louis, The concentration-compactness principle in the calculus of variations. The locally compact case. I, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1, 109-145 (1984). ZBL0541.49009.
The author want to verify the sub-additivity conditions. But I fail to understand it.
First, about the second red line, why distance between the supports of $u_varepsilon,v_varepsilon^n$ go to $+infty$ means the two expression go to zero?
Second, about the third red line, I think there should be a typing error, I think it should be
$$
I_lambdale I_alpha+ I^infty_lambda-alpha+ 2varepsilon
$$
But, I still can't get it. How should I understand it?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
pde
asked Sep 7 at 1:42
lanse7pty
1,7861723
add a comment |Â
up vote
0
down vote
favorite
Picture below is from Lions, Pierre-Louis, The concentration-compactness principle in the calculus of variations. The locally compact case. I, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1, 109-145 (1984). ZBL0541.49009.
The author want to verify the sub-additivity conditions. But I fail to understand it.
First, about the second red line, why distance between the supports of $u_varepsilon,v_varepsilon^n$ go to $+infty$ means the two expression go to zero?
Second, about the third red line, I think there should be a typing error, I think it should be
$$
I_lambdale I_alpha+ I^infty_lambda-alpha+ 2varepsilon
$$
But, I still can't get it. How should I understand it?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
pde
asked Sep 7 at 1:42
lanse7pty
1,7861723
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Picture below is from Lions, Pierre-Louis, The concentration-compactness principle in the calculus of variations. The locally compact case. I, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1, 109-145 (1984). ZBL0541.49009.
The author want to verify the sub-additivity conditions. But I fail to understand it.
First, about the second red line, why distance between the supports of $u_varepsilon,v_varepsilon^n$ go to $+infty$ means the two expression go to zero?
Second, about the third red line, I think there should be a typing error, I think it should be
$$
I_lambdale I_alpha+ I^infty_lambda-alpha+ 2varepsilon
$$
But, I still can't get it. How should I understand it?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
pde
asked Sep 7 at 1:42
lanse7pty
1,7861723
Picture below is from Lions, Pierre-Louis, The concentration-compactness principle in the calculus of variations. The locally compact case. I, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1, 109-145 (1984). ZBL0541.49009.
The author want to verify the sub-additivity conditions. But I fail to understand it.
First, about the second red line, why distance between the supports of $u_varepsilon,v_varepsilon^n$ go to $+infty$ means the two expression go to zero?
Second, about the third red line, I think there should be a typing error, I think it should be
$$
I_lambdale I_alpha+ I^infty_lambda-alpha+ 2varepsilon
$$
But, I still can't get it. How should I understand it?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
pde
pde
asked Sep 7 at 1:42
lanse7pty
1,7861723
asked Sep 7 at 1:42
lanse7pty
1,7861723
asked Sep 7 at 1:42
lanse7pty
1,7861723
asked Sep 7 at 1:42
lanse7pty
1,7861723
asked Sep 7 at 1:42
lanse7pty
1,7861723
1,7861723
add a comment |Â
add a comment |Â
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