Mixing
Question to the Lax-Oleinik formula in Evans PDE
On p. 146 in Evans' book Partial Differential equations (AMS, 1998), it is written:
Now for each $ x ∈ mathbb{R}$ and $t > 0$, define the
point $y(x,t)$ to equal the smallest of those points $y$ giving the minimum of $t L(frac{x−y}{t})+h(y)$.
Then the mapping $x → y(x,t)$ is nondecreasing and is thus continuous for all but at most countably many $x$. At a point $x$ of continuity of $y(·,t)$, $y(x,t)$ is the unique value of $y$ yielding the minimum.
I am trying to see why at a point $x$ of continuity of $y(·,t)$, $y(x,t)$ is the unique value of $y$ yielding the minimum.
Can someone help me with that?
pde optimization
edited Nov 23 '18 at 17:39
Harry49
6,00121031
asked Nov 21 '18 at 23:22
Infinite_28Infinite_28
1978
add a comment |
On p. 146 in Evans' book Partial Differential equations (AMS, 1998), it is written:
Now for each $ x ∈ mathbb{R}$ and $t > 0$, define the
point $y(x,t)$ to equal the smallest of those points $y$ giving the minimum of $t L(frac{x−y}{t})+h(y)$.
Then the mapping $x → y(x,t)$ is nondecreasing and is thus continuous for all but at most countably many $x$. At a point $x$ of continuity of $y(·,t)$, $y(x,t)$ is the unique value of $y$ yielding the minimum.
I am trying to see why at a point $x$ of continuity of $y(·,t)$, $y(x,t)$ is the unique value of $y$ yielding the minimum.
Can someone help me with that?
pde optimization
edited Nov 23 '18 at 17:39
Harry49
6,00121031
asked Nov 21 '18 at 23:22
Infinite_28Infinite_28
1978
add a comment |
On p. 146 in Evans' book Partial Differential equations (AMS, 1998), it is written:
Now for each $ x ∈ mathbb{R}$ and $t > 0$, define the
point $y(x,t)$ to equal the smallest of those points $y$ giving the minimum of $t L(frac{x−y}{t})+h(y)$.
Then the mapping $x → y(x,t)$ is nondecreasing and is thus continuous for all but at most countably many $x$. At a point $x$ of continuity of $y(·,t)$, $y(x,t)$ is the unique value of $y$ yielding the minimum.
I am trying to see why at a point $x$ of continuity of $y(·,t)$, $y(x,t)$ is the unique value of $y$ yielding the minimum.
Can someone help me with that?
pde optimization
edited Nov 23 '18 at 17:39
Harry49
6,00121031
asked Nov 21 '18 at 23:22
Infinite_28Infinite_28
1978
On p. 146 in Evans' book Partial Differential equations (AMS, 1998), it is written:
Now for each $ x ∈ mathbb{R}$ and $t > 0$, define the
point $y(x,t)$ to equal the smallest of those points $y$ giving the minimum of $t L(frac{x−y}{t})+h(y)$.
Then the mapping $x → y(x,t)$ is nondecreasing and is thus continuous for all but at most countably many $x$. At a point $x$ of continuity of $y(·,t)$, $y(x,t)$ is the unique value of $y$ yielding the minimum.
I am trying to see why at a point $x$ of continuity of $y(·,t)$, $y(x,t)$ is the unique value of $y$ yielding the minimum.
Can someone help me with that?
pde optimization
pde optimization
edited Nov 23 '18 at 17:39
Harry49
6,00121031
asked Nov 21 '18 at 23:22
Infinite_28Infinite_28
1978
edited Nov 23 '18 at 17:39
Harry49
6,00121031
asked Nov 21 '18 at 23:22
Infinite_28Infinite_28
1978
edited Nov 23 '18 at 17:39
Harry49
6,00121031
edited Nov 23 '18 at 17:39
Harry49
6,00121031
edited Nov 23 '18 at 17:39
Harry49
6,00121031
6,00121031
asked Nov 21 '18 at 23:22
Infinite_28Infinite_28
1978
asked Nov 21 '18 at 23:22
Infinite_28Infinite_28
1978
asked Nov 21 '18 at 23:22
Infinite_28Infinite_28
1978
1978
add a comment |
add a comment |
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