How shall I identify all assumptions required by a Theorem in Evan's PDE textbook?
$begingroup$
My general question is given in the title. Let us consider Hopf-Lax formula - Theorem 4 of Section 3.3.2 of the PDE book by Evans, Click here.
[Theorem 4] If $xin mathbb R^n$ and $t>0$, then the solution $u= u(x, t)$ of the minimization problem (17) is
$$u(x, t) = ldots$$
The assumption given for Hopf-Lax formula above is "If $xin mathbb R^n$ and $t>0$". But obviously, one shall impose some conditions for the functions $L$ and $g$, at least measurability condition. Therefore, I wonder what exactly the conditions needed to have Hopf-Lax formula valid.
From the context, $L$ is Lagrangian which may be required strongly convex as of (19) on the same page. In addition, $g$ is mentioned to be Lipschitz in (20). So my understanding for Hopf-Lax formula is
[Restatement of Hopf-Lax] $L$ is strongly convex, $g$ is Lipschitz. If $xin mathbb R^n$ and $t>0$, then ....
In fact, I have many similar questions on other Theorems. Is there any unified approach to figure out the exact conditions for theorems? Or otherwise, we have to read the proof back and forth and extract the conditions by ourselves?
pde hamilton-jacobi-equation
edited Dec 11 '18 at 13:29
Harry49
6,21331132
asked Jun 7 '17 at 1:42
user79963user79963
444313
$endgroup$
add a comment |
$begingroup$
My general question is given in the title. Let us consider Hopf-Lax formula - Theorem 4 of Section 3.3.2 of the PDE book by Evans, Click here.
[Theorem 4] If $xin mathbb R^n$ and $t>0$, then the solution $u= u(x, t)$ of the minimization problem (17) is
$$u(x, t) = ldots$$
The assumption given for Hopf-Lax formula above is "If $xin mathbb R^n$ and $t>0$". But obviously, one shall impose some conditions for the functions $L$ and $g$, at least measurability condition. Therefore, I wonder what exactly the conditions needed to have Hopf-Lax formula valid.
From the context, $L$ is Lagrangian which may be required strongly convex as of (19) on the same page. In addition, $g$ is mentioned to be Lipschitz in (20). So my understanding for Hopf-Lax formula is
[Restatement of Hopf-Lax] $L$ is strongly convex, $g$ is Lipschitz. If $xin mathbb R^n$ and $t>0$, then ....
In fact, I have many similar questions on other Theorems. Is there any unified approach to figure out the exact conditions for theorems? Or otherwise, we have to read the proof back and forth and extract the conditions by ourselves?
pde hamilton-jacobi-equation
edited Dec 11 '18 at 13:29
Harry49
6,21331132
asked Jun 7 '17 at 1:42
user79963user79963
444313
$endgroup$
add a comment |
$begingroup$
My general question is given in the title. Let us consider Hopf-Lax formula - Theorem 4 of Section 3.3.2 of the PDE book by Evans, Click here.
[Theorem 4] If $xin mathbb R^n$ and $t>0$, then the solution $u= u(x, t)$ of the minimization problem (17) is
$$u(x, t) = ldots$$
The assumption given for Hopf-Lax formula above is "If $xin mathbb R^n$ and $t>0$". But obviously, one shall impose some conditions for the functions $L$ and $g$, at least measurability condition. Therefore, I wonder what exactly the conditions needed to have Hopf-Lax formula valid.
From the context, $L$ is Lagrangian which may be required strongly convex as of (19) on the same page. In addition, $g$ is mentioned to be Lipschitz in (20). So my understanding for Hopf-Lax formula is
[Restatement of Hopf-Lax] $L$ is strongly convex, $g$ is Lipschitz. If $xin mathbb R^n$ and $t>0$, then ....
In fact, I have many similar questions on other Theorems. Is there any unified approach to figure out the exact conditions for theorems? Or otherwise, we have to read the proof back and forth and extract the conditions by ourselves?
pde hamilton-jacobi-equation
edited Dec 11 '18 at 13:29
Harry49
6,21331132
asked Jun 7 '17 at 1:42
user79963user79963
444313
$endgroup$
My general question is given in the title. Let us consider Hopf-Lax formula - Theorem 4 of Section 3.3.2 of the PDE book by Evans, Click here.
[Theorem 4] If $xin mathbb R^n$ and $t>0$, then the solution $u= u(x, t)$ of the minimization problem (17) is
$$u(x, t) = ldots$$
The assumption given for Hopf-Lax formula above is "If $xin mathbb R^n$ and $t>0$". But obviously, one shall impose some conditions for the functions $L$ and $g$, at least measurability condition. Therefore, I wonder what exactly the conditions needed to have Hopf-Lax formula valid.
From the context, $L$ is Lagrangian which may be required strongly convex as of (19) on the same page. In addition, $g$ is mentioned to be Lipschitz in (20). So my understanding for Hopf-Lax formula is
[Restatement of Hopf-Lax] $L$ is strongly convex, $g$ is Lipschitz. If $xin mathbb R^n$ and $t>0$, then ....
In fact, I have many similar questions on other Theorems. Is there any unified approach to figure out the exact conditions for theorems? Or otherwise, we have to read the proof back and forth and extract the conditions by ourselves?
pde hamilton-jacobi-equation
pde hamilton-jacobi-equation
edited Dec 11 '18 at 13:29
Harry49
6,21331132
asked Jun 7 '17 at 1:42
user79963user79963
444313
edited Dec 11 '18 at 13:29
Harry49
6,21331132
asked Jun 7 '17 at 1:42
user79963user79963
444313
edited Dec 11 '18 at 13:29
Harry49
6,21331132
edited Dec 11 '18 at 13:29
Harry49
6,21331132
edited Dec 11 '18 at 13:29
Harry49
6,21331132
6,21331132
asked Jun 7 '17 at 1:42
user79963user79963
444313
asked Jun 7 '17 at 1:42
user79963user79963
444313
asked Jun 7 '17 at 1:42
user79963user79963
444313
444313
add a comment |
add a comment |
1 Answer
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The hypotheses are listed directly above the theorem statement (e.g., "Recall we are assuming..."). It is common in textbooks or papers to list some of the hypotheses right before the theorem to make the theorem statements more concise. In general, you should aim to understand the proof in sufficient detail that you can deduce on your own what assumptions are necessary.
answered Jun 8 '17 at 17:09
JeffJeff
3,238313
$endgroup$
$begingroup$
Thanks and I agree with you.
$endgroup$
– user79963
Jun 9 '17 at 3:44
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
The hypotheses are listed directly above the theorem statement (e.g., "Recall we are assuming..."). It is common in textbooks or papers to list some of the hypotheses right before the theorem to make the theorem statements more concise. In general, you should aim to understand the proof in sufficient detail that you can deduce on your own what assumptions are necessary.
answered Jun 8 '17 at 17:09
JeffJeff
3,238313
$endgroup$
$begingroup$
Thanks and I agree with you.
$endgroup$
– user79963
Jun 9 '17 at 3:44
add a comment |
$begingroup$
The hypotheses are listed directly above the theorem statement (e.g., "Recall we are assuming..."). It is common in textbooks or papers to list some of the hypotheses right before the theorem to make the theorem statements more concise. In general, you should aim to understand the proof in sufficient detail that you can deduce on your own what assumptions are necessary.
answered Jun 8 '17 at 17:09
JeffJeff
3,238313
$endgroup$
$begingroup$
Thanks and I agree with you.
$endgroup$
– user79963
Jun 9 '17 at 3:44
add a comment |
$begingroup$
The hypotheses are listed directly above the theorem statement (e.g., "Recall we are assuming..."). It is common in textbooks or papers to list some of the hypotheses right before the theorem to make the theorem statements more concise. In general, you should aim to understand the proof in sufficient detail that you can deduce on your own what assumptions are necessary.
answered Jun 8 '17 at 17:09
JeffJeff
3,238313
$endgroup$
The hypotheses are listed directly above the theorem statement (e.g., "Recall we are assuming..."). It is common in textbooks or papers to list some of the hypotheses right before the theorem to make the theorem statements more concise. In general, you should aim to understand the proof in sufficient detail that you can deduce on your own what assumptions are necessary.
answered Jun 8 '17 at 17:09
JeffJeff
3,238313
answered Jun 8 '17 at 17:09
JeffJeff
3,238313
answered Jun 8 '17 at 17:09
JeffJeff
3,238313
answered Jun 8 '17 at 17:09
JeffJeff
3,238313
3,238313
$begingroup$
Thanks and I agree with you.
$endgroup$
– user79963
Jun 9 '17 at 3:44
add a comment |
$begingroup$
Thanks and I agree with you.
$endgroup$
– user79963
Jun 9 '17 at 3:44
$begingroup$
Thanks and I agree with you.
$endgroup$
– user79963
Jun 9 '17 at 3:44
$begingroup$
Thanks and I agree with you.
$endgroup$
– user79963
Jun 9 '17 at 3:44
add a comment |
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