Extension of weak derivatives in Bochner spaces Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Are weak derivatives and distributional derivatives different?Weak Convergence of piecewise constant function (Evans PDE, Section 8.7, Q1b)Evans PDE mappings into better spacesReference about Sobolev spacesWhy L belongs to the dual space $H^-1$Bochner-Sobolev space vs. Sobolev space on product via Fubini-Tonelli?Theorem about difference quotients and weak derivatives in Evans - Why not stronger statement?Inequality between differential quotient and Sobolev normProof of the Sobolev Extension Theorem in Evans' PDE bookConfusions in Evans book regarding weak derivatives in Banach spaces
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Extension of weak derivatives in Bochner spaces
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Are weak derivatives and distributional derivatives different?Weak Convergence of piecewise constant function (Evans PDE, Section 8.7, Q1b)Evans PDE mappings into better spacesReference about Sobolev spacesWhy L belongs to the dual space $H^-1$Bochner-Sobolev space vs. Sobolev space on product via Fubini-Tonelli?Theorem about difference quotients and weak derivatives in Evans - Why not stronger statement?Inequality between differential quotient and Sobolev normProof of the Sobolev Extension Theorem in Evans' PDE bookConfusions in Evans book regarding weak derivatives in Banach spaces
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I am struggling to understand estimate $(15)$ from the following proof from the PDE book by Evans:
He argues that estimate $(15)$ follows from difference quotients, but I can't understand this. In fact, the theory developed in the earlier section of difference quotients is for real-valued functions, not for functions with values in Banach spaces. What theorem did he use in fact? Any ideas?
functional-analysis pde sobolev-spaces bochner-spaces
asked Mar 25 at 19:05
RichardRichard
1,2551724
$endgroup$
add a comment |
$begingroup$
I am struggling to understand estimate $(15)$ from the following proof from the PDE book by Evans:
He argues that estimate $(15)$ follows from difference quotients, but I can't understand this. In fact, the theory developed in the earlier section of difference quotients is for real-valued functions, not for functions with values in Banach spaces. What theorem did he use in fact? Any ideas?
functional-analysis pde sobolev-spaces bochner-spaces
asked Mar 25 at 19:05
RichardRichard
1,2551724
$endgroup$
add a comment |
$begingroup$
I am struggling to understand estimate $(15)$ from the following proof from the PDE book by Evans:
He argues that estimate $(15)$ follows from difference quotients, but I can't understand this. In fact, the theory developed in the earlier section of difference quotients is for real-valued functions, not for functions with values in Banach spaces. What theorem did he use in fact? Any ideas?
functional-analysis pde sobolev-spaces bochner-spaces
asked Mar 25 at 19:05
RichardRichard
1,2551724
$endgroup$
I am struggling to understand estimate $(15)$ from the following proof from the PDE book by Evans:
He argues that estimate $(15)$ follows from difference quotients, but I can't understand this. In fact, the theory developed in the earlier section of difference quotients is for real-valued functions, not for functions with values in Banach spaces. What theorem did he use in fact? Any ideas?
functional-analysis pde sobolev-spaces bochner-spaces
functional-analysis pde sobolev-spaces bochner-spaces
asked Mar 25 at 19:05
RichardRichard
1,2551724
asked Mar 25 at 19:05
RichardRichard
1,2551724
asked Mar 25 at 19:05
RichardRichard
1,2551724
asked Mar 25 at 19:05
RichardRichard
1,2551724
asked Mar 25 at 19:05
RichardRichard
1,2551724
1,2551724
add a comment |
add a comment |
1 Answer
1
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The method of difference quotients is also true for functions with values in $L^2$. It can be found for instance in:
J.L. Lions. Equations différentielles opérationnelles et problèmes aux limites. Die Grundlehren der mathematischen Wissenschaften, Bd. 111 Springer-Verlag, Berlin-Göttingen-Heidelberg, 1961.
answered Mar 25 at 21:55
S. MathsS. Maths
667116
$endgroup$
$begingroup$
Sorry, I can't download the book. May you please state the result being used? I still have no idea.
$endgroup$
– Richard
Mar 25 at 22:12
$begingroup$
Give me your email I will send to you some references on the method.
$endgroup$
– S. Maths
Mar 26 at 20:10
add a comment |
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1 Answer
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1 Answer
1
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$begingroup$
The method of difference quotients is also true for functions with values in $L^2$. It can be found for instance in:
J.L. Lions. Equations différentielles opérationnelles et problèmes aux limites. Die Grundlehren der mathematischen Wissenschaften, Bd. 111 Springer-Verlag, Berlin-Göttingen-Heidelberg, 1961.
answered Mar 25 at 21:55
S. MathsS. Maths
667116
$endgroup$
$begingroup$
Sorry, I can't download the book. May you please state the result being used? I still have no idea.
$endgroup$
– Richard
Mar 25 at 22:12
$begingroup$
Give me your email I will send to you some references on the method.
$endgroup$
– S. Maths
Mar 26 at 20:10
add a comment |
$begingroup$
The method of difference quotients is also true for functions with values in $L^2$. It can be found for instance in:
J.L. Lions. Equations différentielles opérationnelles et problèmes aux limites. Die Grundlehren der mathematischen Wissenschaften, Bd. 111 Springer-Verlag, Berlin-Göttingen-Heidelberg, 1961.
answered Mar 25 at 21:55
S. MathsS. Maths
667116
$endgroup$
$begingroup$
Sorry, I can't download the book. May you please state the result being used? I still have no idea.
$endgroup$
– Richard
Mar 25 at 22:12
$begingroup$
Give me your email I will send to you some references on the method.
$endgroup$
– S. Maths
Mar 26 at 20:10
add a comment |
$begingroup$
The method of difference quotients is also true for functions with values in $L^2$. It can be found for instance in:
J.L. Lions. Equations différentielles opérationnelles et problèmes aux limites. Die Grundlehren der mathematischen Wissenschaften, Bd. 111 Springer-Verlag, Berlin-Göttingen-Heidelberg, 1961.
answered Mar 25 at 21:55
S. MathsS. Maths
667116
$endgroup$
The method of difference quotients is also true for functions with values in $L^2$. It can be found for instance in:
J.L. Lions. Equations différentielles opérationnelles et problèmes aux limites. Die Grundlehren der mathematischen Wissenschaften, Bd. 111 Springer-Verlag, Berlin-Göttingen-Heidelberg, 1961.
answered Mar 25 at 21:55
S. MathsS. Maths
667116
answered Mar 25 at 21:55
S. MathsS. Maths
667116
answered Mar 25 at 21:55
S. MathsS. Maths
667116
answered Mar 25 at 21:55
S. MathsS. Maths
667116
667116
$begingroup$
Sorry, I can't download the book. May you please state the result being used? I still have no idea.
$endgroup$
– Richard
Mar 25 at 22:12
$begingroup$
Give me your email I will send to you some references on the method.
$endgroup$
– S. Maths
Mar 26 at 20:10
add a comment |
$begingroup$
Sorry, I can't download the book. May you please state the result being used? I still have no idea.
$endgroup$
– Richard
Mar 25 at 22:12
$begingroup$
Give me your email I will send to you some references on the method.
$endgroup$
– S. Maths
Mar 26 at 20:10
$begingroup$
Sorry, I can't download the book. May you please state the result being used? I still have no idea.
$endgroup$
– Richard
Mar 25 at 22:12
$begingroup$
Sorry, I can't download the book. May you please state the result being used? I still have no idea.
$endgroup$
– Richard
Mar 25 at 22:12
$begingroup$
Give me your email I will send to you some references on the method.
$endgroup$
– S. Maths
Mar 26 at 20:10
$begingroup$
Give me your email I will send to you some references on the method.
$endgroup$
– S. Maths
Mar 26 at 20:10
add a comment |
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