Approximation of arbitrary convex function
I am currently studying Jürgen Moser's "A New Proof of de Giorgi's Theorem Concerning the Regularity Problem for Elliptic Differential Equations". During the proof of the first lemma, he claims that an arbitrary convex function $f$ can be approximated by a sequence of twice continously differentiable convex functions $f_m$, such that $f_m''(u)=0$ for large $|u|$, $f_m rightarrow f$ and $f_m'(u) rightarrow f'(u)$, where $f'(u)$ exists.
Having looked in several books about convex analysis (Rockafellar), I found out that a convex function should be twice differentiable almost everywhere. I further found in one of Rockafellar's books (convex analysis, thm. 25.7) a Theorem that shows the claim, except for the restriction on $f''$.
Now the question is how can we come up with that restriction on $f''$ and maybe someone has a reference for a book as well, since I would be interested in seeing the proof.
differential-equations pde convex-analysis
asked Dec 5 '18 at 11:21
MaxMax
234
add a comment |
I am currently studying Jürgen Moser's "A New Proof of de Giorgi's Theorem Concerning the Regularity Problem for Elliptic Differential Equations". During the proof of the first lemma, he claims that an arbitrary convex function $f$ can be approximated by a sequence of twice continously differentiable convex functions $f_m$, such that $f_m''(u)=0$ for large $|u|$, $f_m rightarrow f$ and $f_m'(u) rightarrow f'(u)$, where $f'(u)$ exists.
Having looked in several books about convex analysis (Rockafellar), I found out that a convex function should be twice differentiable almost everywhere. I further found in one of Rockafellar's books (convex analysis, thm. 25.7) a Theorem that shows the claim, except for the restriction on $f''$.
Now the question is how can we come up with that restriction on $f''$ and maybe someone has a reference for a book as well, since I would be interested in seeing the proof.
differential-equations pde convex-analysis
asked Dec 5 '18 at 11:21
MaxMax
234
1
Is this about pointwise convergence? Then you could just cut off at say $|u|>m$ and replace that bit by a linear function.
– Michal Adamaszek
Dec 5 '18 at 11:52
I then need to apply Fatou's Lemma so I think pointwise convergence almost everywhere should be enough. Then interpolation would actually do the job..
– Max
Dec 5 '18 at 15:28
I thought about it again and I am not certain that Interpolation would do the job. The function's derivate could only be defined on $mathbb{R}$ $ mathbb{Q}$. Then interpolation on the rational numbers wouldn't necessarily do the job, I guess. Any thoughts on that?
– Max
Dec 5 '18 at 15:41
add a comment |
I am currently studying Jürgen Moser's "A New Proof of de Giorgi's Theorem Concerning the Regularity Problem for Elliptic Differential Equations". During the proof of the first lemma, he claims that an arbitrary convex function $f$ can be approximated by a sequence of twice continously differentiable convex functions $f_m$, such that $f_m''(u)=0$ for large $|u|$, $f_m rightarrow f$ and $f_m'(u) rightarrow f'(u)$, where $f'(u)$ exists.
Having looked in several books about convex analysis (Rockafellar), I found out that a convex function should be twice differentiable almost everywhere. I further found in one of Rockafellar's books (convex analysis, thm. 25.7) a Theorem that shows the claim, except for the restriction on $f''$.
Now the question is how can we come up with that restriction on $f''$ and maybe someone has a reference for a book as well, since I would be interested in seeing the proof.
differential-equations pde convex-analysis
asked Dec 5 '18 at 11:21
MaxMax
234
I am currently studying Jürgen Moser's "A New Proof of de Giorgi's Theorem Concerning the Regularity Problem for Elliptic Differential Equations". During the proof of the first lemma, he claims that an arbitrary convex function $f$ can be approximated by a sequence of twice continously differentiable convex functions $f_m$, such that $f_m''(u)=0$ for large $|u|$, $f_m rightarrow f$ and $f_m'(u) rightarrow f'(u)$, where $f'(u)$ exists.
Having looked in several books about convex analysis (Rockafellar), I found out that a convex function should be twice differentiable almost everywhere. I further found in one of Rockafellar's books (convex analysis, thm. 25.7) a Theorem that shows the claim, except for the restriction on $f''$.
Now the question is how can we come up with that restriction on $f''$ and maybe someone has a reference for a book as well, since I would be interested in seeing the proof.
differential-equations pde convex-analysis
differential-equations pde convex-analysis
asked Dec 5 '18 at 11:21
MaxMax
234
asked Dec 5 '18 at 11:21
MaxMax
234
edited Dec 5 '18 at 11:29
Max
asked Dec 5 '18 at 11:21
MaxMax
234
asked Dec 5 '18 at 11:21
MaxMax
234
asked Dec 5 '18 at 11:21
MaxMax
234
234
1
Is this about pointwise convergence? Then you could just cut off at say $|u|>m$ and replace that bit by a linear function.
– Michal Adamaszek
Dec 5 '18 at 11:52
I then need to apply Fatou's Lemma so I think pointwise convergence almost everywhere should be enough. Then interpolation would actually do the job..
– Max
Dec 5 '18 at 15:28
I thought about it again and I am not certain that Interpolation would do the job. The function's derivate could only be defined on $mathbb{R}$ $ mathbb{Q}$. Then interpolation on the rational numbers wouldn't necessarily do the job, I guess. Any thoughts on that?
– Max
Dec 5 '18 at 15:41
add a comment |
1
Is this about pointwise convergence? Then you could just cut off at say $|u|>m$ and replace that bit by a linear function.
– Michal Adamaszek
Dec 5 '18 at 11:52
I then need to apply Fatou's Lemma so I think pointwise convergence almost everywhere should be enough. Then interpolation would actually do the job..
– Max
Dec 5 '18 at 15:28
I thought about it again and I am not certain that Interpolation would do the job. The function's derivate could only be defined on $mathbb{R}$ $ mathbb{Q}$. Then interpolation on the rational numbers wouldn't necessarily do the job, I guess. Any thoughts on that?
– Max
Dec 5 '18 at 15:41
1
1
Is this about pointwise convergence? Then you could just cut off at say $|u|>m$ and replace that bit by a linear function.
– Michal Adamaszek
Dec 5 '18 at 11:52
Is this about pointwise convergence? Then you could just cut off at say $|u|>m$ and replace that bit by a linear function.
– Michal Adamaszek
Dec 5 '18 at 11:52
I then need to apply Fatou's Lemma so I think pointwise convergence almost everywhere should be enough. Then interpolation would actually do the job..
– Max
Dec 5 '18 at 15:28
I then need to apply Fatou's Lemma so I think pointwise convergence almost everywhere should be enough. Then interpolation would actually do the job..
– Max
Dec 5 '18 at 15:28
I thought about it again and I am not certain that Interpolation would do the job. The function's derivate could only be defined on $mathbb{R}$ $ mathbb{Q}$. Then interpolation on the rational numbers wouldn't necessarily do the job, I guess. Any thoughts on that?
– Max
Dec 5 '18 at 15:41
I thought about it again and I am not certain that Interpolation would do the job. The function's derivate could only be defined on $mathbb{R}$ $ mathbb{Q}$. Then interpolation on the rational numbers wouldn't necessarily do the job, I guess. Any thoughts on that?
– Max
Dec 5 '18 at 15:41
add a comment |
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1
Is this about pointwise convergence? Then you could just cut off at say $|u|>m$ and replace that bit by a linear function.
– Michal Adamaszek
Dec 5 '18 at 11:52
I then need to apply Fatou's Lemma so I think pointwise convergence almost everywhere should be enough. Then interpolation would actually do the job..
– Max
Dec 5 '18 at 15:28
I thought about it again and I am not certain that Interpolation would do the job. The function's derivate could only be defined on $mathbb{R}$ $ mathbb{Q}$. Then interpolation on the rational numbers wouldn't necessarily do the job, I guess. Any thoughts on that?
– Max
Dec 5 '18 at 15:41