Skip to main content
MathOverflow

Return to Question

added 346 characters in body
Source Link
User1723
User1723
  • 337
  • 1
  • 5

I am in trouble in understanding one step of theorem 1.1 in the paper by Ciraolo-Figalli-Maggi: arXiv link, link at Figalli's page.

Namely in equation \eqref{1}, it is written: "Since $u=\sigma+\rho$, a the Talyor expansion yields $$ \int_{\mathbb{R}^n}u^p\rho=\int_{\mathbb{R}^n}\sigma^p\rho+p\int_{\mathbb{R}^n}\sigma^{p-1}\rho^2+O\left(\int_{\mathbb{R}^n}|\nabla\rho|^2\right)^{1+\gamma}\label{1}\tag{2.8}$$ where $\gamma=\min(\frac{1}{2},\frac{2}{n-2})$."
The problem is the big O term since I do not get it: how this is coming with power $1+\gamma$?

When $\frac{2^*}{3}>1$ then using Holder inequality and Sobolev inequality we can bound the term $\int_{\mathbb{R}^n}\sigma^{p-2}\rho^3$ by $O(\int_{\mathbb{R}^n}{|\nabla\rho|^2})^{\frac{3}{2}}$ but I am not getting the other exponent of $\gamma$ i.e $\frac{2}{n-2}$ and the minimum over both the two would be the exponent. Any help is very much appreciated.

 

I am in trouble in understanding one step of theorem 1.1 in the paper by Ciraolo-Figalli-Maggi: arXiv link, link at Figalli's page.

Namely in equation \eqref{1}, it is written: "Since $u=\sigma+\rho$, a the Talyor expansion yields $$ \int_{\mathbb{R}^n}u^p\rho=\int_{\mathbb{R}^n}\sigma^p\rho+p\int_{\mathbb{R}^n}\sigma^{p-1}\rho^2+O\left(\int_{\mathbb{R}^n}|\nabla\rho|^2\right)^{1+\gamma}\label{1}\tag{2.8}$$ where $\gamma=\min(\frac{1}{2},\frac{2}{n-2})$."
The problem is the big O term since I do not get it: how this is coming with power $1+\gamma$? Any help is very much appreciated.

I am in trouble in understanding one step of theorem 1.1 in the paper by Ciraolo-Figalli-Maggi: arXiv link, link at Figalli's page.

Namely in equation \eqref{1}, it is written: "Since $u=\sigma+\rho$, a the Talyor expansion yields $$ \int_{\mathbb{R}^n}u^p\rho=\int_{\mathbb{R}^n}\sigma^p\rho+p\int_{\mathbb{R}^n}\sigma^{p-1}\rho^2+O\left(\int_{\mathbb{R}^n}|\nabla\rho|^2\right)^{1+\gamma}\label{1}\tag{2.8}$$ where $\gamma=\min(\frac{1}{2},\frac{2}{n-2})$."
The problem is the big O term since I do not get it: how this is coming with power $1+\gamma$?

When $\frac{2^*}{3}>1$ then using Holder inequality and Sobolev inequality we can bound the term $\int_{\mathbb{R}^n}\sigma^{p-2}\rho^3$ by $O(\int_{\mathbb{R}^n}{|\nabla\rho|^2})^{\frac{3}{2}}$ but I am not getting the other exponent of $\gamma$ i.e $\frac{2}{n-2}$ and the minimum over both the two would be the exponent. Any help is very much appreciated.

 
formatting
Source Link
YCor
YCor
  • 63.9k
  • 5
  • 187
  • 286

Difficulty in Understandingunderstanding one step in Ciraolo-Figalli-Maggi paper

https://people.math.ethz.ch/~afigalli/papers-pdf/A-quantitative-analysis-of-metrics-on-Rn-with-almost-constant-positive-scalar-curvature-with-applications-to-fast-diffusion-flows.pdf

I am in trouble in understanding one step of theorem 1.1 in the above mentioned paper by Ciraolo-Figalli-Maggi: arXiv link, link at Figalli's page. 

Namely in equation \eqref{1}, it is written: "Since $u=\sigma+\rho$, a the Talyor expansion yields $$ \int_{\mathbb{R}^n}u^p\rho=\int_{\mathbb{R}^n}\sigma^p\rho+p\int_{\mathbb{R}^n}\sigma^{p-1}\rho^2+O\left(\int_{\mathbb{R}^n}|\nabla\rho|^2\right)^{1+\gamma}\label{1}\tag{2.8}$$ where $\gamma=\min(\frac{1}{2},\frac{2}{n-2})$."
The problem is the big O term since I do not get it: how this is coming with power $1+\gamma$? Any help is very much appreciated.

Difficulty in Understanding one step in Ciraolo-Figalli-Maggi paper

https://people.math.ethz.ch/~afigalli/papers-pdf/A-quantitative-analysis-of-metrics-on-Rn-with-almost-constant-positive-scalar-curvature-with-applications-to-fast-diffusion-flows.pdf

I am in trouble in understanding one step of theorem 1.1 in the above mentioned paper. Namely in equation \eqref{1}, it is written: "Since $u=\sigma+\rho$, a the Talyor expansion yields $$ \int_{\mathbb{R}^n}u^p\rho=\int_{\mathbb{R}^n}\sigma^p\rho+p\int_{\mathbb{R}^n}\sigma^{p-1}\rho^2+O\left(\int_{\mathbb{R}^n}|\nabla\rho|^2\right)^{1+\gamma}\label{1}\tag{2.8}$$ where $\gamma=\min(\frac{1}{2},\frac{2}{n-2})$."
The problem is the big O term since I do not get it: how this is coming with power $1+\gamma$? Any help is very much appreciated.

Difficulty in understanding one step in Ciraolo-Figalli-Maggi paper

I am in trouble in understanding one step of theorem 1.1 in the paper by Ciraolo-Figalli-Maggi: arXiv link, link at Figalli's page. 

Namely in equation \eqref{1}, it is written: "Since $u=\sigma+\rho$, a the Talyor expansion yields $$ \int_{\mathbb{R}^n}u^p\rho=\int_{\mathbb{R}^n}\sigma^p\rho+p\int_{\mathbb{R}^n}\sigma^{p-1}\rho^2+O\left(\int_{\mathbb{R}^n}|\nabla\rho|^2\right)^{1+\gamma}\label{1}\tag{2.8}$$ where $\gamma=\min(\frac{1}{2},\frac{2}{n-2})$."
The problem is the big O term since I do not get it: how this is coming with power $1+\gamma$? Any help is very much appreciated.

Minor Math Jaxing, typo fixed + minor grammar improvements
Source Link
Daniele Tampieri
Daniele Tampieri
  • 6.4k
  • 7
  • 30
  • 45

Difficulty in Understanding one step in CiraloCiraolo-Figalli-Maggi paper

https://people.math.ethz.ch/~afigalli/papers-pdf/A-quantitative-analysis-of-metrics-on-Rn-with-almost-constant-positive-scalar-curvature-with-applications-to-fast-diffusion-flows.pdf

I am in trouble in understanding one step of theorem 1.1 in the above mentioned paper. Namely in the equation (2.8) \eqref{1}, it is written that since: "Since $u=\sigma+\rho$ so, a the Talyor expansion yields $\int_{\mathbb{R}^n}u^p\rho=\int_{\mathbb{R}^n}\sigma^p\rho+p\int_{\mathbb{R}^n}\sigma^{p-1}\rho+O(\int_{\mathbb{R}^n}|\nabla\rho|^{1+\gamma})..........(2.8)$$$ \int_{\mathbb{R}^n}u^p\rho=\int_{\mathbb{R}^n}\sigma^p\rho+p\int_{\mathbb{R}^n}\sigma^{p-1}\rho^2+O\left(\int_{\mathbb{R}^n}|\nabla\rho|^2\right)^{1+\gamma}\label{1}\tag{2.8}$$ where $\gamma=\min(\frac{1}{2},\frac{2}{n-2})$;."
The problem is the big OhO term since I amdo not gettingget it: how this is coming with power $1+\gamma$? Any help is very much appreciated.

Difficulty in Understanding one step in Ciralo-Figalli-Maggi paper

https://people.math.ethz.ch/~afigalli/papers-pdf/A-quantitative-analysis-of-metrics-on-Rn-with-almost-constant-positive-scalar-curvature-with-applications-to-fast-diffusion-flows.pdf

I am in trouble in understanding one step of theorem 1.1 in the above mentioned paper. Namely in the equation (2.8) , it is written that since $u=\sigma+\rho$ so the Talyor expansion yields $\int_{\mathbb{R}^n}u^p\rho=\int_{\mathbb{R}^n}\sigma^p\rho+p\int_{\mathbb{R}^n}\sigma^{p-1}\rho+O(\int_{\mathbb{R}^n}|\nabla\rho|^{1+\gamma})..........(2.8)$ where $\gamma=\min(\frac{1}{2},\frac{2}{n-2})$; the big Oh term I am not getting how this is coming with power $1+\gamma$? Any help is very much appreciated.

Difficulty in Understanding one step in Ciraolo-Figalli-Maggi paper

https://people.math.ethz.ch/~afigalli/papers-pdf/A-quantitative-analysis-of-metrics-on-Rn-with-almost-constant-positive-scalar-curvature-with-applications-to-fast-diffusion-flows.pdf

I am in trouble in understanding one step of theorem 1.1 in the above mentioned paper. Namely in equation \eqref{1}, it is written: "Since $u=\sigma+\rho$, a the Talyor expansion yields $$ \int_{\mathbb{R}^n}u^p\rho=\int_{\mathbb{R}^n}\sigma^p\rho+p\int_{\mathbb{R}^n}\sigma^{p-1}\rho^2+O\left(\int_{\mathbb{R}^n}|\nabla\rho|^2\right)^{1+\gamma}\label{1}\tag{2.8}$$ where $\gamma=\min(\frac{1}{2},\frac{2}{n-2})$."
The problem is the big O term since I do not get it: how this is coming with power $1+\gamma$? Any help is very much appreciated.

Source Link
User1723
User1723
  • 337
  • 1
  • 5