Activity for Richardâ€
| Type | On... | Excerpt | Status | Date |
|---|---|---|---|---|
| Edit | Post #292628 | Initial revision | — | 29 days ago |
| Question | — |
How to prove that solutions of semilinear differential equations is even function? My question comes from the book Stable Solutions of Elliptic Partial Differential Equations Louis Dupaigne, pages 30-32. \ Summary: Which uniqueness theorem to use for this differential equation ? I am working with the following semilinear differential equation \begin{equation} -u^{\prime... (more) |
— | 29 days ago |
| Edit | Post #292410 | Initial revision | — | about 2 months ago |
| Question | — |
Why $\gamma\cdot\operatorname{grad}u<0$ in the Theorem? (Nirenberg academic paper) I am working on the following academic paper Symmetry and Related Properties via the Maximum Principle, which is a classic by Louis Nirenberg. I am trying to understand the next theorem, which is on page 211 and 216: > Theorem 2. Let $u>0$ be a $C^2$ solution of (1.1) in a ring-shaped domain ... (more) |
— | about 2 months ago |
| Comment | Post #292320 |
Thanks for the reply. When I have more time I will read this in detail. On first reading I saw it was correct. (more) |
— | about 2 months ago |
| Edit | Post #292317 | Initial revision | — | about 2 months ago |
| Question | — |
Reflection in the plane with polar coordinates My question comes from the academic paper: Symmetry of solutions to semilinear elliptic equations via Morse Index. The author is Filomena Pacella. Let $S$ be the vector of the $n$-dimensional unit sphere in $\mathbb{R}^N$, $S=\{x \in \mathbb{R}^N : |x|=1\}$. For a unit vector $e \in S$, we consid... (more) |
— | about 2 months ago |
| Edit | Post #292241 |
Post edited: |
— | 2 months ago |
| Comment | Post #292231 |
Thanks for your advice. I made a mistake with the calculations. The correct change of variable was $u=r^2$. I leave the correct result below. (more) |
— | 2 months ago |
| Edit | Post #292241 | Initial revision | — | 2 months ago |
| Answer | — |
A: Complex functions and inner product $\langle \frac{\partial f}{\partial z} , g\rangle $ The first case $\langle \frac{\partial f}{\partial z} , g \rangle$. Change of variable \begin{align} u =r^2 \\ \qquad du =2r dr \end{align} \begin{align} (k+1)\int{0}^{\infty} e^{-r^2}r^{2k+1}dr=\int{0}^{\infty} e^{-u}u^{k +1/2}\frac{1}{2\sqrt{u}}du = \frac{1}{2}\int{0}^{\infty}e... (more) |
— | 2 months ago |
| Comment | Post #292225 |
No the complex conjugation is correct in each inner product used. Any advice on how to get it? You can read the paper directly, I have left the link (more) |
— | 2 months ago |
| Edit | Post #292225 | Initial revision | — | 2 months ago |
| Question | — |
Complex functions and inner product $\langle \frac{\partial f}{\partial z} , g\rangle $ I'm working through this academic paper: Stability of the Solutions of Differential Equations whose author is Bernard Beauzamy. A link to paper. In the academic paper, it works with the norm \begin{equation} \left\Vert f\right\Vert= \left( \int{0}^{\infty}\int{0}^{2\pi} e^{-r^2} |f(re^{i\theta})... (more) |
— | 2 months ago |