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Ordinary or partial differential equations. Delay differential equations, neutral equations, integro-differential equations. Well-posedness, asymptotic behavior, and related questions.

3 votes

The linearization problem of fully nonlinear equation on standard sphere

Here is my favorite way to do this computation. Let $$ \delta_{i_1 \dotsm i_k}^{j_1 \dotsm j_k} = \begin{cases} \mathrm{sgn}\, \sigma, & \text{if $j_k = i_{\sigma(k)}$}, \\ 0, & \text{otherwise} \end{ …
Jeffrey Case's user avatar
  • 1,713
7 votes

Nirenberg problem in conformal change

Yes, it is true. This is a consequence of the conformal invariance of the conformal Laplacian. Let $(M^n,g)$ be a Riemannian manifold. The conformal Laplacian is $$ L_2^g = -\Delta + \frac{n-2}{4(n- …
Jeffrey Case's user avatar
  • 1,713
2 votes

Kelvin transformation in fully nonlinear equation

This follows easily from the fact that the Kelvin transform is a conformal diffeomorphism and the naturality of the Schouten tensor. Let $\Phi(z) := \frac{z}{\lvert z\rvert^2}$ denote the Kelvin trans …
Jeffrey Case's user avatar
  • 1,713