| Abstract: |
| Abstruct: In this talk we study a functional associated with Boltzmann-Poisson equation involving probability measure, that is,
$$
J_{\lambda}(v)=\frac{1}{2}\|\nabla v\|_2^2-\lambda\int_{I_+}\log\Big(\int_{\Omega} e^{\alpha v} dx \Big)\mathcal P(d\alpha), \quad v \in H_0^1(\Omega)
$$
where $\lambda>0$ is a constant, $\Omega \subset \mathbb R^2$ is a smooth bounded domain and $\mathcal P(d\alpha)$ is a Borel probability measure on $I_+=[0, 1]$. We show the boundedness of $J_{\lambda}$ from below with the extremal case for $\lambda$ when $\mathcal P(d\alpha)$ is continuous case and satisfies the suitable assumptions. To show this, we have to consider the behavior of minimizing sequence for the above functional near the blow up point. This work is supported by JSPS Grant-in-Aid for Scientific Research (A) 26247013. |
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