Perturbation theory of eigenvalue problems

On Certain Nonlinear Eigenvalue Problems

The Petrowsky type equation y e tt + ey e xxxx − y e xx = 0, e > 0 encountered in linear beams theory is null controllable through Neumann boundary controls. Due to the boundary layer of size of order √ e occurring at the extremities, these boundary controls get singular as e goes to 0. Using the matched asymptotic method, we describe the boundary layer of the solution y e then derive a rigorous second order asymptotic expansion of the control of minimal L 2 −norm, with respect to the parameter e. In particular, we recover that the leading term of the expansion is a null Dirichlet control for the limit hyperbolic wave equation, in agreement with earlier results du to J-.L. Lions in the eighties. Numerical experiments support the analysis.

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https://hal.archives-ouvertes.fr/hal-02169764/document

Singular asymptotic expansion of the exact control for a linear model of the Rayleigh beam

On the Caffarelli-Kohn-Niremberg type inequalities and related topics

For $N\geq 5$ and $0<\mu<N-4$, we first show a non-degenerate result of the extremal for the following Rellich-Sobolev type inequality \begin{align*}&\int_{\mathbb{R}^N}|\Delta u|^2 \mathrm{d}x -C_{\mu,1}\int_{\mathbb{R}^N}\frac{|\nabla u|^2}{|x|^2} \mathrm{d}x +C_{\mu,2}\int_{\mathbb{R}^N}\frac{u^2}{|x|^4} \mathrm{d}x \geq \mathcal{S}_\mu\left(\int_{\mathbb{R}^N}|u|^{\frac{2N}{N-4}} \mathrm{d}x\right)^\frac{N-4}{N},\quad u\in C^\infty_0(\mathbb{R}^N), \end{align*} where $C_{\mu,1}$, $C_{\mu,1}$ and $\mathcal{S}_\mu$ are constants depending on $\mu$, furthermore equality only holds for some radial functions, which is a key ingredient in analyzing the blow-up phenomena of solutions to various elliptic equations on bounded or unbounded domain. Moreover, by using spectral estimate combined with a compactness argument, we give the remainder term of above inequality.

Stability of Rellich-Sobolev type inequality involving Hardy term for bi-Laplacian

This dissertation is mainly devoted to the research of two problems - the continuous-time portfolio optimization in different Wishart models and the effects of discrete rebalancing on portfolio wealth distribution and optimal portfolio strategy.

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Perturbation theory of eigenvalue problems

Citations

On Certain Nonlinear Eigenvalue Problems

Singular asymptotic expansion of the exact control for a linear model of the Rayleigh beam

On the Caffarelli-Kohn-Niremberg type inequalities and related topics

Stability of Rellich-Sobolev type inequality involving Hardy term for bi-Laplacian

Optimal Portfolios in Wishart Models and Effects of Discrete Rebalancing on Portfolio Distribution and Strategy Selection

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