On the optimization problems of the principal eigenvalues of measure differential equations with indefinite measures

Zhiyuan Wen; Meirong Zhang

doi:10.3934/dcdsb.2020061

Article Contents

2020, Volume 25, Issue 8: 3257-3274. Doi: 10.3934/dcdsb.2020061

This issue Previous Article Random attractor for stochastic Hindmarsh-Rose equations with multiplicative noise Next Article Null controllability for distributed systems with time-varying constraint and applications to parabolic-like equations

On the optimization problems of the principal eigenvalues of measure differential equations with indefinite measures

Zhiyuan Wen^1, , and
Meirong Zhang^2,

1.
School of Mathematical Sciences, Inner Mongolia University, Huhhot, 010021, China
2.
Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China

^* Corresponding author: Zhiyuan Wen
^* Corresponding author: Zhiyuan Wen

Received: September 2019

Early access: February 2020

Published: August 2020

The first author is supported the Scientific Starting Research Foundation of Inner Mongolia University (No. 21200-5175108) and the National Natural Science Foundation of China (No. 11901321). The second author is supported by the National Natural Science Foundation of China (No. 11790273)

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Abstract

In this paper, we will first establish the necessary and sufficient conditions for the existence of the principal eigenvalues of second-order measure differential equations with indefinite weighted measures subject to the Neumann boundary condition. Then we will show the principal eigenvalues are continuously dependent on the weighted measures when the weak$^*$ topology is considered for measures. As applications, we will finally solve several optimization problems on principal eigenvalues, including some isospectral problems.

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Mathematics Subject Classification: Primary: 34L05, 34L15; Secondary: 49K15, 58J53.

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