Two-step minimization approach to an $L^\infty$-constrained variational problem with a generalized potential
Year: 2024
Type: preprint
Abstract: We study a variational problem on $H^1({\mathbb R})$ under an $L^\infty$-constraint related to Sobolev-type inequalities for a class of generalized potentials, including $L^p$-potentials, non-positive potentials, and signed Radon measures. We establish various essential tools for this variational problem, including the decomposition principle, the comparison principle, and the perturbation theorem, which are the basis of the two-step minimization method. As for their applications, we present precise results for minimizers of minimization problems, such as the study of potentials of Dirac's delta measure type and the analysis of trapped modes in potential wells.
Source: arXiv (Cornell University)
Institutions: Kanazawa University, Universitas Islam Negeri Ar-Raniry
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: Mathematical analysis on the linear response for solutions of mean field equations, New trends of energy gradient flow: mathematical aspects and various phenomena and applications, Evolution equations describing non-standard irreversible processes --Analysis on singularities emerging in the dynamics of solutions--
OpenAlex ID: W4396653493