2011/03/03 by Benjamin Jaye, Jaye, Benjamin J., Vladimir Maz’ya +3
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.1103.0698
openalex publication_date 2011/03/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove the existence of positive solutions with optimal local regularity to\nhomogeneous elliptic equations of Schr "odinger type, under only a form\nboundedness assumption on \σ \∈ D'(\Ω) and ellipticity assumption on\n\A\∈ L^\∞(\Ω)n\× n, for an arbitrary open set\n\Ω\⊆ \Rn. We demonstrate that there is a two way\ncorrespondence between the form boundedness and the existence of positive\nsolutions to this equation, as well as weak solutions to certain elliptic\nequations with quadratic nonlinearity in the gradient. As a consequence, we\nobtain necessary and sufficient conditions for both the form-boundedness (with\na sharp upper form bound) and the positivity of the quadratic form of the\nSchr "odinger type operator with arbitrary distributional potential \σ\n\∈ D'(\Ω), and give examples clarifying the relationship between these\ntwo properties.\n