Integral means of solutions of the one-dimensional Poisson equation with Robin boundary conditions

2023/12/19 by Christos H. Papadimitriou, Papadimitriou, Christos · 1 citation
Engineering · Mathematics · #34B08 #34C10 #Classical Analysis and ODEs (math.CA) #Differential Equations and Boundary Problems #FOS: Mathematics #Numerical methods in engineering #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2312.12535

openalex publication_date 2023/12/19 · openalex created_date 2023/12/22 · openalex updated_date 2026/07/28

Abstract

Consider the one-dimentional Poisson equation \(-u''=f\) on the interval \([-π,π]\), where \(f\) is an non-negative integrable function, with Robin boundary conditions \(-u'(-π)+αu(-π)=u'(π)+αu(π)=0\), where \(α>0\) is a constant. In this paper we prove inequalities for the convex integral means of solutions of this problem, using the polarization of functions and its properties. We find solutions with maximal convex integral means and prove their uniqueness.

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