Strict positivity for the principal eigenfunction of elliptic operators\n with various boundary conditions

2019/09/26 by Wolfgang Arendt, Arendt, Wolfgang, A. F. M. ter Elst +3
Computer Science · Mathematics · #35B50 #35K08 #35P15 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Nonlinear Partial Differential Equations #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.1909.12194

openalex publication_date 2019/09/26 · openalex created_date 2022/07/19 · openalex updated_date 2026/07/28

Abstract

We consider elliptic operators with measurable coefficients and Robin\nboundary conditions on a bounded domain \Ω \⊂ \ℝ^d and show\nthat the first eigenfunction v satisfies v(x) \≥ \δ > 0 for all x \∈\n\\Ω, even if the boundary \∂ \Ω is only Lipschitz\ncontinuous. Under such weak regularity assumptions the Hopf-Ole u inik\nboundary lemma is not available; instead we use a new approach based on an\nabstract positivity improving condition for semigroups that map L_p(\Ω)\ninto C(\\Ω). The same tool also yields corresponding results\nfor Dirichlet or mixed boundary conditions.\n Finally, we show that our results can be used to derive strong minimum and\nmaximum principles for parabolic and elliptic equations.\n

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