Stability estimates in H¹₀ for solutions of elliptic equations in\n varying domains

2012/05/09 by José M. Arrieta, Arrieta, José M., Gerassimos Barbatis +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1205.2027

openalex publication_date 2012/05/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider second-order uniformly elliptic operators subject to Dirichlet\nboundary conditions. Such operators are considered on a bounded domain \Ω\nand on the domain \φ(\Ω) resulting from \Ω by means of a\nbi-Lipschitz map \φ. We consider the solutions u and u of the\ncorresponding elliptic equations with the same right-hand side f\∈\nL²(\Ω\∪\φ(\Ω)). Under certain assumptions we estimate the\ndifference \‖\∇ u-\∇ u\‖_{L²(\Ω\∪\φ(\Ω))} in terms\nof certain measure of vicinity of \φ to the identity map. For domains\nwithin a certain class this provides estimates in terms of the Lebesgue measure\nof the symmetric difference of \φ(\Ω) and \Ω, that is\n|\φ(\Ω) triangle \Ω|. We provide an example which shows that the\nestimates obtained are in a certain sense sharp.\n

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