2022/01/31 by Jiřı́ Benedikt, Petr Girg, Benedikt, Jiří +5
Computer Science · Engineering · Mathematics · #35B51 #35K55 #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Primary 35B09 #Secondary 35K20
paper · pdf · doi:10.48550/arxiv.2202.00106
openalex publication_date 2022/01/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
The weak and strong comparison principles, respectively, are investigated for quasi-linear elliptic boundary value problems with the p-Laplacian in one space dimension. We treat the degenerate case of 2 < p < ∞ and allow also for the nontrivial convection velocity b\colon [-1,1]→ ℝ in the underlying domain (-1,1). We establish the weak comparison principle under a rather general, natural sufficient condition on the convection velocity, b(x), and the reaction function, φ(x,u). Furthermore, we establish also the strong comparison principle under a number of various additional hypotheses. In contrast, with these hypotheses being violated, we construct also a few rather natural counterexamples to the strong comparison principle and discuss their applications to an interesting classical problem of fluid flow in porous medium, seepage flow of fluids in inclined bed. Our methods are based on a mixture of classical and new techniques.