On the asymptotic behavior of solutions of quasilinear elliptic equations

Sunto

È studiato il comportamento asintotico di soluzioni di equazioni alle derivate parziali del secondo ordine quasilineari ellittiche e noniperboliche, definite in domini nonlimitati diR n contenuti in\(\{ x_1 ,...,x_n :\left| {x_n } \right|< \lambda \sqrt {x_1^2 + ...x_{n - 1}^2 } \) per certe funzioni sublineari λ, quando tali soluzioni soddisfano condizioni di Dirichlet al bordo e il dato al contorno di Dirichlet ha un assegnato comportamento asintotico all'infinito. Proviamo teoremi di Pragmen-Lindelöf per una larga classe di operatori noniperbolici, senza «termini di ordine inferiore», inclusi gli operatori uniformemente ellittici ed operatori congenere ben befinito, usando speciali funzioni barriera che sono costruite considerando un operatore associato al nostro operatore originale. Inoltre, stimiamo la velocità con la quale una soluzione converge alla sua funzione limite all'infinito in termini di proprietà del coefficiente di ordine più altoa nn dell'operatore e la velocità con la quale il valore al bordo converge alla rispettiva funzione limite: questi risultati sono provati usando appropriate funzioni barriera che dipendono dal comportamento dei coefficienti dell'operatore e la velocità di convergenza dei valori al bordo.

Abstract

The asymptotic behavior of solutions of second-order quasilinear elliptic and nonhyperbolic partial differential equations defined on unbounded domains inR n contained in\(\{ x_1 ,...,x_n :\left| {x_n } \right|< \lambda \sqrt {x_1^2 + ...x_{n - 1}^2 } \) for certain sublinear functions λ is investigated when such solutions satisfy Dirichlet boundary conditions and the Dirichlet boundary data has appropriate asymptotic behavior at infinity. We prove Phragmèn-Lindelöf theorems for large classes of nonhyperbolic operators, without «lower order terms”, including uniformly elliptic operators and operators with well-definedgenre, using special barrier functions which are constructed by considering an operator associated to our original operator. We also estimate the rate at which a solution converges to its limiting function at infinity in terms of properties of the top order coefficienta nn of the operator and the rate at which the boundary values converge to their limiting function; these results are proven using appropriate barrier functions which depend on the behavior of the coefficients of the operator and the rate of convergence of boundary values.

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Lancaster, K., Stanley, J. On the asymptotic behavior of solutions of quasilinear elliptic equations. Ann. Univ. Ferrara 49, 85 (2003). https://doi.org/10.1007/BF02844912

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Key words

  • Phragmèn-Lindelöf theorem
  • behavior at infinity
  • degenerate elliptic equation
  • elliptic equation
  • parabolic equation

Mathematics Subject Classification

  • Primary 35B40
  • Secondary 35J25
  • 35K20