Some theorems of Phragmen-Lindelof type for nonlinear partial differential equations

Quintanilla, R.

doi:10.5565/publmat_37293_15

Cited by 11 publications

(7 citation statements)

References 11 publications

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“…The Phragmén-Lindelöf principle, which has connections to elasticity theory, see e.g. Horgan [27], Quintanilla [52], has been frequently studied during the last century. To mention few papers, Ahlfors [4] extended results from [51] to the upper half space of R n , Gilbarg [21] and Serrin [53] considered more general elliptic equations of second order and Vitolo [54] considered the problem in angular sectors.…”

Section: Introductionmentioning

confidence: 99%

Phragmén-Lindelöf Theorems and p-harmonic Measures for Sets Near Low-dimensional Hyperplanes

Lundström

2015

Potential Anal

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We prove estimates of a p-harmonic measure, p ∈ (n − m, ∞], for sets in R n which are close to an m-dimensional hyperplane Λ ⊂ R n , m ∈ [0, n − 1]. Using these estimates, we derive results of Phragmén-Lindelöf type in unbounded domains Ω ⊂ R n \ Λ for p-subharmonic functions. Moreover, we give local and global growth estimates for p-harmonic functions, vanishing on sets in R n , which are close to an m-dimensional hyperplane.

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Section: Introductionmentioning

confidence: 99%

Phragmén-Lindelöf Theorems and p-harmonic Measures for Sets Near Low-dimensional Hyperplanes

Lundström

2015

Potential Anal

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“…Decay estimates were usually obtained for particular classes of operators in special geometries, including strips (e.g. [18,19,22,30,34,37]) and cylinders (e.g. [6,11,15,31]).…”

Section: Introductionmentioning

confidence: 99%

Phragmen–Lindelöf theorems and the asymptotic behaviour of solutions of quasilinear elliptic equations in slabs

Jin

Lancaster

2000

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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The asymptotic behaviour of solutions of second-order quasilinear elliptic partial differential equations defined on unbounded domains in Rn contained in strips (when n = 2) or slabs (when n > 2) is investigated when such solutions satisfy Dirichlet boundary conditions and the Dirichlet boundary data have appropriate asymptotic behaviour at infinity. We prove Phragmèn–Lindelöf theorems for large classes of elliptic operators, including uniformly elliptic operators and operators with well-defined genre, establish exponential decay estimates for uniformly elliptic operators when the Dirichlet boundary data vanish outside a compact set, establish the uniqueness of solutions, and give examples of solutions for non-uniformly elliptic operators which decay but do not decay exponentially. Our principal theorems are proven using special barrier functions; these barriers are constructed by considering an operator associated to our original operator.

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“…, n, are the eigenvalues of the matrix X ∈ S n and Γ, Ψ : [0, ∞) → [0, ∞) are continuous and nondecreasing functions such that Γ(s) ≤ s ≤ Ψ(s), see Capuzzo-Dolcetta-Vitolo [10]. The Phragmén-Lindelöf principle and results of Phragmén-Lindelöf type, which has connections to elasticity theory (Horgan [20], Quintanilla [36]), have been frequently studied during the last century. To mention few papers (without giving a complete summary), Ahlfors [2] extended results from Phragmén-Lindelöf [35] to the upper half space of R n , Gilbarg [15], Serrin [37] and Herzog [18] considered more general elliptic equations of second order.…”

Section: Introductionmentioning

confidence: 99%

Growth of subsolutions to fully nonlinear equations in halfspaces

Lundström¹

2021

Preprint

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We characterize lower growth estimates for subsolutions in halfspaces of fully nonlinear partial differential equations on the formin terms of solutions to ordinary differential equations built solely upon a growth assumption on F . Using this characterization we derive several sharp Phragmen-Lindelöf-type theorems for certain classes of well known PDEs. The equation need not be uniformly elliptic nor homogeneous and we obtain results both in case the subsolution is bounded or unbounded. Among our results we retrieve classical estimates in the halfspace for p-subharmonic functions and extend those to more general equations; we prove sharp growth estimates, in terms of k and the asymptotic behaviour of R 0 C(s)ds, for subsolutions of equations allowing for sublinear growth in the gradient of the form C(|x|)|Du| k with k ≥ 1; we establish a Phragmen-Lindelöf theorem for weak subsolutions of the variable exponent p-Laplace equation in halfspaces, 1 < p(x) < ∞, p(x) ∈ C 1 , of which we conclude sharpness by finding the "slowest growing" p(x)-harmonic function together with its corresponding family of p(x)-exponents. The paper ends with a discussion of our results from the point of view of a spatially dependent diffusion problem.

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Some theorems of Phragmen-Lindelof type for nonlinear partial differential equations

Cited by 11 publications

References 11 publications

Phragmén-Lindelöf Theorems and p-harmonic Measures for Sets Near Low-dimensional Hyperplanes

Phragmén-Lindelöf Theorems and p-harmonic Measures for Sets Near Low-dimensional Hyperplanes

Phragmen–Lindelöf theorems and the asymptotic behaviour of solutions of quasilinear elliptic equations in slabs

Growth of subsolutions to fully nonlinear equations in halfspaces

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