On Phragmen — Lindelof Principle for Non-Divergence Type Elliptic Equations and Mixed Boundary Conditions

We investigate the qualitative properties of solution to the Zaremba type problem in unbounded domain for the non-divergence elliptic equation with possible degeneration at infinity. The main result is Phragmén-Lindelöf type principle on growth/decay of a solution at infinity depending on both the structure of the Neumann portion of the boundary and the "thickness" of its Dirichlet portion. The result is formulated in terms of so-called s-capacity of the Dirichlet portion of the boundary, while the Neumann boundary should satisfy certain "admissibility" condition in the sequence of layers converging to infinity.

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“…The following definition of barrier and Growth Lemma for mixed boundary problem was introduced in [8,Sec. 2].…”

Section: Preliminairy Resultsmentioning

confidence: 99%

“…Roughly speaking it states that if the Wiener type series diverges then a positive sub-elliptic function, which vanishes on the Dirichlet boundary in a neighborhood of infinity tends either to zero or to infinity with prescribed speed as x 1 → ∞. Corresponding result in bounded domains for non-degenerate equation was obtained in [8].…”

Section: Introductionmentioning

confidence: 94%

Mixed boundary value problems for non-divergence type elliptic equations in unbounded domains

Cao

Ibraguimov

Nazarov

2018

Asymptotic Analysis

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“…A mixed problem for linear equations of nondivergence type was considered in Cao-Ibragimov-Nazarov [5] and Ibragimov-Nazarov [18].…”

Section: Introductionmentioning

confidence: 99%

Behaviour at infinity for solutions of a mixed boundary value problem via inversion

Björn¹,

Mwasa²

2021

Preprint

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We study a mixed boundary value problem for the quasilinear elliptic equation div A(x, ∇u(x)) = 0 in an open infinite circular half-cylinder with prescribed continuous Dirichlet data on a part of the boundary and zero conormal derivative on the rest. We prove the existence and uniqueness of bounded weak solutions to the mixed problem and characterize the regularity of the point at infinity in terms of p-capacities. For solutions with only Neumann data near the point at infinity we show that they behave in exactly one of three possible ways, similar to the alternatives in the Phragmén-Lindelöf principle.

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A Survey of Results of St. Petersburg State University Research School on Nonlinear Partial Differential Equations. I

Apushkinskaya,

Arkhipova,

Nazarov

et al. 2024

Vestnik St.Petersb. Univ.Math.

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On Phragmen — Lindelof Principle for Non-Divergence Type Elliptic Equations and Mixed Boundary Conditions

Cited by 3 publications

References 9 publications

Mixed boundary value problems for non-divergence type elliptic equations in unbounded domains

Mixed boundary value problems for non-divergence type elliptic equations in unbounded domains

Behaviour at infinity for solutions of a mixed boundary value problem via inversion

A Survey of Results of St. Petersburg State University Research School on Nonlinear Partial Differential Equations. I

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