Best possible estimates of weak solutions of boundary value problems for quasi-linear elliptic equations in unbounded domains

AbstractWe investigate the behaviour of weak solutions of boundary value problems for quasi-linear elliptic divergence second order equations in unbounded domains. We show the boundedness of weak solutions to our problem. Using barrier function and applying the comparison principle, we find the exact exponent of weak solutions decreasing rate near the infinity.

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The Behaviour of Weak Solutions of Boundary Value Problems for Linear Elliptic Second Order Equations in Unbounded Cone-Like Domains

Annales Mathematicae Silesianae ◽

10.1515/amsil-2016-0009 ◽

2016 ◽

Vol 30 (1) ◽

pp. 203-217

Author(s):

Damian Wiśniewski

Keyword(s):

Weak Solution ◽

Boundary Value Problems ◽

Weak Solutions ◽

Boundary Value ◽

Continuity Condition ◽

Second Order ◽

Second Order Equations ◽

Elliptic Divergence

AbstractWe investigate the behaviour of weak solutions of boundary value problems (Dirichlet, Neumann, Robin and mixed) for linear elliptic divergence second order equations in domains extending to infinity along a cone. We find an exponent of the solution decreasing rate: we derive the estimate of the weak solution modulus for our problems near the infinity under assumption that leading coefficients of the equations do not satisfy the Dini-continuity condition.

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Boundary value problems for quasi-linear elliptic second order equations in unbounded cone-like domains

Open Mathematics ◽

10.2478/s11533-012-0127-2 ◽

2012 ◽

Vol 10 (6) ◽

Author(s):

Mikhail Borsuk ◽

Damian Wiśniewski

Keyword(s):

Boundary Value Problems ◽

Weak Solutions ◽

Boundary Value ◽

Second Order ◽

Second Order Equations ◽

Elliptic Divergence ◽

Divergence Equations

AbstractWe study the behaviour of weak solutions (as well as their gradients) of boundary value problems for quasi-linear elliptic divergence equations in domains extending to infinity along a cone.

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On the uniqueness of the solutions of boundary-value problems in unbounded domains, and on isolated singular points of solutions of the system of elasticity theory and second-order elliptic equations

Russian Mathematical Surveys ◽

10.1070/rm1987v042n04abeh001449 ◽

1987 ◽

Vol 42 (4) ◽

pp. 143-144

Author(s):

V A Kondrat'ev ◽

O A Oleinik

Keyword(s):

Boundary Value Problems ◽

Elasticity Theory ◽

Elliptic Equations ◽

Boundary Value ◽

Unbounded Domains ◽

Second Order ◽

Singular Points ◽

Second Order Elliptic Equations

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Boundary value problems for singular second order equations

Fixed Point Theory and Applications ◽

10.1186/s13663-018-0645-0 ◽

2018 ◽

Vol 2018 (1) ◽

Cited By ~ 5

Author(s):

Alessandro Calamai ◽

Cristina Marcelli ◽

Francesca Papalini

Keyword(s):

Boundary Value Problems ◽

Boundary Value ◽

Second Order ◽

Second Order Equations

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Linear second order elliptic equations with Venttsel boundary conditions

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500029048 ◽

1991 ◽

Vol 118 (3-4) ◽

pp. 193-207 ◽

Cited By ~ 28

Author(s):

Yousong Luo ◽

Neil S. Trudinger

Keyword(s):

Boundary Conditions ◽

Boundary Value Problems ◽

Elliptic Equations ◽

Existence Result ◽

Boundary Value ◽

Second Order ◽

Classical Solutions ◽

Schauder Estimate ◽

Second Order Elliptic Equations

SynopsisWe prove a Schauder estimate for solutions of linear second order elliptic equations with linear Venttsel boundary conditions, and establish an existence result for classical solutions for such boundary value problems.

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