Behavior of Weak Solutions to the Boundary Value Problems for Second Order Elliptic Quasi-Linear Equation with Constant and Variable Nonlinearity Exponent in a Neighborhood of a Conical Boundary Point

2017 ◽  
pp. 3-14
Author(s):  
Yury Alkhutov ◽  
Mikhail Borsuk ◽  
Sebastian Jankowski
Keyword(s):  
Linear Equation ◽  
Boundary Value Problems ◽  
Weak Solutions ◽  
Boundary Point ◽  
Boundary Value ◽  
Second Order

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

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.

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

2017 ◽  
Vol 25 (2) ◽  
pp. 201-224
Author(s):  
Damian Wiśniewski
Keyword(s):  
Boundary Value Problems ◽  
Comparison Principle ◽  
Weak Solutions ◽  
Barrier Function ◽  
Elliptic Equations ◽  
Boundary Value ◽  
Unbounded Domains ◽  
Second Order ◽  
Second Order Equations ◽  
Elliptic Divergence

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.

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

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.

scholarly journals Nonlinear conservation laws for the Schrödinger boundary value problems of second order

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Ming Ren ◽  
Shiwei Yun ◽  
Zhenping Li
Keyword(s):  
Cauchy Problem ◽  
Conservation Laws ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Maximum Modulus ◽  
Second Order ◽  
Nonlinear Conservation Laws ◽  
The Cauchy Problem ◽  
Modulus Method ◽  
Semilinear Schrödinger Equation

AbstractIn this paper, we apply a reliable combination of maximum modulus method with respect to the Schrödinger operator and Phragmén–Lindelöf method to investigate nonlinear conservation laws for the Schrödinger boundary value problems of second order. As an application, we prove the global existence to the solution for the Cauchy problem of the semilinear Schrödinger equation. The results reveal that this method is effective and simple.

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