A Neumann-type problem for the biharmonic equation

Abstract In this paper a new class of well-posed boundary value problems for the biharmonic equation is studied. The considered problems are nonlocal boundary value problems of Bitsadze- -Samarskii type. These problems are solved by reducing them to Dirichlet and Neumann type problems. Theorems on existence and uniqueness of the solution are proved and exact solvability conditions of the considered problems are found. In addition, the integral representations of solutions are obtained.

Download Full-text

Two non-trivial solutions for a non-homogeneous Neumann problem: an Orlicz–Sobolev space setting

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s030821050700025x ◽

2009 ◽

Vol 139 (2) ◽

pp. 367-379 ◽

Cited By ~ 21

Author(s):

Alexandru Kristály ◽

Mihai Mihăilescu ◽

Vicenţiu Rădulescu

Keyword(s):

Variational Principle ◽

Sobolev Space ◽

Neumann Problem ◽

Growth Condition ◽

Type Problem ◽

Space Setting ◽

Standard Growth Condition ◽

Neumann Type

In this paper we study a non-homogeneous Neumann-type problem which involves a nonlinearity satisfying a non-standard growth condition. By using a recent variational principle of Ricceri, we establish the existence of at least two non-trivial solutions in an appropriate Orlicz–Sobolev space.

Download Full-text

On a Boundary Value Problem for the Biharmonic Equation with Multiple Involutions

Mathematics ◽

10.3390/math9172020 ◽

2021 ◽

Vol 9 (17) ◽

pp. 2020

Author(s):

Batirkhan Turmetov ◽

Valery Karachik ◽

Moldir Muratbekova

Keyword(s):

Boundary Value Problem ◽

Boundary Conditions ◽

Existence And Uniqueness ◽

Biharmonic Equation ◽

Boundary Value ◽

Boundary Operator ◽

Biharmonic Operator ◽

Uniqueness Of Solutions ◽

Neumann Type ◽

Type Boundary

A nonlocal analogue of the biharmonic operator with involution-type transformations was considered. For the corresponding biharmonic equation with involution, we investigated the solvability of boundary value problems with a fractional-order boundary operator having a derivative of the Hadamard-type. First, transformations of the involution type were considered. The properties of the matrices of these transformations were investigated. As applications of the considered transformations, the questions about the solvability of a boundary value problem for a nonlocal biharmonic equation were studied. Modified Hadamard derivatives were considered as the boundary operator. The considered problems covered the Dirichlet and Neumann-type boundary conditions. Theorems on the existence and uniqueness of solutions to the studied problems were proven.

Download Full-text

The Initial and Neumann Boundary Value Problem for a Class Parabolic Monge-Ampère Equation

Abstract and Applied Analysis ◽

10.1155/2013/535629 ◽

2013 ◽

Vol 2013 ◽

pp. 1-8

Author(s):

Juan Wang ◽

Jinlin Yang ◽

Xinzhi Liu

Keyword(s):

Boundary Value Problem ◽

Nonlinear Elliptic Equation ◽

Boundary Value ◽

Neumann Boundary ◽

Type Problem ◽

Neumann Boundary Value Problem ◽

Nonlinear Elliptic ◽

Nonlinear Parabolic ◽

Neumann Type ◽

Monge Ampere

We consider the existence, uniqueness, and asymptotic behavior of a classical solution to the initial and Neumann boundary value problem for a class nonlinear parabolic equation of Monge-Ampère type. We show that such solution exists for all times and is unique. It converges eventually to a solution that satisfies a Neumann type problem for nonlinear elliptic equation of Monge-Ampère type.

Download Full-text