Nonexistence for the Laplace equation with a dynamical boundary condition of fractional type

Mokhtar Kirane; Nasser Eddine Tatar

doi:10.1007/s11202-007-0086-1

Nonexistence for the Laplace equation with a dynamical boundary condition of fractional type

Mokhtar Kirane, Nasser Eddine Tatar

Mathematics

King Fahd University for Petroleum and Minerals (KFUPM)

Research output: Contribution to journal › Article › peer-review

15 Scopus citations

Abstract

We consider the Laplace equation in ℝ^d-1 × ℝ⁺ × (0,+∞) with a dynamical nonlinear boundary condition of order between 1 and 2. Namely, the boundary condition is a fractional differential inequality involving derivatives of noninteger order as well as a nonlinear source. Nonexistence results and necessary conditions are established for local and global existence. In particular, we show that the critical exponent depends only on the fractional derivative of the least order.

Original language	British English
Pages (from-to)	849-856
Number of pages	8
Journal	Siberian Mathematical Journal
Volume	48
Issue number	5
DOIs	https://doi.org/10.1007/s11202-007-0086-1
State	Published - Sep 2007

Keywords

Critical exponent
Dynamical boundary condition
Fractional derivative
Laplace equation

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10.1007/s11202-007-0086-1

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abstract = "We consider the Laplace equation in ℝd-1 × ℝ+ × (0,+∞) with a dynamical nonlinear boundary condition of order between 1 and 2. Namely, the boundary condition is a fractional differential inequality involving derivatives of noninteger order as well as a nonlinear source. Nonexistence results and necessary conditions are established for local and global existence. In particular, we show that the critical exponent depends only on the fractional derivative of the least order.",

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author = "Mokhtar Kirane and Tatar, {Nasser Eddine}",

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AU - Kirane, Mokhtar

AU - Tatar, Nasser Eddine

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N2 - We consider the Laplace equation in ℝd-1 × ℝ+ × (0,+∞) with a dynamical nonlinear boundary condition of order between 1 and 2. Namely, the boundary condition is a fractional differential inequality involving derivatives of noninteger order as well as a nonlinear source. Nonexistence results and necessary conditions are established for local and global existence. In particular, we show that the critical exponent depends only on the fractional derivative of the least order.

AB - We consider the Laplace equation in ℝd-1 × ℝ+ × (0,+∞) with a dynamical nonlinear boundary condition of order between 1 and 2. Namely, the boundary condition is a fractional differential inequality involving derivatives of noninteger order as well as a nonlinear source. Nonexistence results and necessary conditions are established for local and global existence. In particular, we show that the critical exponent depends only on the fractional derivative of the least order.

KW - Critical exponent

KW - Dynamical boundary condition

KW - Fractional derivative

KW - Laplace equation

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DO - 10.1007/s11202-007-0086-1

M3 - Article

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SN - 0037-4466

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SP - 849

EP - 856

JO - Siberian Mathematical Journal

JF - Siberian Mathematical Journal

IS - 5

ER -

Nonexistence for the Laplace equation with a dynamical boundary condition of fractional type

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Keywords

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