Nonexistence for the Laplace equation with a dynamical boundary condition of fractional type. (Russian, English) Zbl 1164.35360

Kirane, Mokhtar; Tatar, Nasser-Eddine

Sib. Mat. Zh. 48, No. 5, 1056-1064 (2007); translation in Sib. Math. J. 48, No. 5, 849-856 (2007).

Summary: We consider the Laplace equation in \(\mathbb R^{d-1}\times\mathbb R^+\times (0,+\infty)\) 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.

MSC:

35J65	Nonlinear boundary value problems for linear elliptic equations
35B33	Critical exponents in context of PDEs
35J05	Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation

Keywords:

critical exponent; dynamical boundary condition; fractional derivative; Laplace equation