On Solvability of the Neumann Problem in an Energy Space for a Domain with Peak

Vladimir G. Maz'ya; Sergei V. Poborchi

doi:10.1515/GMJ.2007.499

Published by De Gruyter March 10, 2010

On Solvability of the Neumann Problem in an Energy Space for a Domain with Peak

Vladimir G. Maz'ya and Sergei V. Poborchi

From the journal Georgian Mathematical Journal

https://doi.org/10.1515/GMJ.2007.499

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Abstract

We describe the dual space of the boundary trace space for functions with a finite Dirichlet integral for a domain with a vertex of an isolated cusp at the boundary. This leads to conditions of solvability of the Neumann problem for elliptic equations of second order. In particular, we give an explicit necessary and sufficient condition for 𝑞 such that the Neumann problem is solvable if the boundary function is in 𝐿_𝑞 over the boundary of a domain with an outer peak.

Key words and phrases:: Traces of Sobolev functions; irregular domains; Neumann problem

Received: 2006-11-17

Published Online: 2010-03-10

Published in Print: 2007-September

On Solvability of the Neumann Problem in an Energy Space for a Domain with Peak

Abstract

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