Abstract
We study a mixed boundary value problem for the p-Laplace equation Δpu=0 in an open infinite circular half-cylinder with prescribed Dirichlet boundary data on a part of the boundary and zero Neumann boundary data on the rest. Existence of weak solutions to the mixed problem is proved both for Sobolev and for continuous data on the Dirichlet part of the boundary. We also obtain a boundary regularity result for the point at infinity in terms of a variational capacity adapted to the cylinder.
Highlights
When solving the Dirichlet problem for a given partial differential equation in a nonempty open setΩ ⊂ Rn one primarily seeks a solution u which is constructed from the boundary data f ∈ C(∂Ω ) so that lim u(x) = f (x0) for x0 ∈ ∂Ω
The solution u is often found in a suitable Sobolev space associated with the studied equation and the boundary data are only attained in a weak sense
In Theorem 6.3, we prove the existence of weak solutions to the mixed boundary value problem for (1.2) with Sobolev type Dirichlet data
Summary
Lemmas 7.7 and 7.8 show that for K ⊂ Gt \ Gt+1, the two capacities are comparable, but this is not true for general K ⊂ Gt−1 To obtain these results, we use the change of variables introduced in Bjorn [1] to transform the infinite half-cylinder G and the p-Laplace equation (1.2) into a unit half-ball and a weighted elliptic quasilinear equation div A(ξ, ∇u(ξ)) = 0,. We use the Wiener criterion for such equations, together with tools from [6], to determine the regularity of the point at infinity and to prove the existence of continuous weak solutions to the mixed boundary value problem for (1.2). This makes it possible to use the tools developed for weighted elliptic quasilinear equations in [6].
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
