Quasi-linear boundary value problems with generalized nonlocal boundary conditions

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We investigate a quasi-linear boundary value problem of the form − div ( α | ∇ u | p − 2 ∇ u ) = 0 involving a general boundary map and mixed Neumann boundary conditions on a bounded Lipschitz domain. We show existence, uniqueness, and Hölder continuity of the weak solution of this mixed boundary value problem, and obtain maximum principles for this class of mixed equations. As a consequence, we obtain uniform continuity up to the boundary to solutions associated with a class of electrical models described by Maxwell’s equations with nonlocal boundary conditions. An extension to boundary value problems with generalized nonlocal Robin boundary conditions is also achieved.