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We employ a variational approach to study the Neumann boundary value problemfor the $p$-Laplacian on bounded smooth-enough domains in the metric setting,and show that solutions exist and are bounded. The boundary data considered areBorel measurable bounded functions. We also study boundary continuityproperties of the solutions. One of the key tools utilized is the trace theoremfor Newton-Sobolev functions, and another is an analog of the De Giorgiinequality adapted to the Neumann problem.

Neumann problem for p-Laplace equation in metric spaces using a variational approach: existence, boundedness, and boundary regularity