In a 2D free boundary problem for the A-Laplacian, the free boundary is locally the graph of continuous functions.
Saadi: Coninuity of the free boundary in elliptic problems with Ne uman boundary condition: Electronic Journal of Differential Equations, Vol
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Continuity of the Free Boundary in a Problem involving the A-Laplacian
In a 2D free boundary problem for the A-Laplacian, the free boundary is locally the graph of continuous functions.