4-Harmonic functions and beyond
Abstract
The family of partial differential equations −∆4u−ε∆∞u = 0 (ε > 0) is studied in a bounded domain Ω for given boundary data. We show that for each ε > 0 the problem has a unique viscosity solution which is exactly the (4+ε)-harmonic map with the given boundary data. We also explore the connections between the solutions of these problems and infinity harmonic and 4-harmonic maps by studying the limiting behavior of the solutions as ε → ∞ and ε → 0+, respectively.
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