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Positive powers of the Laplacian in the half-space under Dirichlet boundary conditions

  • * Corresponding author: Sven Jarohs

    * Corresponding author: Sven Jarohs 
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  • We present some explicit formulas for solutions to nonhomogeneous boundary value problems involving any positive power of the Laplacian in the half-space. For non-integer powers the operator becomes nonlocal and this requires a suitable extension of Dirichlet-type boundary conditions. A key ingredient in our proofs is a point inversion transformation which preserves harmonicity and allows us to use known results for the ball. We include uniqueness statements, regularity estimates, and describe the growth or decay of solutions at infinity and at the boundary.

    Mathematics Subject Classification: Primary: 35C05, 35C15, 35S15; Secondary: 35B50.


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