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

2018/06/13 by Nicola Abatangelo, Serena Dipierro, Abatangelo, Nicola +7
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1806.05128

openalex publication_date 2018/06/13 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28

Abstract

We present explicit formulas for solutions to nonhomogeneous boundary value\nproblems involving any positive power of the Laplacian in the half-space. For\nnon-integer powers the operator becomes nonlocal and this requires a suitable\nextension of Dirichlet-type boundary conditions. A key ingredient in our proofs\nis a point inversion transformation which preserves harmonicity and allows us\nto use known results for the ball. We include uniqueness statements, regularity\nestimates, and describe the growth or decay of solutions at infinity and at the\nboundary.\n

Related

• Classification of solutions to Δu = u^-γ in the half-space
• Nonhomogeneous Boundary Value Problems of Nonlinear Schrödinger Equations in a Half Plane
• Positive solutions of nonlinear problems involving the square root of the Laplacian
• Boundary regularity, Pohozaev identities, and nonexistence results
• Nonlinear boundary problem for Harmonic functions in higher dimensional Euclidean half-spaces