On solutions of the Dirichlet problem for the polyharmonic equation in unbounded domains

Abstract

We study the unique solvability of the Dirichlet problem for the polyharmonic equation in unbounded domains under the assumption that a generalized solution of this problem has a bounded Dirichlet integral with weight |x| a . Depending on the value of the parameter a, we prove uniqueness theorem or present exact formulas for the dimension of the solution space of the Dirichlet problem in the exterior of a compact set and in a half space.

Mixed Biharmonic Dirichlet-Neumann Problem in Exterior Domains

“…In various classes of unbounded domains with finite weighted Dirichlet (energy) integral, one of the author [10][11][12][13][14][15][16][17][18][19][20][21][22][23] studied uniqueness (non-uniqueness) problem and found the dimensions of the spaces of solutions of boundary value problems for the elasticity system and the biharmonic (polyharmonic) equation.…”

On the Mixed Dirichlet–Steklov-Type and Steklov-Type Biharmonic Problems in Weighted Spaces

“…In various classes of unbounded domains with finite weighted Dirichlet (energy) integral, one of the authors [16][17][18][19][20][21][22][23][24][25][26][27][28][29] studied uniqueness (non-uniqueness) problem and found the dimensions of the spaces of solutions of boundary value problems for the elasticity system and the biharmonic (polyharmonic) equation.…”