Liouville Theorem For Harmonic Functions Research Articles

Yau and Souplet-Zhang type gradient estimates on Riemannian manifolds with boundary under Dirichlet boundary condition

In this paper, on Riemannian manifolds with boundary, we establish a Yau type gradient estimate and Liouville theorem for harmonic functions under Dirichlet boundary condition. Under a similar setting, we also formulate a Souplet-Zhang type gradient estimate and Liouville theorem for ancient solutions to the heat equation.

Some Riemann boundary value problems in Clifford analysis

In this paper, we mainly study the Rm (m>0) Riemann boundary value problems for functions with values in a Clifford algebra Cl(V3, 3). We prove a generalized Liouville-type theorem for harmonic functions and biharmonic functions by combining the growth behaviour estimates with the series expansions for k-monogenic functions. We obtain the result under only one growth condition at infinity by using the integral representation formulas for harmonic functions and biharmonic functions. By using the Plemelj formula and the integral representation formulas, a more generalized Liouville theorem for harmonic functions and biharmonic functions are presented. Combining the Plemelj formula and the integral representation formulas with the above generalized Liouville theorem, we prove that the Rm (m>0) Riemann boundary value problems for monogenic functions, harmonic functions and biharmonic functions are solvable. Explicit representation formulas of the solutions are given. Copyright © 2009 John Wiley & Sons, Ltd.

The Liouville property and a conjecture of De Giorgi

We consider bounded entire solutions of the nonlinear PDE Δu + u u 3 = 0i n R d and prove that under certain monotonicity conditions these solutions must be constant on hyperplanes. The proof uses a Liouville theorem for harmonic functions associated with a nonuniformly elliptic divergence form operator. c 2000 John Wiley & Sons, Inc.