Generators of the space of bounded biharmonic functions

1972 ◽  
Vol 127 (3) ◽  
pp. 273-280 ◽  
Author(s):  
Leo Sario ◽  
Cecilia Wang
Keyword(s):  
Biharmonic Functions

On Liouville’s Theorem for Biharmonic Functions

1971 ◽  
Vol 20 (1) ◽  
pp. 37-39 ◽  
Author(s):  
R. R. Huilgol
Keyword(s):  
Biharmonic Functions ◽  
Liouville’S Theorem

Parabolicity and Existence of Dirichlet Finite Biharmonic Functions

1974 ◽  
Vol s2-8 (1) ◽  
pp. 145-148 ◽  
Author(s):  
Leo Sario ◽  
Cecilia Wang
Keyword(s):  
Biharmonic Functions

Continuation of biharmonic functions by reflection

1955 ◽  
Vol 22 (2) ◽  
pp. 313-324 ◽  
Author(s):  
R. J. Duffin
Keyword(s):  
Biharmonic Functions

Biharmonic functions and Brownian motion

1967 ◽  
Vol 4 (1) ◽  
pp. 130-136 ◽  
Author(s):  
L. L. Helms
Keyword(s):  
Brownian Motion ◽  
Boundary Conditions ◽  
Boundary Function ◽  
Normal Derivative ◽  
Dimensional Euclidean Space ◽  
Biharmonic Functions ◽  
Neumann Type ◽  
Bounded Open Subset ◽  
Image Position ◽  
First Time

Let R be a bounded open subset of N-dimensional Euclidean space EN,N ≧ 1, let {xt: t ≧ 0} be a separable Brownian motion starting at a point x ɛ R, and let τ = τR be the first time the motion hits the complement of R. It is known [1] that if g is a bounded measurable function on the boundary ∂R of R, then h(x) = Ex[g(xτ)] is a harmonic function of x ɛ R which “solves” the Dirichlet problem for the boundary function g; i.e., Δh = 0 on R, where Δ is the Laplacian. In elastic plate problems, one must solve the biharmonic equation subject to certain boundary conditions. For the more important applications, these boundary conditions involve the values of u and the normal derivative of u at points of ∂R. Even though a treatment of this Neumann type problem is not available at this time, some things can be said about biharmonic functions and their relationship to Brownian motion. We will show, in fact, that u(x)= Ex[τ(xτ)] is a biharmonic function on R which “satisfies” the boundary conditions (i) u=0 on ∂R and (ii) Δu= −2g on ∂R, provided g satisfies certain hypotheses. More generally, we will show that u(x)=Ex[Δkg(XΔ)] is polyharmonic of order k + 1 on R (i.e., Δk + 1u = Δ(Δku) = 0 on R) and that it satisfies certain boundary conditions. A treatment of the special case g ≡ 1 on ∂R can be found in [3].

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