### Chapter 3. High-order elliptic equations [chapter]

1997 Spherical Means for PDEs
In this section we give a general scheme of construction of the spherical mean value relations for high-order elliptic equations (see  and ). Let X be a locally compact Hausdorf space, and let CQ(X) be a space of continuous functions / : X -* H with a compact support. We denote by R(X) the space of Radon measures FI : CQ{X) -• JR. Let B(xo, r) be a ball, and let IIjB(xo, r) be the balayage operator for the problem Lu -0 in the ball B(xo, r ) C G where r is sufficiently small. Let cr£o
more » ... y small. Let cr£o be a measure uniformly distributed on the sphere S(xo,r): S(x 0,r) where c(r) is a normalizing factor. Throughout this chapter we shall derive mainly spherical mean value relations generated by the measure + f^\2*ife!m(m + 2)---(m + 2fc-2) A ^J ' k=2 Using the expansion (3.10) we get the following mean value relation : Unauthenticated Download Date | 2/25/20 10:51 PM 3.2. The biharmonic equation Unauthenticated Download Date | 2/25/20 10:51 PM xq = (x^x^), y = (2/1,2/2)-Then ,.2 ruj 2 J p 2 r 2 u>2 J p S(xo,r) S(®o,»") + fc(xi,®?) J A(x h x\,yi)gi(y)dS y S(x 0 ,r) + k(x2,x°2) J A(x u xl,y 2 )gi{y)dS y S(x 0 ,r) ir 2~2 t>) J r -^^g o{y)dSy r 2 u>2 S(x Q ,r)