LOCAL AND NONLOCAL WEIGHTED p-LAPLACIAN EVOLUTION EQUATIONS WITH NEUMANN BOUNDARY CONDITIONS

作者：Andreu F^*; Mazon J M; Rossi J D; Toledo J

来源：Publicacions Matematiques, 2011, 55(1): 27-66.

DOI：10.5565/PUBLMAT_55111_03

• Nonlocal diffusion
• p-Laplacian
• total variation flow
• Neumann boundary conditions

In this paper we study existence and uniqueness of solutions to the local diffusion equation with Neumann boundary conditions and a bounded nonhomogeneous diffusion coefficient g >= 0,
{u(t) = div (g vertical bar del u vertical bar(p-2)del u) in ]0,T[x Omega,
g vertical bar del u vertical bar(p-2)del u . eta = 0 on ]0,T[x partial derivative Omega,
for 1 <= p < infinity. We show that a nonlocal counterpart of this diffusion problem is
u(t)(t,x) = integral(Omega)J(x-y)g(x+y/2)vertical bar u(t,y)-u(t,x)vertical bar(p-2)(u(t,y)-u(t,x)) dy in ]0,T[x Omega,
where the diffusion coefficient has been reinterpreted by means of the values of g at the point x+y/2 in the integral operator. The fact that g >= 0 is allowed to vanish in a set of positive measure involves subtle difficulties, specially in the case p = 1.

• 出版日期2011