AN ANISOTROPIC INFINITY LAPLACIAN OBTAINED AS THE LIMIT OF THE ANISOTROPIC (p, q)-LAPLACIAN

作者：Perez Llanos Mayte^*; Rossi Julio D

来源：Communications in Contemporary Mathematics, 2011, 13(6): 1057-1076.

DOI：10.1142/S0219199711004543

• (p, q)-Laplacian
• infinity-Laplacian
• viscosity solutions

In this work we study the behavior of the solutions to the following Dirichlet problem related to the anisotropic (p, q)-Laplacian operator
{-div(x)(vertical bar del(x)u vertical bar(p-2)del(x)u) - div(y)(vertical bar del(y)u vertical bar(q-2)del(y)u) = 0 in Omega,
u = g on partial derivative Omega,
as p, q -> infinity. Here Omega. R(N) x R(K) and del(x)u = (partial derivative u/partial derivative x(1), partial derivative u/partial derivative x(2), ... , partial derivative u/partial derivative x(N)) and del(y)u = (partial derivative u/partial derivative y(1), partial derivative u/partial derivative y(2), ... , partial derivative u/partial derivative y(K)) denote the gradient of u with respect to the first N variables (x variables) and with respect to the last K variables (y variables). We consider a sequence of exponents (p(n), q(n)) that goes to infinity with p(n)/q(n) -> R. We prove that un, the solution with p = p(n), q = q(n), verifies u(n) -> u(infinity) uniformly in (Omega) over bar, where u(infinity) is the unique viscosity solution to
{-Delta(infinity,x)u(infinity) = 0 for vertical bar del(y)u infinity vertical bar(R) < vertical bar del(x)u(infinity)vertical bar,
-R Delta(infinity,y)u(infinity) = 0 for vertical bar del(y)u(infinity)vertical bar(R) < vertical bar del(x)u(infinity)vertical bar,
-Delta(infinity,x)u(infinity) - R Delta(infinity,y)u(infinity) = 0 for vertical bar del(y)u(infinity)vertical bar(R) < vertical bar del(x)u(infinity)vertical bar,
u(infinity) = g on partial derivative Omega.
Here Delta(infinity,x)u = del(x)uD(x)(2)u(del(x)u)(t) and Delta(infinity,y)u = del(y)uD(y)(2)u(del(y)u)(t) are the infinity Laplacian in x variables and in y variables, respectively.

• 出版日期2011-12