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Limits as p(x) of p(x)-harmonic functions
Manfredi, Juan J.; Rossi, Julio D.; Urbano, José MiguelIn this note we study the limit as p(x) ! 1of solutions to − p(x)u = 0 in a domain , with Dirichlet boundary conditions. Our approach consists in considering sequences of variable exponents converging uniformly to +1 and analyzing how the corresponding solutions of the problem converge and what equation is satisfied by the limit.
p(x)-Harmonic functions with unbounded exponent in a subdomain
Manfredi, Juan J.; Rossi, Julio D.; Urbano, José MiguelWe study the Dirichlet problem −div( ; ∇u ; p(x)−2∇u) = 0 in , with u = f on @ and p(x) = ∞ in D, a subdomain of the reference domain . The main issue is to give a proper sense to what a solution is. To this end, we consider the limit as n → ∞ of the solutions un to the corresponding problem when pn(x) = p(x)∧ n, in particular, with p = n in D. Under suitable assumptions on the data, we find that such a li...
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