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A strongly degenerate quasilinear equation : the elliptic case

Fuensanta Andreu;  Vicent Caselles;  José Mazón — 2004

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We prove existence and uniqueness of entropy solutions for the Neumann problem for the quasilinear elliptic equation u - div 𝐚 ( u , D u ) = v , where v ∈ L 1 , 𝐚 ( z , ξ ) = ∇ ξ f ( z , ξ ) , and f is a convex function of ξ with linear growth as ∥ ξ ∥ → ∞ , satisfying other additional assumptions. In particular, this class includes the case where f ( z , ξ ) = ϕ ( z ) ψ ( ξ ) , ϕ > 0 , ψ being a convex function with linear growth as ∥ ξ ∥ → ∞ . In the second part of this work, using Crandall-Ligget’s iteration scheme, this result will permit us to prove existence and uniqueness of entropy solutions for the...

On optimal matching measures for matching problems related to the Euclidean distance

José Manuel Mazón;  Julio Daniel Rossi;  Julián Toledo — 2014

Mathematica Bohemica

We deal with an optimal matching problem, that is, we want to transport two measures to a given place (the target set) where they will match, minimizing the total transport cost that in our case is given by the sum of two different multiples of the Euclidean distance that each measure is transported. We show that such a problem has a solution with an optimal matching measure supported in the target set. This result can be proved by an approximation procedure using a p -Laplacian system. We prove...

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