A strongly degenerate quasilinear elliptic equation

The Cauchy Problem for a Strongly Degenerate Quasilinear Equation

“…The purpose of the present paper is to extend the results in [4] and [5] to the case where the Lagrangian f satisfies (1.7) and the initial condition is in (…”

“…according to [6]. We assume that the Lagrangian f : R × R N → R + satisfies the following assumptions, to which we refer collectively as (H):…”

“…But, as far as we know, existence and uniqueness results for higher dimensional problems have not been considered in the literature. This was the purpose of our papers [4] and [5] in which we studied the Neumann problem for Lagrangians f satisfying the following coercivity and linear growth condition:…”

“…By introducing a notion of entropy solution for (1.9) we proved in [6] the existence and uniqueness of an entropy solution of (1.…”

“…In Section 3 we state the main assumptions on the Lagrangian f , recall the meaning of expressions of type f (u, Du) for functions u in BV(R N ) and define an associated functional calculus. We also recall the notion of entropy solution for the elliptic problem (1.9) and the existence and uniqueness results for it proved in [6]. Then we translate this result into the language of accretive operators to be able to apply Crandall-Liggett's iteration scheme to prove existence of solutions of (…”