Moshe Marcus

Existence and uniqueness results for semi-linear Dirichlet problems in annuli

A foot clamping device for ski boots comprises, inside the boot body a presser member at the foot heel region and a threaded peg extending from the presser member and rotatably engaged in a threaded bush associated with a boss on the outside of the boot body. When the boss is rotated by a strap rigid therewith and constituting a closure element for the boot, the presser member is caused to traverse.

A characterization of first order nonlinear partial differential operators on Sobolev spaces

Nonlinear partial differential operators G: W1,p(Ω) → Lq(Ω) (1 ⩽ p, q ∞) having the form G(u) = g(u, D1u,…, DNu), with g ϵ C(R × RN), are here shown to be precisely those operators which are local, (locally) uniformly continuous on, W1,∞(Ω), and (roughly speaking) translation invariant. It is also shown that all such partial differential operators are necessarily bounded and continuous with respect to the norm topologies of W1,p(Ω) and Lq(Ω).

A Radon-Nikodym type theorem for functionals

Let (Ω, τ, m) be a finite, nonatomic, separable measure space. This paper extends the Radon-Nikodym theorem to odd, disjointly additive, m-continuous functionals whose domain consists of all differences of characteristic functions which belong to a given subspace of L∞(m). Such a functional will possess a density in L1(m) provided that the subspace is weak∗-closed and separates sets; the conclusion can fail if the latter hypothesis is removed. Analogous results are obtained for functionals which are not necessarily odd.

Extension theorems of Hahn-Banach type for nonlinear disjointly additive functionals and operators in Lebesgue spaces

Abstract Let (Ω, τ, M ) be a nonatomic separable finite measure space. Every continuous functional N on L p ( m ), 1 ⩽ p N ( u + v ) = N ( u ) + N ( v ) whenever uv = 0, is known to be representable by an integral with a nonlinear Caratheodory kernel. Such functionals share several regularity properties with continuous linear functionals. Here we study the question of whether every continuous, disjointly additive functional defined on a closed subspace of L p ( m ) possesses an extension to L p ( m ) with these same properties. This question has applications to the study of nonlinear functionals on Sobolev spaces. It is shown that for a class of subspaces, including those of finite codimension, such an extension always exists, but there are also closed subspaces not possessing this extension property. Analogous results are obtained for disjointly additive mappings from closed subspaces of L p ( m ) into L 1 ( m ) and for functionals defined on subspaces of L ∞ ( m ). The techniques depend heavily on the utilization of Lyapunov vector measures.