Non-convexity of level sets in convex rings for semilinear elliptic problems

Non-convexity of level sets in convex rings for semilinear elliptic problems

We show that there is a convex ring R = Ω \Ω ⊂ R in which there exists a solution u to a semilinear partial differential equation ∆u = f(u), u = −1 on ∂Ω, u = 1 on ∂Ω, with level sets, not all convex. Moreover every bounded solution u has at least one nonconvex level set. In our construction, the nonlinearity f , is non-positive, and smooth. AMS Classification: 35J60, 35R35.

An algorithm for convexity of semilinear sets over ordered fields

In this paper we study semilinear sets over arbitrary ordered fields. In particular we show that open (resp. closed) semilinear sets are convex if and only if they are basic, that is, a finite intersection of open (resp. closed) halfspaces. As a corollary we get that any bounded convex closed semilinear set is the convex hull of a finite family of points and we obtain a separation result for co...

Compact Convex Sets and Complex Convexity

We construct a quasi-Banach space which cannot be given an equivalent plurisubharmonic quasi-norm, but such that it has a quotient by a onedimensional space which is a Banach space. We then use this example to construct a compact convex set in a quasi-Banach space which cannot be atfinely embedded into the space L0 of all measurable functions.

Optimal Control of Semilinear Elliptic Obstacle Problems

Convexity Conditions for Non - Locally Convex Lattices

for any x 1 ( . . . , x,, GX. A theorem of Aolci and Rolewicz (see [18]) asserts that if in (1.3) C = 2~\ then X is p-normable. We can then equivalently re-norm X so that in (1.4) JB = 1. If in addition X is a vector lattice and ||x||<||y|| whenever |x|<|y| we say that X is a quasi-Banach lattice. As in the case of Banach lattices [13] we may make the following definitions. We shall say that X ...