69,141 research outputs found
Operator inclusions and operator-differential inclusions
In Chapter 2, we first introduce a generalized inverse differentiability for set-valued mappings and consider some of its properties. Then, we use this differentiability, Ekeland's Variational Principle and some fixed point theorems to consider constrained implicit function and open mapping theorems and surjectivity problems of set-valued mappings. The mapping considered is of the form F(x, u) + G (x, u). The inverse derivative condition is only imposed on the mapping x F(x, u), and the mapping x G(x, u) is supposed to be Lipschitz. The constraint made to the variable x is a closed convex cone if x F(x, u) is only a closed mapping, and in case x F(x, u) is also Lipschitz, the constraint needs only to be a closed subset. We obtain some constrained implicit function theorems and open mapping theorems. Pseudo-Lipschitz property and surjectivity of the implicit functions are also obtained. As applications of the obtained results, we also consider both local constrained controllability of nonlinear systems and constrained global controllability of semilinear systems. The constraint made to the control is a time-dependent closed convex cone with possibly empty interior. Our results show that the controllability will be realized if some suitable associated linear systems are constrained controllable.
In Chapter 3, without defining topological degree for set-valued mappings of monotone type, we consider the solvability of the operator inclusion y0 N1(x) + N2 (x) on bounded subsets in Banach spaces with N1 a demicontinuous set-valued mapping which is either of class (S+) or pseudo-monotone or quasi-monotone, and N2 is a set-valued quasi-monotone mapping. Conclusions similar to the invariance under admissible homotopy of topological degree are obtained. Some concrete existence results and applications to some boundary value problems, integral inclusions and controllability of a nonlinear system are also given.
In Chapter 4, we will suppose u A (t,u) is a set-valued pseudo-monotone mapping and consider the evolution inclusions
x' (t) + A(t,x((t)) f (t) a.e. and (d)/(dt) (Bx(t)) + A (t,x((t)) f(t) a.e.
in an evolution triple (V,H,V*), as well as perturbation problems of those two inclusions
Pseudo-holomorphic functions at the critical exponent
We study Hardy classes on the disk associated to the equation \bar\d
w=\alpha\bar w for with . The paper seems to
be the first to deal with the case . We prove an analog of the M.~Riesz
theorem and a topological converse to the Bers similarity principle. Using the
connection between pseudo-holomorphic functions and conjugate Beltrami
equations, we deduce well-posedness on smooth domains of the Dirichlet problem
with weighted boundary data for 2-D isotropic conductivity equations
whose coefficients have logarithm in . In particular these are not
strictly elliptic. Our results depend on a new multiplier theorem for
-functions.Comment: 43 pages; to appear in the Journal of the European Mathematical
Societ
Pr\"ufer intersection of valuation domains of a field of rational functions
Let be a rank one valuation domain with quotient field . We
characterize the subsets of for which the ring of integer-valued
polynomials is a Pr\"ufer
domain. The characterization is obtained by means of the notion of
pseudo-monotone sequence and pseudo-limit in the sense of Chabert, which
generalize the classical notions of pseudo-convergent sequence and pseudo-limit
by Ostrowski and Kaplansky, respectively. We show that is
Pr\"ufer if and only if no element of the algebraic closure of
is a pseudo-limit of a pseudo-monotone sequence contained in , with
respect to some extension of to . This result expands a
recent result by Loper and Werner.Comment: to appear in J. Algebra. All comments are welcome. Keywords: Pr\"ufer
domain, pseudo-convergent sequence, pseudo-limit, residually transcendental
extension, integer-valued polynomia
Product structure of heat phase space and branching Brownian motion
A generical formalism for the discussion of Brownian processes with
non-constant particle number is developed, based on the observation that the
phase space of heat possesses a product structure that can be encoded in a
commutative unit ring. A single Brownian particle is discussed in a Hilbert
module theory, with the underlying ring structure seen to be intimately linked
to the non-differentiability of Brownian paths. Multi-particle systems with
interactions are explicitly constructed using a Fock space approach. The
resulting ring-valued quantum field theory is applied to binary branching
Brownian motion, whose Dyson-Schwinger equations can be exactly solved. The
presented formalism permits the application of the full machinery of quantum
field theory to Brownian processes.Comment: 32 pages, journal version. Annals of Physics, N.Y. (to appear
- …