A general setting for constrained Z,1-approximation is presented. Let Un be a finite dimensional subspace of C[a, b] and L be a linear operator from Un to C(K) (r = 0, 1) where K is a finite union of disjoint, closed, bounded intervals. For v , u e C(K) with v < u, the approximating set is Univ, u) = {p e Un : v < Lp < u on K} and the norm is \\f\\w = Xf \f\wdx where w a positive continuous fun...

1. Introduction. Let J be a closed linear interval ao^tSfo. Let r(/) = (#(/), y(t) } z(t)), tSI, represent a vector function whose three components x(t), y(i) t z(t) are of bounded variation and continuous on I. This vector function determines in Euclidean 3-space a curve x~x{t), y~y(t), z=*z(t) whose length we denote by LQç). By convergence in length of a sequence of such vector functions $ n ...

Let I be an open bounded interval of R and W a non-negative continuous function vanishing only at α, β ∈ R. We investigate the asymptotic behaviour in terms of Γ-convergence of the following functional Gε(u) := ε p−2 ∫∫ I×I ∣∣∣∣u(x)− u(y) x− y ∣∣∣∣pdxdy + 1ε ∫ I W (u) dx (p > 2), as ε→ 0. Mathematics Subject Classification (2000). Primary 82B26, 49J45 ; Secondary 49Q20.

For any semifinite von Neumann algebra ${\mathcal M}$ and $1\leq p<\infty$, we introduce a natutal $S^1$-valued noncommutative $L^p$-space $L^p({\mathcal M};S^1)$. We say that bounded map $T\colon L^p({\mathcal M})\to N})$ is $S^1$-bounded (resp. $S^1$-contractive) if $T\otimes I_{S^1}$ extends to contractive) $T\overline{\otimes} from $ M};S^1)$ into N};S^1)$. show completely positive $S^1$-bo...

We consider the optimization problem of minimizing R Ω |∇u| + λχ{u>0} dx in the class of functions W 1,p(·)(Ω) with u − φ0 ∈ W 1,p(·) 0 (Ω), for a given φ0 ≥ 0 and bounded. W 1,p(·)(Ω) is the class of weakly differentiable functions with R Ω |∇u| dx < ∞. We prove that every solution u is locally Lipschitz continuous, that it is a solution to a free boundary problem and that the free boundary, Ω...

In this paper we study regularity and free boundary regularity, below the continuous threshold, for the p Laplace equation in Lipschitz and C domains. To formulate our results we let Ω ⊂ R be a bounded Lipschitz domain with constant M . Given p, 1 < p < ∞, w ∈ ∂Ω, 0 < r < r0, suppose that u is a positive p harmonic function in Ω ∩ B(w, 4r), that u is continuous in Ω̄ ∩ B̄(w, 4r) and u = 0 on ∆(w,...

The positive definiteness of the exponential transform of a planar domain is proved by elementary means. This direct approach avoids the heavy machinery of the theory of hyponormal operators and leads to a better understanding of the linear data associated in previous works to a quadrature domain. Version: September 10, 2003. 1. The exponential transform Let Ω be a bounded open subset of the co...

where E: D(E) —• S, D(E) C S, is a linear not necessarily bounded operator in a real Hilbert space S with 1 < dim W < °°, W = ketE, and N: S —> S is a continuous nonlinear operator. In terms of the alternative method (see Cesari [1] , [2]), let P: S —> S be the orthogonal projector with range PS = S0 = W = ker E, let St = (I -P)S, and let S1 be also the range of E. Let H: Sx -—• 5'1 denote a li...

Let K be a compact subset of R " with K=~nt K. Necessary conditions on an n-dimensional subspace U, of C(K) are given so that for each f~ C(K) there exists a unique best L'(w)-approximation from U,,, for every fixed positive weight function w. Let K be a compact subset of R ". For convenience we assume that K= int K. W will denote the set of bounded, integrable functions on K for which inf(w(x)...

Let pj, q,, denote the self-adjoint operators -iajax, and multiplication by x, on L’(W), respectively. Let H,, = (l/2) xyr= r o,(pf +qj-l)whereo,>O,andletPbe a polynomial in the q’s that is bounded below. The existence and uniqueness problems are considered for H, + IF’(q) renormalized relative to its own ground state. For example, suppose that P has degree 2d and P(q) >~(qf + ... +qz)“-k for s...