The present paper is devoted to the study of the “quality” of the compactness of the trace operator. More precisely, we characterize the asymptotic behaviour of entropy numbers of the compact map , where Γ is a d-set with 0 < d < n and a weight of type near Γ with ϰ > -(n-d). There are parallel results for approximation numbers.
Let id be the natural embedding of the Sobolev space in the Zygmund space , where , 1 < p < ∞, l ∈ ℕ, 1/p = 1/q + l/n and a < 0, a ≠ -l/n. We consider the entropy numbers of this embedding and show that , where η = min(-a,l/n). Extensions to more general spaces are given. The results are applied to give information about the behaviour of the eigenvalues of certain operators of elliptic type.
We determine the asymptotic behavior of the entropy numbers of diagonal operators D: lp → lq, (xk) → (skxk), 0 < p,q ≤ ∞, under mild regularity and decay conditions on the generating sequence (σk). Our results extend the known estimates for polynomial and logarithmic diagonals (σk). Moreover, we also consider some exotic intermediate examples like (σk)=exp(-√log k).
We study a class of anisotropic nonlinear elliptic equations with variable exponent p⃗(·) growth. We obtain the existence of entropy solutions by using the truncation technique and some a priori estimates.
We discuss the existence of entropy solution for the strongly nonlinear unilateral parabolic inequalities associated to the nonlinear parabolic equations ∂u/∂t - div(a(x,t,u,∇u) + Φ(u)) + g(u)M(|∇u|) = μ in Q, in the framework of Orlicz-Sobolev spaces without any restriction on the N-function of the Orlicz spaces, where -div(a(x,t,u,∇u)) is a Leray-Lions operator and . The function g(u)M(|∇u|) is a nonlinear lower order term with natural growth with respect to |∇u|, without satisfying the sign...
We prove an existence result of entropy solutions for a class of strongly nonlinear parabolic problems in Musielak-Sobolev spaces, without using the sign condition on the nonlinearities and with measure data.
We introduce a notion of disjoint envelope functions to study asymptotic structures of Banach spaces. The main result gives a new characterization of asymptotic- spaces in terms of the -behavior of “disjoint-permissible” vectors of constant coefficients. Applying this result to Tirilman spaces we obtain a negative solution to a conjecture of Casazza and Shura. Further investigation of the disjoint envelopes leads to a finite-representability result in the spirit of the Maurey-Pisier theorem.