We prove an inequality ofthe form fen a(lxl)7,-(dx) _> fen a(lxl)T,_ (dx), where Q is a bounded domain in R" with smooth boundary, B is a ball centered in the origin having the same measure as f. From this we derive inequalities comparing a weighted $obolev norm of a given function with the norm ofits symmetric decreasing rearrangement. Furthermore, we use the inequality to obtain comparison results for elliptic boundary value problems.

We solve a class of isoperimetric problems on R 2 + := (x, y) ∈ R 2 : y > 0 with respect to monomial weights. Let α and β be real numbers such that 0 ≤ α < β +1, β ≤ 2α. We show that, among all smooth sets Ω in R 2 + with fixed weighted measure Ω y β dxdy, the weighted perimeter ∂Ω y α ds achieves its minimum for a smooth set which is symmetric w.r.t. to the y-axis, and is explicitly given. Our results also imply an estimate of a weighted Cheeger constant and a lower bound for the first eigenvalue of a class of nonlinear problems.

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In the present paper we study the behaviour as p goes to 1 of the weak solutions to the problems 8 < :where Ω is a bounded open set of R N (N ≥ 2) with Lipschitz boundary and p > 1. As far as the datum f is concerned, we analyze several cases: the most general one is f ∈ W −1,∞ (Ω). We also illustrate our results by means of remarks and examples. IntroductionIn the present paper we study the behaviour, when p goes to 1, of the solutions u p ∈ W 1,p 0 (Ω) to the problemswhere p > 1 and Ω is a bounded open set of R N (N ≥ 2) with Lipschitz boundary. We analyze the case where Ω is a ball and the datum f is a non-negative radially decreasing function belonging to the Lorentz space L N,∞ (Ω) and the case where the datum f belongs to the dual space W −1,∞ (Ω). We are interested in finding the pointwise limit of u p as p goes to 1 and in proving that such a limit is a solution to the "limit equation" of (1.1), namely:2000 Mathematics Subject Classification. 35J20, 35J70.

We study isoperimetric inequalities for a certain class of probability measures on ℝn together with applications to integral inequalities for weighted rearrangements. Furthermore, we compare the solution to a linear elliptic problem with the solution to some “rearranged” problem defined in the domain {x: x1 < α (x2, …, xn)} with a suitable function α (x2, …, xn). (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)