Abstract. We prove that the Steiner symmetrization of a function can be approximated in L p (R n ) by a sequence of very simple rearrangements which are called polarizations. This result is exploited to develop elementary proofs of many inequalities, including the isoperimetric inequality in Euclidean space. In this way we also obtain new symmetry results for solutions of some variational problems. Furthermore we compare the solutions of two boundary value problems, one of them having a "polarized" geometry and we show some pointwise inequalities between the solutions. This leads to new proofs of well-known functional inequalities which compare the solutions of two elliptic or parabolic problems, one of them having a "Steiner-symmetrized" geometry. The method also allows us to investigate the case of equality in the inequalities. Roughly speaking we prove that the equality sign is valid only if the original problem has the symmetrized geometry.

No abstract

Let Ω be a bounded smooth domain in ℝn and let 0 = λ1 ≤ λ2 ≤ … denote the eigenvalues of the Stekloff problem: Δu = 0 in Ω and (∥u)/(∥ν) = λi on ∥Ω. We show that ${\textstyle \sum \limits _{i=2} ^{n+1}} \lambda _{i} ^{-1} \geq n /\lambda ^{\ast}_{2} $, where $\lambda ^{\ast}_{2}$ denotes the second eigenvalue of the Stekloff problem in a ball having the same measure as Ω. The proof is based on a weighted isoperimetric inequality.

For nonnegative L-measurable functions u : R" -+ R a continuous homotopy u', 0 < t 5 + co, is constructed, connecting u with its Steinersymmetrization u*. It is shown that a number of familiar relations between u and u* including some integral inequalities are also valid for u and u'. The method is applicable to prove symmetry properties of stationary solutions of variational problems. R(1) where K is a convex subset of some functions space, e.g. W$j' (SZ) or Lp(sl), p > 1, and SZ c R" is a domain lying symmetrically to the hyperplane {y = 0}, {x = (x', y), x' E R"-', y E R). If u E K , we often also have u* E K , where u* denotes the Steiner-symmetrization of u with respect to y, and J(u*) I J(u) .It can be proved for the absolute minimum u of (1) that u = u*.This argumentation fails for local minima or stationary points w of the functional J . Therefore the following question is natural: Is there a (in the norm of X ) continuous homotopy t H d , O I t < + 0 3 , u 0 = u , Urn = u * , such that for u E K we also have u' E K and

We consider the integral functionalwhere Ω ⊂ R n , n ≥ 2, is a nonempty bounded connected open subset of R n with smooth boundary, and R s → f (|s|) is a convex, differentiable function. We prove that if J admits a minimizer in W 1,1 0 (Ω) depending only on the distance from the boundary of Ω, then Ω must be a ball.

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.

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)