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Si illustra attraverso qualche esempio significativo come alcuni campi della matematica e dell'ingegneria si siano sviluppati attraverso una proficua interazione tra gli aspetti teorici e applicativi. In particolare, si fa riferimento alla geometria descrittiva e alla teoria del trasporto ottimo.

Proceedings of "The Renato Caccioppoli Centenary Conference", Napoli, 2004

We prove a sharp upper bound on convex domains, in terms of the diameter alone, of the best constant in a class of weighted Poincar�� inequalities. The key point is the study of an optimal weighted Wirtinger inequality., Few typos have been corrected. 10 pages. No figures. arXiv admin note: substantial text overlap with arXiv:1110.2960

Atti del Convegno Nonlinear Analysis and Calculus of Variations, Perugia, 1993

Moser-type estimates for solutions of linear elliptic equations with lower-order terms are given. The results are proved under various assumptions on the coefficients and on the known term in the equations. Also solutions of Hamilton-Jacobi equations are considered.

Let Ω be an open convex set in Rm with finite width, and with boundary ∂Ω. Let vΩ be the torsion function for Ω, i.e., the solution of −Δv=1,v|∂Ω=0. An upper bound is obtained for the product of ∥vΩ∥L∞(Ω)λ(Ω), where λ(Ω) is the bottom of the spectrum of the Dirichlet Laplacian acting in L2(Ω). The upper bound is sharp in the limit of a thinning sequence of convex sets. For planar rhombi and isosceles triangles with area 1, it is shown that ∥vΩ∥L1(Ω)λ(Ω)≥π2/24, and that this bound is sharp.

Given a smooth simply connected planar domain, the area is bounded away from zero in terms of the maximal curvature alone. We show that in higher dimensions this is not true, and for a given maximal mean curvature we provide smooth embeddings of the ball with arbitrary small volume.

For a given bounded Lipschitz set $\Omega$, we consider a Steklov--type eigenvalue problem for the Laplacian operator whose solutions provide extremal functions for the compact embedding $H^1(\Omega)\hookrightarrow L^2(\partial \Omega)$. We prove that a conjectured reverse Faber--Krahn inequality holds true at least in the class of Lipschitz sets which are ``close'' to a ball in a Hausdorff metric sense. The result implies that among sets of prescribed measure, balls are local minimizers of the embedding constant.

It was recently proved in [D. Bucur and A. Henrot, J. Eur. Math. Soc. (JEMS) 19, No. 11, 3355–3376] and [V. Ferone, B. Kawohl & C. Nitsch, Math. Ann., 365, No. 3–4, 987–1015] that the elastic energy E(γ)=1/2int_γ κ^2 ds of a closed curve γ with curvature κ has a minimizer among all plane, simple, regular and closed curves of given enclosed area A(γ), and that the minimum is attained only for circles. In particular, the proof used in [V. Ferone, B. Kawohl & C. Nitsch, loc. cit.] is of a geometric nature, and here, we show under which hypothesis it can be extended to other functionals involving the curvature. As an example, we show that the optimal shape remains a circle for the p-elastic energy int_γ |κ|^p ds, whenever p>1.

We prove a quantitative form of the Faber–Krahn inequality for the first eigenvalue of the Laplace operator with Robin boundary conditions. The asymmetry term involves the square power of the Fraenkel asymmetry, multiplied by a constant depending on the Robin parameter, the dimension of the space and the measure of the set.

We prove a long standing conjecture concerning the fencing problem in the plane: among planar convex sets of given area, the disc, and only the disc, maximizes the length of the shortest area-bisecting curve. Although it may look intuitive, the result is by no means trivial since we also prove that among planar convex sets of given area the set which maximizes the length of the shortest bisecting chords is the so-called Auerbach triangle.

In this paper we deal with blow-up solutions to p-Laplacian equations with a nonlinear gradient term. We prove comparison results for the solutions in terms of the solutions to suitable symmetrized problems defined in a ball. We analyze two cases where the form of the term which depends on the gradient plays different roles.

In this paper the question of finding the best constant in a Hardy-Sobolev inequality is addressed. The inequality under investigation gives the possibility to bound a weighted norm of a function u , with a weight which depends only on a group of variables, in terms of the L^p -norm of its gradient. A complete answer is obtained in a special case.

In the present paper we prove some comparison results via Steiner symmetrization for solutions to the Neumann problem where T > 0, Ω is a smooth connected open bounded subset of ℝ n , the coefficients a ij (x, y) and the datum f are smooth functions such that a ij (x, y)ξ i ξ j ≥ |ξ|2, for any (x, y) ∈ G, for any ξ ∈ ℝ n and ∈ t G f dx dy = 0.

We investigate the first eigenvalue of a highly nonlinear class of elliptic operators which includes the p--Laplace operator $\Delta_p u=\sum_i {{\partial}\over{\partial x_i}} (\vert\nabla u \vert^{p-2}{{\partial u}\over{\partial x_i}})$, the pseudo-p-Laplace operator $\tilde\Delta_p u=\sum_i {{\partial}\over{\partial x_i}} (\vert {{\partial u}\over{\partial x_i}} \vert^{p-2} {{\partial u}\over{\partial x_i}})$ and others. We derive the positivity of the first eingefunction, simlicity of the first eigenvalue, Faber-Krahn and Payne-Rayner type inequalities. In another chapter we address the question of symmetry for positive solutions to more general equations. Using a Pohozaev-type inequality and isoperimetric inequalities as well as convex rearrangement methods we generalize a symmetry result of Kesavan and Pacella. Our optimal domains are level sets of a convex function H o. They have the so-called Wulff shape associated with H and only in special cases they are Euclidean balls.

In Newton's problem of minimal resistance one seeks to minimize the functional R(v)= ∫ Ω 1/1+|⊇v(x)| 2 dx over a suitable class A of admissible functions. Here Ω ⊂ IR 2 is the maximal cross section of a body travelling through a rarefied liquid. This variational problem can be derived from first principles in mechanics, see [10]. Various classes of admissible functions have been discussed for instance in [2, 5]. Since R is not coercive, one has to introduce some bound on the class of admissible functions, and one of the bounds that was suggested in [4 p. 259 for the case of radial functions was the surface area of the body. In this case EGGERS was able to conclude that a minimizer of R had to have conical shape. In the present paper we return to this optimal shap problem (as well as to closely related questions) for a base domain Ω ⊂ R n which is not necessarily a disk or ball. Throughout the paper Ω is assumed to be bounded and simply connected.

Let Ω be an open bounded set of R n with Lipschitz boundary and let u be a smooth nonnegative function defined in a cylinder G = Ω × (0, T ), T > 0. We give a derivation formula for a function of the type F ( s, y ) = ∫ u ( x,y )> u *( s,y ) ƒ( x,y )d x , where ƒ is a Lipschitz function on Ḡ and u *( s,y ) denotes the decreasing rearrangement of u (·, y ) for any fixed y .

We give a “generalized” version of the isoperimetric inequality when the perimeter is defined with respect to a convex, positively homogeneous function on R n. We use it to prove that, for any function u compactly supported in R n, the integral of a convex function of Du decreases when u is rearranged in the corresponding “convex” way. Similar arguments allow us, for example, to prove comparison results for solutions of the Dirichlet problem for elliptic equations when the differential operator satisfies suitable structure assumptions.

Balls are shown to have the smallest optimal constant, among all admissible Euclidean domains, in Poincar\'e type boundary trace inequalities for functions of bounded variation with vanishing median or mean value., Comment: 20 pages, no figures

We prove an inequality concerning the decreasing rearrangement of functions. The inequality also provides a comparison result between the viscosity solution of a Cauchy problem for a Hamilton-Jacobi equation and the viscosity solution of a symmetrized problem

We consider a solution u of the homogeneous Dirichlet problem for a class of nonlinear elliptic equations in the form A(u) = g(x, u) + f, where the principal term is a Leray-Lions operator defined on W^{1,p}_0(Ω). The function g(x, u) satisfies suitable growth assumptions, but no sign hypothesis on it is assumed. We prove that the rearrangement of u can be estimated by the solution of a problem whose data are radially symmetric.

The purpose of this paper is to prove the existence of a solution for a nonlinear parabolic equation in the form u t - div( a ( t , x , u , D u )) = H ( t , x , u , D u ) - div( g ( t , x )) in Q T =]0, T [×Ω, Ω ⊂ R N , with an initial condition u (0) = u 0 , where u 0 is not bounded, | H ( t , x , u , ξ)⩽ β|ξ| p + f ( t , x ) + βe λ 1 |u| f , |g| p/(p-1) ∈ L r (Q T ) for some r = r {N) ⩾ 1, and - div( a ( t , x , u , D u )) is the usual Leray-Lions operator.

A recent comparison result ([3]) allows to get pointwise estimates for the solution of the Dirichlet problem for linear second–order elliptic equations in terms of the solution of a suitable "symmetrized" problem. In this paper we prove that equality is achieved only in the spherically symmetric situation.

The asymptotic behaviour of all the moments of a class of symmetric one-dimensional random flights, at the first few orders, are calculated. Different symmetric random flights are considered and it is shown that they simply differ up to a scaling in the number of jumps and that important statistical quantities are universal.

In this paper we study the Dirichlet problem for a class of nonlinear elliptic equations in the form A(u) = H(x, u, Du), where the principal term is a Leray-Lions operator defined on W0 1,p (Ω), and H(x, u, Du) grows with respect to Du at most like |Du|q, p-1 ≤ q ≤ p. Comparison results are obtained between the rearrangement of a solution u of Dirichlet problem quoted above and the rearrangement of the solution of a problem whose data are radially symmetric.

We consider the eigenvalue problem for a nonlinear equation involving the p-Laplacian with homogeneous Dirichlet conditions. We give someproperties of the first eigenvalue and of the corresponding eigenfunction. In particular we prove inequalities similar to the well-known Payne--Rayner inequality, which holds in the linear case.