On theL p-bounds for maximal functions associated to convex bodies inR n

The dimension-freeL ²-maximal inequality for convex symmetric bodies obtained in [2] is extended forp>3/2.     

Wulff clusters inR 2

We give the first existence and regularity results on the cheapest way to enclose and separate planar regions of prescribed areas, where cost is given by a general norm ϕ, thus generalizing the Wulff shape for enclosing a single region. As an example, we classify the cheapest way to enclose and separate two planar regions of prescribed areas for the ℓ¹ norm (“Manhattan metric”) into three distinct types, according to the relative size of the prescribed areas.     

Level surfaces of non-degenerate functions inR n+1

We study level surfaces of non-degenerate functions inR ⁿ⁺¹. Such level surfaces are non-degenerate in the sense of affine differential geometry. In affine differential geometry, the affine normal plays an important role for the study of a non-degenerate hypersurface. In this note, being motivated by Koszul's work we take a canonical vector field for level surfaces of a non-degenerate function and give certain characterizations of when is transversal, by the shape operatorS, the transversal connection , and consider the difference between and the affine normal.     

On the stationary and nonstationary Navier-Stokes equations inR n

Sunto Si dimostra che le soluzioni dette equazioni di Navier-Stokes stazionarie considerate in tutto lo spazio Rⁿ sono asintoticamente stabili rispetto a piccole perturbazioni dei dati iniziali nella norma di L^p.     

Visible shorelines inR d

Summary LetC be a closed set inR ^d and letj be a fixed integer,j 1. The setS R ^d ~C is said to have aj-partition relative toC if there existj or fewer pointsc ₁,, c _j ofC such that each point ofS sees via the complement ofC at least one pointc _i. For every triple of integersd, p, j withd 0, p d + 1, j 1, there exists a smallest integerf(d, p, j) such that the following is true: IfC is a convexd-polytope inR ^d havingp vertices and ifS R ^d ~C, S has aj-partition relative toC if and only if everyf(d, p, j)-member subset of S has such a partition.ForC a convex polytope inR ² andS R ² ~C, all points ofS see via the complement ofC a common neighborhood in the boundary ofC if and only if every three points ofS see via the complement ofC such a neighborhood.A weak analogue of this result holds for arbitrary compact convex sets inR ^d .     

On the index of elliptic partial differential operators inR n

Summary Let A= be an elliptic differential operator inR _u, If, for |α|=l, the coefficients a_α are ? nearly constant ? and, for |α|<l, they tend to zero at infinity with a certain swiftness, it is proved that A is a Fredholm operator with index_x(A)=0 between a suitable weighted Sobolev space M contained in W^l,p (R n) and L^p(R ⁿ, (1+|x|)^lp)== . It is shown, by counterexamples, that the above result, holds only if n>l, p>n/(n−l) and that isomorphism results can be obtained, in general only if the coefficients a_α(|α|<l) are assumed to be ? sufficiently small ? also on compact sets. Then a Sturm-Liouville type problem is studied and a class of negative and falling off at infinity potentials V(x) is constructed in such a way that the Schr?dinger operator H=−Δ+V(x), in L²(R ⁿ), has a zero eigenvalue.

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Sunto Sia un operatore differenziale ellittico inR ⁿ. Se, per |α|=l, i coefficienti a_α sono ? quasi costanti ? e, per |α|<l, tendono a zero all'infinito con una certa rapidità, si dimostra che A è un operatore di Fredholm con indice_X(A)=0 tra un opportuno spazio di Sobolev con peso M contenuto in W^l,p(R ⁿ) ed L^p(R ⁿ, (1+|x|)^lp)== . Si prova, mediante controesempi, che tale risultato è valido solo se n>l, p>n/(n−l) e che teoremi di isomorfismo si possono ottenere, in generale, solo se si assume che i coefficienti a_α (|α|<l) sono ? sufficientemente piccoli ? anche su insiemi compatti. Si studia quindi un problema del tipo Sturm-Liouville e si costruisce una classe di potenziali V(x) negativi e convergenti a zero all'infinito, tali che l'operatore di Schr?dinger H=−δ+V(x) in L²(R ⁿ) abbia un autovalore nullo.

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Entrata in Redazione il 10 agosto 1977.

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Work supported by C.N.R. (G.N.A.F.A.).     

On Liouville’s theorem for locally quasiregular mappings inR n

Letf:R ⁿ→Rⁿ be locally quasiregular in the sense that the restriction off to any ball |x|<r has finite inner dilatationK ₁(r). Suppose that the growth condition ∫^∞r^-1K₁(r)^1/(1-n) holds. Then Liouville’s theorem is valid:If f is bounded, f is a constant. An example shows that this growth condition is relatively sharp.     

Approximation of plurisubharmonic functions by multipole Green functions

For a strongly hyperconvex domain we prove that multipole pluricomplex Green functions are dense in the cone in of negative plurisubharmonic functions with zero boundary values.

     

Invariant measures ofC 2-diffeomorphisms of Markov type inR d

A class of uniformly expanding, piecewiseC ²-diffeomorphisms from domainsIR ^d (bounded or not) into themselves is considered. It is shown that the number of the extreme points of Fix (P )={gG:Pg=g} whereP is the Frobenius-Perron operator associated with andG={gL ¹: g0 g=1}, can be determined in an effective way. Moreover, it is shown that the sequence {P ^j g} is convergent inL ¹ for anygG, and in the topology of uniform convergence for anygG(1). The limit is a linear projectionR inL ¹ (defined by (3.1)) which mapsG onto Fix (P ) (see Th. 3.1).Dedicated to professor A. Lasota on the occasion of his 60th birthday