The Mountain-Pass Theorem of Ambrosetti and Rabinowitz (see [1]) and the Saddle Point Theorem of Rabinowitz (see [21]) are very important tools in the critical point theory of C1-functional. That is why it is natural to ask us what happens if the functional fails to be differentiable. The first who considered such a case were Aubin and Clarke (see [6]) and Chang (see [12]),who gave suitable variants of the Mountain-Pass Theorem for locally Lipschitz functionals which are denned on reflexive Banach spaces. For this aim they replaced the usual gradient with a generalized one, which was firstly defined by Clarke (see [13], [14]).As observed by Brezis (see [12, p. 114]), these abstract critical point theorems remain valid in non-reflexive Banach spaces.

Let Ω be a bounded domain in Rn(n ≧ 3) with Lipschitz-continuous boundary, ∂Ω = Γ0∪Γ1. In this paper we consider the following problem:

where φ ∈ L2 (Γ1), φ ≢ 0 on Γ1 and γ is the unit outward normal and p = 2n/(n − 2) = 2* is the critical exponent for the Sobolev embedding . We prove that for φ ∈ L2(Γ1) satisfying suitable conditions, the problem admits two solutions.

In this paper we prove the global existence of the solutions of the Riemann problem for a class of 2 × 2 hyperbolic conservation laws, which is neither necessarily strictly hyperbolic nor necessarily genuinely nonlinear.

The starting point for the present paper is the following question, which asks whether points can be replaced by flats (translates of linear subspaces of arbitrary dimension) as the basic objects in a convexity structure on ℝd.

In Section 1 of this note we will construct an example of a subset of R × Rn such that the parabolic capacity with respect to the heat equation is zero although its orthogonal projection onto {0} × Rn is the whole space. Such examples were already given by R. Kaufman and J.-M. Wu in [5] and [6]. However, our probabilistic approach seems to be more transparent since it does not depend on explicit formulas for Green functions.

Let

where φ is the notation used by Ramanujan in his notebooks [15], and is the familiar notation of Whittaker and Watson [20, p. 464]. It is well known that [1, p. 102] (with a misprint corrected)

where denotes the ordinary or Gaussian hypergeometric function; k, 0 < k < 1, is the modulus; K is the complete elliptic integral of the first kind; and

where K′=K(k′) and is the complementary modulus. Thus, an evaluation of any one of the functions φ, , or K yields an evaluation of the other two functions. However, such evaluations may not be very explicit. For example, if K(k) is known for a certain value of k, it may be difficult or impossible to explicitly determine K′, and so q cannot be explicitly determined. Conversely, it may be possible to evaluate φ(q) for a certain value of q, but it may be impossible to determine the corresponding value of k. (Recall that [1, p. 102].)

For a sequence of polynomials (Pn) orthonormal on the interval [−1, 1], we consider the sequence of transforms (gn) of the series given by . We establish necessary and sufficient conditions on the matrix (bnk) for the sequence (gn) to converge uniformly on compact subsets of the interior of an appropriate ellipse to a function holomorphic on that interior.

For finite coverings in euclidean d-space Ed we introduce a parametric density function. Here the parameter controls the influence of the boundary of the covered region to the density. This definition gives a new approach to covering which is similar to the approach for packing in [BHW1], [BHW2]. In this way we obtain a unified theory for finite and infinite covering and generalize similar results, which were developed by various authors since 1950 for d=2, to all dimensions.

Let Δ = {z:|Z|<1}, Γ={z:|z|=1}, and ℳ denote the set of complex-valued Borel measures on Γ. Let Kα(z)=(1−z)−α for α>0 and K0(z)=log 1/(1−z). For α > 0 let ℱα denote the family of functions f on Δ having the property that there exists a measure μ∈ℳ such that

for |z|<1. When α=0, this condition is replaced by

The dynamics of cluster growth can be modelled by the following infinite system of ordinary differential equations, first proposed by Smoluchowski, [8],

where cj=cj(t) represents the physical concentration of j-clusters (aggregates of j identical particles), aj,k=aj,k≥0 are the time-independent coagulation coefficients, measuring the effectiveness of the coagulation process between a j-cluster and a k-cluster, and the first sum in the right-hand side of (1) is defined to be zero if j = 1.

In the paper [2] Hsia noted that the forms x2+xy+y2+9z2 and x2+3y2+3yz+3z2 constitute a genus and that both forms are regular; he asked whether there exist any other genera containing two or more regular forms. In this note it is proved that the forms

are regular. They constitute a genus with discriminant 27 (in the normalization used by Brandt and Intrau in [1]). It is noteworthy that Hsia's genus has the same discriminant.

For arbitrary f: R → R and ϒ ⊂ Z × R we define the set of quantized observations of f relative to ϒ as follows: for each integer n and each y∈R we write

(the supremum of an empty set is taken to be −∞ ) and we put

Thus for example and , where [x] (without subscript) denotes as usual the integer part of x.

One of the most beautiful and important results in geometric convexity is Hadwiger's characterization theorem for the quermassintegrals. Hadwiger's theorem classifies all continuous rigid motion invariant valuations on convex bodies as consisting of the linear span of the quermassintegrals (or, equivalently, of the intrinsic volumes) [4]. Hadwiger's characterization leads to effortless proofs of numerous results in integral geometry, including various kinematic formulas [7, 9] and the mean projection formulas for convex bodies [10]. Hadwiger's result also provides a connection between rigid motion invariant set functions and symmetric polynomials [1, 7].

A double elastic panel, set in a light compressible fluid, is excited by a time harmonic force applied along a line on one of the plates. The double panel consists of two parallel elastic plates, with different elastic properties, each of width a and separated by distance d≪a, set in rigid plane baffles with acoustically soft adjoining side walls. Each plate is taken to have infinite extent in the z-direction, so that the problem treated is a two-dimensional one. The radiated acoustic power is estimated asymptotically, averaged over a small frequency band and over all line force positions, for frequencies that are sufficiently high to ensure the excitation of many panel modes, with ka≫1 and kd >> 1, where k is the acoustic wave number. Transition formulae are given for frequencies ω that are near to either of the coincidence frequencies, ω1 and ω2, of the individual panels.

We prove a generalization of a theorem of Ryshkov relating the Voronoï vectors of lattices to the defining conditions for the Minkowski fundamental domain . This is then used to prove that a Minkowski reduced basis of a lattice of dimension n < 7 consists of strict Voronoï vectors.

Suppose X and Y are spaces of analytic functions in the open unit disk D in the complex plane C. A sequence {λ} is called a coefficient multiplier from X to Y if the function belongs to Y whenever the function belongs to X.

We prove that for every strictly convex body C in the Euclidean space of dimension d≥3, some aflfine image of C admits a non-lattice covering of the space, thinner than any lattice covering. We illustrate the general construction with an example of a thin non-lattice covering of with certain congruent ellipsoids.

In this paper we study various classes of centrally symmetric sets in d-dimensional Euclidean space Rd. As we will see, it is appropriate to focus our attention on those sets which have interior points.