Solutions to nonlinear Neumann problems with an inverse square potential

Let Ω be an open bounded domain in with smooth boundary . We are concerned with the critical Neumann problem

Existence theorems for nonlinear functional differential equations of neutral type

Conditions are found upon satisfaction of which the differential equation

Uniqueness of solutions for some nonlinear Dirichlet problems

We consider here a class of nonlinear Dirichlet problems, in a bounded domain , of the form

Some connections between results and problems of De Giorgi, Moser and Bangert

Using theorems of Bangert, we prove a rigidity result which shows how a question raised by Bangert for elliptic integrands of Moser type is connected, in the case of minimal solutions without self-intersections, to a famous conjecture of De Giorgi for phase transitions. The work of Enrico Valdinoci was supported by MIUR Variational Methods and Nonlinear Differential Equations. Diese Zusammenarbeit wurde bei einem sehr angenehmen Besuch von EV in Freiburg begonnen.     

Loewner matrices and operator convexity

Let f be a function from \({\mathbb{R}_{+}}\) into itself. A classic theorem of K. Löwner says that f is operator monotone if and only if all matrices of the form \({\left [\frac{f(p_i) - f(p_j)}{p_i-p_j}\right ]_{\vphantom {X_{X_1}}}}\) are positive semidefinite. We show that f is operator convex if and only if all such matrices are conditionally negative definite and that f (t) = t g(t) for some operator convex function g if and only if these matrices are conditionally positive definite. Elementary proofs are given for the most interesting special cases f (t) = t ^r , and f (t) = t log t. Several consequences are derived.     

Domains of Existence of Polymonogenic Functions

The paper addresses the Levi problem for a system of n Fueter equations in a domain in quaternionic space . This problem relates to various conditions of convexity and pseudoconvexity of the boundary of the domain. Received: October, 2007, Accepted: February, 2008.     

p-convergence in measure of a sequence of measurable functions and corresponding minimal elements of c 0

We characterize convergence in measure of a sequence (f_n)_n of measurable functions to a measurable function f by elements of c₀, which express the quality of convergence of (f_n)_n to f. This characterization motivates the introduction of a new notion of convergence, called “p-convergence in measure” (p > 0), which is stronger than convergence in measure. We prove the existence of “minimal” elements in c₀ which characterize the convergence in measure of (f_n)_n to f.      

Singular solutions of some nonlinear parabolic equations with spatially inhomogeneous absorption

We study the limit behaviour of solutions of with initial data k δ ₀ when k → ∞, where h is a positive nondecreasing function and p > 1. If h(r) = r ^β , β > N(p − 1) − 2, we prove that the limit function u _∞ is an explicit very singular solution, while such a solution does not exist if β ≤  N(p − 1) − 2. If lim inf _{r→ 0} r ² ln (1/h(r))  >  0, u _∞ has a persistent singularity at (0, t) (t ≥  0). If , u _∞ has a pointwise singularity localized at (0, 0).     

The complexity of the Specht modules corresponding to hook partitions

We show that the complexity of the Specht module corresponding to any hook partition is the p-weight of the partition. We calculate the variety and the complexity of the signed permutation modules. Let E _s be a representative of the conjugacy class containing an elementary abelian p-subgroup of a symmetric group generated by s disjoint p-cycles. We give formulae for the generic Jordan types of signed permutation modules restricted to E _s and of Specht modules corresponding to hook partitions μ restricted to E _s where s is the p-weight of μ.      

A colorful theorem on transversal lines to plane convex sets

We prove a colorful version of Hadwiger’s transversal line theorem: if a family of colored and numbered convex sets in the plane has the property that any three differently colored members have a transversal line that meet the sets consistently with the numbering, then there exists a color such that all the convex sets of that color have a transversal line. All authors are partially supported by CONACYT research grant 5040017.     

Free membrane eigenvalues

For a simply connected and normalized domain D in the plane it was proven by Pólya and Schiffer in 1954 for the fixed membrane eigenvalues

The parity problem of polymatroids without double circuits

According to the present state of the theory of the matroid parity problem, the existence of a good characterization to the size of a maximum matching depends on the behavior of certain substructures, called double circuits. In this paper we prove that if a polymatroid has no double circuits then a partition type min-max formula characterizes the size of a maximum matching. Applications to parity constrained orientations and to a rigidity problem are given. Research is supported by OTKA grants K60802, TS049788 and by European MCRTN Adonet, Contract Grant No. 504438.     

Stabilized boundary element methods for exterior Helmholtz problems

In this paper we describe and analyze some modified boundary element methods to solve the exterior Dirichlet boundary value problem for the Helmholtz equation. As in classical combined field integral equations also the proposed approach avoids spurious modes. Moreover, the stability of related modified boundary element methods can be shown even in the case of Lipschitz boundaries. The proposed regularization is done based on boundary integral operators which are already included in standard boundary element formulations. Numerical examples are given to compare the proposed approach with other already existing regularized formulations.     

Existence for a degenerate Cauchy problem

We prove the existence of a solution to the degenerate parabolic Cauchy problem with a possibly unbounded Radon measure as an initial data. To accomplish this, we establish a priori estimates and derive a compactness result. We also show that the result is optimal in the Euclidian setting.     

Positive solutions for second-order superlinear repulsive singular Neumann boundary value problems

In this paper we establish the multiplicity of positive solutions to second-order superlinear repulsive singular Neumann boundary value problems. It is proved that such a problem has at least two positive solutions under reasonable conditions. Our nonlinearity may be repulsive singular in its dependent variable and superlinear at infinity. The proof relies on a nonlinear alternative of Leray-Schauder type and on Krasnoselskii fixed point theorem on compression and expansion of cones.      

Independent systems of representatives in weighted graphs

The following conjecture may have never been explicitly stated, but seems to have been floating around: if the vertex set of a graph with maximal degree Δ is partitioned into sets V _i of size 2Δ, then there exists a coloring of the graph by 2Δ colors, where each color class meets each V _i at precisely one vertex. We shall name it the strong 2Δ-colorability conjecture. We prove a fractional version of this conjecture. For this purpose, we prove a weighted generalization of a theorem of Haxell, on independent systems of representatives (ISR’s). En route, we give a survey of some recent developments in the theory of ISR’s. The research of the first author was supported by grant no 780/04 from the Israel Science Foundation, and grants from the M. & M. L. Bank Mathematics Research Fund and the fund for the promotion of research at the Technion. The research of the third author was supported by the Sacta-Rashi Foundation.     

Proof of the bandwidth conjecture of Bollobás and Komlós

In this paper we prove the following conjecture by Bollobás and Komlós: For every γ > 0 and integers r ≥ 1 and Δ, there exists β > 0 with the following property. If G is a sufficiently large graph with n vertices and minimum degree at least ((r ? 1)/r + γ)n and H is an r-chromatic graph with n vertices, bandwidth at most β n and maximum degree at most Δ, then G contains a copy of H.     

The mixed problem in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> for some two-dimensional Lipschitz domains

We consider the mixed problem,

Note making a <Emphasis Type="Italic">K</Emphasis><Subscript>4</Subscript>-free graph bipartite

We show that every K ₄-free graph G with n vertices can be made bipartite by deleting at most n ²/9 edges. Moreover, the only extremal graph which requires deletion of that many edges is a complete 3-partite graph with parts of size n/3. This proves an old conjecture of P. Erdős. Research supported in part by NSF CAREER award DMS-0546523, NSF grant DMS-0355497, USA-Israeli BSF grant, and by an Alfred P. Sloan fellowship.     

Hyperfunctions and (analytic) hypoellipticity

In this work we discuss the problem of smooth and analytic regularity for hyperfunction solutions to linear partial differential equations with analytic coefficients. In particular we show that some well known “sum of squares” operators, which satisfy Hörmander’s condition and consequently are hypoelliptic, admit hyperfunction solutions that are not smooth (in particular they are not distributions).