On an obstacle problem for degenerate elliptic operators involving the critical Sobolev exponent

We establish the existence of a solution to the variational inequality (the obstacle problem) (1.1) which involves the critical Sobolev exponent. This result is also extended to an obstacle problem with a lower order perturbation. Dedicated to Professor F. Browder on the occasion of his 80-th birthday     

Non-variational elliptic systems in dimension two: a priori bounds and existence of positive solutions

We establish a priori bounds for positive solutions of semilinear elliptic systems of the form

Radial solutions with a vortex to an asymptotically linear elliptic equation

In this paper, we study a two-dimensional nonlinear elliptic equation:

Local regularity of weak solutions of semilinear parabolic systems with critical growth

We show that, under so called controllable growth conditions, any weak solution in the energy class of the semilinear parabolic system

Singular integral operators on Riemann surfaces and elliptic differential systems

The derivatives of the Cauchy kernels on compact Riemann surfaces generate singular integral operators analogous to the Calderón-Zigmund operators with the kernel (t - z)² on the complex plane. These operators play an important role in studying elliptic differential equations, boundary value problems, etc. We consider here the most important case of the multi-valued Cauchy kernel with real normalization of periods. In the opposite plane case, such an operator is not unitary. Nevertheless, its norm in L₂ is equal to one. This result is used to study multi-valued solutions of elliptic differential systems.     

Sharp asymptotics and compactness for local low energy solutions of critical elliptic systems in potential form

Let (M, g) be a smooth compact Riemannian n-manifold, n ≥ 3. Let also p ≥ 1 be an integer, and be the vector space of symmetrical p × p real matrix. We consider critical elliptic systems of equations which we write in condensed form as

Bifurcation problems for superlinear elliptic indefinite equations with exponential growth

This paper deals with the existence and the behaviour of global connected branches of positive solutions of the problem

High-accuracy schemes of the finite element method for systems of degenerate elliptic equations on an interval

In this paper for the finite elementmethod for systems of degenerate elliptic equations we develop high-accuracy schemes based on multiplicative singularity extraction. We prove theorems about the smoothness of a solution. Based on these theorems we estimate the error of the proposed method.     

Effect of topology on the multiplicity of solutions for some semilinear elliptic systems with critical Sobolev exponent

In this paper, we are concerned with the effect of the domain topology on the multiplicity of solutions for some semilinear elliptic systems involving critical Sobolev exponent. We show that if the interaction term is sufficiently small, then the number of solutions of the system is estimated from below by 2 catΩ. The proof of this fact requires analysis of the structure of the Nehari manifold associated with the system. This research was partially supported by the Grant-in-Aid for Young Scientists (B), The Ministry of Education, Culture, Sports, Science and Technology, Japan.     

Multiplicity of solutions for a noncooperative elliptic system with critical Sobolev exponent

In this paper, we study a system of elliptic equations by applying the Limit Index Theory. Under some assumptions on nonlinear part, we can obtain the existence of multiple solutions for the equations. The research is supported by NNSF of China (10471024) and Fujian Provincial Natural Science Foundation of China (A0410015).     

Comparison theorem and uniqueness of positive solutions for sublinear elliptic equations

In semilinear elliptic equations, we prove that the necessary and sufficient condition for the comparison theorem of positive solutions to be valid is that the nonlinear term is sublinear. Our theorem needs neither any regularity of the nonlinear term nor the smoothness of the boundary. Applying this theorem, we prove the uniqueness of positive solutions for the Dirichlet problem. Received: 9 April 2008     

Transferring strong boundedness among Laguerre orthogonal systems

Given the family of Laguerre polynomials, it is known that several orthonormal systems of Laguerre functions can be considered. In this paper we prove that an exhaustive knowledge of the boundedness in weighted L ^p of the heat and Poisson semigroups, Riesz transforms and g-functions associated to a particular Laguerre orthonormal system of functions, implies a complete knowledge of the boundedness of the corresponding operators on the other Laguerre orthonormal system of functions. As a byproduct, new weighted L ^p boundedness are obtained. The method also allows us to get new weighted estimates for operators related with Laguerre polynomials. Carlos Segovia passed away on April 3, 2007.     

Invariant subspaces of submarkovian semigroups

We characterize the invariance under a submarkovian semigroup of a measurable subset by capacity conditions on its boundary.      

Some generalizations of Knopp's identity*

For integers a, b and n > 0, define

The critical value for some non-attractive long range nearest particle systems

We consider the nearest particle system which gives birth rate to each vacant interval, concentrated on the interval's midpoint(s). We prove that a critical value for exists and equals one. The proof extends to a large class of nearest particle systems. This paper solves a problem suggested by Liggett (1985).In the following we deal with nearest particle systems { _t :t0}. These can be described as particle systems with the following flip rates:

Estimation on curves of Dirichlet series with a convex growth majorant

For a class of Dirichlet series defined by a certain convex growth majorant we establish conditions for a sequence of indices which provide the implementation of precise estimates for their increase and decrease on curves that tend to infinity in a special way.     

Distance and fractional isomorphism in Steiner triple systems

In [8], Quattrochi and Rinaldi introduced the idea ofn ⁻¹-isomorphism between Steiner systems. In this paper we study this concept in the context of Steiner triple systems. The main result is that for any positive integerN, there existsv ₀(N) such that for all admissiblev≥v ₀(N) and for each STS(v) (sayS), there exists an STS(v) (sayS′) such that for somen>N, S is strictlyn ⁻¹-isomorphic toS′. We also prove that for all admissiblev≥13, there exist two STS(v)s which are strictly 2⁻¹-isomorphic. Define the distance between two Steiner triple systemsS andS′ of the same order to be the minimum volume of a tradeT which transformsS into a system isomorphic toS′. We determine the distance between any two Steiner triple systems of order 15 and, further, give a complete classification of strictly 2⁻¹-isomorphic and 3⁻¹-isomorphic pairs of STS(15)s.     

Energy Concentration and Sommerfeld Condition for Helmholtz Equation with Variable Index at Infinity

We consider the Helmholtz equation with a variable index of refraction n(x), which is not necessarily constant at infinity but can have an angular dependency like as . Under some appropriate assumptions on this convergence and on n _∞ we prove that the Sommerfeld condition at infinity still holds true under the explicit form

Schauder estimates for elliptic operators with applications to nodal sets

In this paper we first give a priori estimates on asymptotic polynomials of solutions to elliptic equations at nodal points. This leads to a pointwise version of Schauder estimates. As an application we discuss the structure of nodal sets of solutions to elliptic equations with nonsmooth coefficients.     

Solutions of elliptic problems with nonlinearities of linear growth

In this paper, we study existence of nontrivial solutions to the elliptic equation