Weak solutions for equations defined by accretive operators II: relaxation limits

In the first part of this paper we define solutions for certain nonlinear equations defined by accretive operators, “dissipative solution”. This kind of solution is equivalent to the viscosity solutions for Hamilton-Jacobi equations and to the entropy solutions for conservation laws.In this paper we use dissipative solutions to obtain several relaxation limits for systems of semilinear transport equations and quasilinear conservation laws. These converge to diffusion second-order equations and in one case to a single conservation law. The relaxation limit is obtained using a version of the perturbed test function method to pass to the limit. This guarantees existence for the considered equations.     

An integration by parts formula in submanifolds of positive codimension

We prove a general integration by parts formula for first‐order differential operators on submanifolds of arbitrary codimension of a given Riemannian manifold. The differential operators in question are assumed to satisfy a suitable tangentiality condition. Several particular cases, particularly relevant to applications to PDE arising in mathematical physics are discussed. Copyright © 2004 John Wiley & Sons, Ltd.     

On some pointwise inequalities

In a recent paper (Studia Math. 138 (2000) 285-291) we proved pointwise estimates relating some classical maximal and singular integral operators. Here we show that, in a sense, there are more flexible inequalities which not only imply the previously known results but also give something new. In particular, they hold for the multilinear Calderón-Zygmund operators. This result gives a new approach to a recent work by Grafakos and Torres, and unifies some classical inequalities by Cotlar and Coifman and Fefferman.     

Generalizing the L-fuzzy unit interval

Given an arbitrary connected topological space, an L-fuzzy space is constructed. When the original space is the real line (unit interval) the constructed space is the L-fuzzy real line (L-fuzzy unit interval). For some spaces, including Rⁿ for all n, the original space is embedded as a subspace of the constructed space. Lastly, the construction yields a non-trivial fuzzy topology on certain classically important spaces of monotone mappings.     

关于强伪压缩算子与强增生算子的Ishikawa型迭代序列收敛性的特征

本文结果表征了用于构造强增生算子方程解,m-增生算子方程解及强伪压缩算子不动点的(带误差的)Ishikawa型迭代序列的收敛性,推广与改进了Chidume与Osilike的定理1,定理2及定理3(Nonlinear Anal.TMA,1999,36(7):863-872)。     

Minimal projections in spaces of functions of N variables

We will construct a minimal and co-minimal projection from L^p([0,1]ⁿ) onto L^p([0,1]^n₁)++L^p([0,1]^n_k), where n=n₁++n_k (see Theorem 2.9). This is a generalization of a result of Cheney, Halton and Light from (Approximation Theory in Tensor Product Spaces, Lecture Notes in Mathematics, Springer, Berlin, 1985; Math. Proc. Cambridge Philos. Soc. 97 (1985) 127; Math. Z. 191 (1986) 633) where they proved the minimality in the case n=2. We provide also some further generalizations (see Theorems 2.10 and 2.11 (Orlicz spaces) and Theorem 2.8). Also a discrete case (Theorem 2.2) is considered. Our approach differs from methods used in [8,13,20].     

Bounds for singular fractional integrals and related Fourier integral operators

We prove sharp L^p−L^q endpoint bounds for singular fractional integral operators and related Fourier integral operators, under the nonvanishing rotational curvature hypothesis.     

Optimal Control Problems for a Class of Semilinear Multisolution Elliptic Equations

We study the optimal control problem for a class of elliptic problems that may possess multiple solutions. We obtain necessary conditions for optimal control by constructing a related parabolic problem and using known results for the parabolic problem.     

A multiplier theorem using the Schechter's method of interpolation

Let m be a measurable bounded function and let us assume that there exists a bounded functions S so that m(ξ)S(ξ)^it−1 is a Fourier multiplier on L^p uniformly in . Then, using the analytic interpolation theorem of Stein, one can show that necessarily m is a L^p multiplier. The purpose of this work is to show that under the above conditions, it holds that, for every , m(log S)^kM_p. The technique is based on the Schechter's interpolation method.