Approximation theorems for the solution of Fourier's problem and Dirichlet's problem

Mathematische Annalen -     

Uniqueness theorems for a nonlinear Tricomi problem and the related evolution problem

We consider an initial-boundary value problem for the non-linear evolution equation in a cylinder Q_t = Ω × (0, t), where T[u] = yu_xx + u_yy is the Tricomi operator and l(u) a special differential operator of first order. In [10] we proved the existence of a generalized solution of problem (1) and the existence of a generalized solution of the corresponding stationary boundary value problem (non-linear Tricomi problem) In this paper we give sufficient conditions for the uniqueness of these solutions.     

On the symmetric travelling salesman problem II: Lifting theorems and facets

Four lifting theorems are derived for the symmetric travelling salesman polytope. They provide constructions and state conditions under which a linear inequality which defines a facet of then-city travelling salesman polytope retains its facetial property for the (n + m)-city travelling salesman polytope, wherem 1 is an arbitrary integer. In particular, they permit a proof that all subtour-elimination as well as comb inequalities define facets of the convex hull of tours of then-city travelling salesman problem, wheren is an arbitrary integer.     

Some duality theorems for the non-linear vector maximum problem

Summary In this paper the well-knownLegendre transform-type of non-linear duality is extended to the general vector maximum problem. Duality is studied in terms of the primal and dual objective sets rather than in terms of the underlying feasible solutions. The structure of the objective sets is fully explored. The main result in duality is that under reasonable regularity assumptions there are no duality gaps between the primal and dual objective sets.

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Zusammenfassung In dieser Arbeit wird das bekannte, derLegendre-Transformation nachgebildete Konzept der nichtlinearen Dualität auf das allgemeine Vektor-Maximum-Problem übertragen. Hierbei wird die Dualität der Zielmengen (nicht aber der zulässigen Lösungen) des primalen und dualen Programms untersucht. Die Struktur der Zielmengen wird eingehend studiert. Hauptergebnis ist, daß unter plausiblen Regularitätsbedingungen keine Lücken zwischen der primalen und der dualen Zielmenge auftreten.

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This paper was originally written when the author was at the Center for Operations Research and Econometrics, University of Louvain. I wish to thank ProfessorsG. de Ghellinck, J. Drèze, of Louvain, andW. Szwarc, of Wrocaw, for stimulating discussions of the subject. I alone am responsible for all remaining errors.

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Vorgel. v.:W. Wittmann     

Proof of existence theorems for the two-parameter martingale problem

Translated from Sibirskii Matemaiicheskii, Vol. 36, No. 2, pp. 248–265, March–April, 1995.     

Precedence theorems and dynamic programming for the single-machine weighted tardiness problem

We tackle precedence-constrained sequencing on a single machine in order to minimize total weighted tardiness. Classic dynamic programming (DP) methods for this problem are limited in performance due to excessive memory requirements, particularly when the precedence network is not sufficiently dense. Over the last decades, a number of precedence theorems have been proposed, which distinguish dominant precedence constraints for a job pool that is initially without precedence relation. In this paper, we connect and extend the findings of the foregoing two strands of literature. We develop a framework for applying the precedence theorems to the precedence-constrained problem to tighten the search space, and we propose an exact DP algorithm that utilizes a new efficient memory management technique. Our procedure outperforms the state-of-the-art algorithm for instances with medium to high network density. We also empirically verify the computational gain of using different sets of precedence theorems.     

Trace theorems for holomorphic semigroups and the second order Cauchy problem

We use the theory of boundary values (also called traces) of holomorphic semigroups as developed by Boyadzhiev-deLaubenfels (1993) and El-Mennaoui (1992) to study the second order Cauchy problem for certain generators of holomorphic semigroups. Our results contain in particular the result of Hieber (Math. Ann. 291 (1991), 1--16) for the Laplace operator on .

     

Liouville theorems for stable solutions of biharmonic problem

We prove some Liouville type results for stable solutions to the biharmonic problem $\Delta ^2 u= u^q, \,u>0$ in $\mathbb{R }^n$ where $1 < q < \infty $ . For example, for $n \ge 5$ , we show that there are no stable classical solution in $\mathbb{R }^n$ when $\frac{n+4}{n-4} < q \le \left(\frac{n-8}{n}\right)_+^{-1}$ .     

Some existence theorems for the Dirichlet problem for quasilinear elliptic equations

Summary The purpose of this paper is to establish the existence of a solution of the Dirichlet problem for a quasilinear elliptic equation. We prove that the problem (1), (2) is solvable for all _j (j=1,...), where _j are eigenvalues of a linear elliptic operator that is associated with a given quasilinear elliptic operator by a limiting process. We also discuss this problem with a boundary data in L². This leads in a natural way to the Dirichlet problem in a weighted Sobolev space.     

Tight bounds for periodicity theorems on the unbounded Knapsack problem

Three new bounds for periodicity theorems on the unbounded Knapsack problem are developed. Periodicity theorems specify when it is optimal to pack one unit of the best item (the one with the highest profit-to-weight ratio). The successive applications of periodicity theorems can drastically reduce the size of the Knapsack problem under analysis, theoretical or empirical. We prove that each new bound is tight in the sense that no smaller bound exists under the given condition.     

Local existence and uniqueness theorems for a free boundary problem

Free boundary problems are considered, where the tangential and normal components u_t and u_n of an otherwise unknown plane harmonic vector field are prescribed along the unknown boundary curve as a function of the coordinates x, y and the tangent angle θ. The vector field is required to exist either in the interior region G⁺ or in the exterior G^?. In each case the free boundary is characterized by a nonlinear integral equation. A linearised version of this equation is a one-dimensional singular integral equation. Under rather general hypotheses which are easy to check, the properties of the linear equation are described by Noether's theorems. The regularity of the solution is studied and the effect of the nonlinear terms is estimated. A variant of the Nash-Moser implicit-function theorem can be applied. This yields local existence and uniqueness theorems for the free boundary problem in Hölder-classes H^2+μ. The boundary curve depends continuously on the defining data. Finally some examples are given, where the linearised equation can be completely discussed.     

Local regularity theorems for the stationary thermistor problem with oscillating degeneracy

In this paper we present some regularity results for solutions to the system −Δu=σ(u)²|∇φ|, div(σ(u)∇φ)=0 in the case where σ(u) is allowed to oscillate between 0 and a positive number as u→∞. In particular, we show that u is locally bounded if σ(u) is bounded below by a suitable exponential function.     

Uniqueness theorems for an impulsive Sturm-Liouville boundary value problem

In this study, an impulsive boundary value problem, generated by Sturm-Liouville differential equation with the eigenvalue parameter contained in one boundary condition is considered. It is shown that the coefficients of the problem are uniquely determined either by the Weyl function or by two given spectra.     

Convergence theorems for equilibrium problem, variational inequality problem and countably infinite relatively quasi-nonexpansive mappings

In this paper, we introduce an iterative process which converges strongly to a common element of set of common fixed points of countably infinite family of closed relatively quasi- nonexpansive mappings, the solution set of generalized equilibrium problem and the solution set of the variational inequality problem for a γ-inverse strongly monotone mapping in Banach spaces. Our theorems improve, generalize, unify and extend several results recently announced.     

On comparison theorems for elliptic inequalities

We obtain estimates for non-negative solutions of quasilinear elliptic inequalities with the lowest terms.