Solvability Results Research Articles

Global solvability for double-diffusive convection system based on Brinkman--Forchheimer equation in general domains

In this paper, we are concerned with the solvability of the initial boundary value problem of a system which describes double-diffusive convection phenomena in some porous medium under general domains, especially unbounded domains. In previous works where the boundedness of the space domain is imposed, some global solvability results have been already derived. However, when we consider our problem in general domains, some compactness theorems are not available. Hence it becomes difficult to follow the same strategies as before. Nevertheless, we can assure the global existence of a unique solution via the contraction method. Moreover, it is revealed that the global solvability holds for higher space dimension and larger class of the initial data than those assumed in previous works.

Qualitative and approximate characteristics of solutions of Beltrami-type systems

We consider Beltrami-type system in and obtain the solvability results. We give separation estimate for its solution and sufficient conditions for the discreteness of spectrum. We also establish bilateral estimates of s-numbers of the resolvent of operator corresponding to this system.

ON SOLVABILITY OF THE DAMPED FUČÍK TYPE PROBLEM WITH INTEGRAL CONDITION

The solvability results are established for the boundary value problem with a damping term , x(0) = 0, where x + = max{x, 0}, x - = max{-x, 0}, h is a bounded nonlinearity, µ, λ real parameters. The existence results are based of the knowledge of the Fučík type spectrum for the problem with h ≡ 0

On the existence of a solution of a second‐order singular initial value problem

In this paper, we consider a class of second‐order initial value problems. We extend the family of second‐order initial value problems for which the existence and uniqueness of a solution is known. To handle the singularity, we introduce the more general Caratheodory type of restrictions. The solvability results of the Emden–Fowler type of equations is established as a special case.Copyright © 2014 John Wiley & Sons, Ltd.

The complexity of multi-agent plan recognition

Multi-agent plan recognition (MAPR) seeks to identify the dynamic team structures and team plans from observations of the action sequences of a set of intelligent agents, based on a library of known team plans (plan library), and an evaluation function. It has important applications in decision support, team work, analyzing data from automated monitoring, surveillance, and intelligence analysis in general. We introduce a general model for MAPR that accommodates different representations of the plan library, and includes single agent plan recognition as a special case. Thus it provides an ideal substrate to investigate and contrast the complexities of single and multi-agent plan recognition. Using this model we generate theoretical insights on hardness, with practical implications. A key feature of these results is that they are baseline, i.e., the polynomial solvability results are given in terms of a compact and expressive plan language (context free language), while the hardness results are given in terms of a less compact language. Consequently the hardness results continue to hold in virtually all realistic plan languages, while the polynomial solvability results extend to the subsets of the context free plan language. In particular, we show that MAPR is in P (polynomial in the size of the plan library and the observation trace) if the number of agents is fixed (in particular 1) but NP-complete otherwise. If the number of agents is a variable, then even the one step MAPR problem is NP-complete. While these results pertain to abduction, we also investigate a related question: adaptation, i.e., the problem of refining the evaluation function based on feedback. We show that adaptation is also NP-hard for a variable number of agents, but easy for a single agent. These results establish a clear distinction between the hardnesses of single and multi-agent plan recognition even in idealized settings, indicating the necessity of a fundamentally different set of techniques for the latter.

DISCONTINUOUS OBLIQUE DERIVATIVE PROBLEM FOR NONLINEAR ELLIPTIC SYSTEM OF SECOND ORDER EQUATIONS IN MULTIPLY CONNECTED DOMAINS

In this article, we discuss that the discontinuous oblique derivative boundary value problem for nonlinear uniformly elliptic system of second order equations in multiply connected domains. We first propose the discontinuous oblique derivative problem and its new modified well-posedness. Next we give a priori estimates of solutions of the modified discontinuous boundary value problem for corresponding elliptic system of first order complex equations and verify its solvability by the above estimates of solutions and the Leray-Schauder theorem. Finally the solvability results of the original discontinuous oblique derivative problem can be derived. Here we mention the discontinuous boundary value problems possess many applications in mechanics and physics etc.

Viscous incompressible free-surface flow down an inclined perturbed plane

The plane stationary free boundary value problem for the Navier-Stokes equations is studied. This problem models the viscous fluid free-surface flow down a perturbed inclined plane. For sufficiently small data the solvability and uniqueness results are proved in Holder spaces. The asymptotic behavior of the solution is investigated.

A New Well-Posed Version of Riemann–Hilbert Problem for Elliptic Complex Equations of First Order in Multiply Connected Domains

In this article, we first propose the Riemann–Hilbert problem for uniformly elliptic complex equations of first order in multiply connected domains and its new well-posed-ness with continuous solutions. Then we obtain a priori estimates of solutions of the modified Riemann–Hilbert problem, and derive the solvability results of the modified and original Riemann–Hilbert problems. Finally we shall verify the equivalence of two kinds of well-posed-ness for the Riemann–Hilbert boundary value problem.

The discontinuous Riemann-Hilbert problem for elliptic complex equations of first order

In this article, we first transform the general uniformly elliptic systems of first order equations with certain conditions into the complex equations, and propose the discontinuous Riemann-Hilbert problem and its modified well-posedness for the complex equations. Then we give a priori estimates of solutions of the modified discontinuous Riemann-Hilbert problem for the complex equations and verify its solvability. Finally the solvability results of the original discontinuous Riemann-Hilbert boundary value problem can be derived. The discontinuous boundary value problem possesses many applications in mechanics and physics etc.

Approximate method for solving discontinuous boundary value problem of non-linear elliptic equations of second order

This article deals with the approximate solving method of general discontinuous boundary value problem for non-linear uniformly elliptic equations of second order in multiply connected domains. We first propose the formulation of the boundary value problem and corresponding modified well-posedness. Next we obtain a priori estimates of solutions for the modified problem. Finally, by the above estimates of solutions and the continuity method, the solvability results and error estimates of approximate solutions of the above-modified problem for the elliptic equations of second order can be obtained. The above problem possesses many applications in mechanics and physics etc.

Solvability of the Laplace equation in a solid with boundary reinforcement

We consider the solvability of a highly non-standard boundary value problem for Laplace’s equation arising when separate boundary elasticity is incorporated in the description of anti-plane deformations of a linearly elastic solid. The boundary elasticity consists of a distinct thin elastic coating bonded to part of the boundary of the solid. Interior and exterior mixed boundary problems are formulated and reduced to singular integro-differential equations from which we deduce the corresponding solvability results.

Existence of Solutions of New Generalized Mixed Vector Variational-Like Inequalities in Reflexive Banach Spaces

In this paper, we extend the concept of monotonicity for a vector set-valued mapping to semimonotonicity for a vector set-valued mapping. Then, we prove solvability results for a class of new generalized mixed vector variational-like inequalities by applying the Fan-KKM theorem and Nadler's result. On the other hand, we introduce the concepts of complete semicontinuity and strong semicontinuity for vector multivalued mappings. Moreover, by using the Brouwer fixed point theorem, we prove the solvability for the class of generalized vector variational-like inequalities without monotonicity assumption. Using this result, we obtain a theorem and corollary that improve and extend some known results.

THE REGIONS OF SOLVABILITY FOR SOME THREE POINT PROBLEM

The solvability results are established for the boundary value problem , where x + = max{x, 0}, x - = max{-x, 0}, h is a bounded nonlinearity. The existence results are based of the knowledge of the spectrum for the problem with h ≡ 0.

On L 2 Solvability of BVPs for Elliptic Systems

In this article we prove solvability results for L 2 boundary value problems of some elliptic systems Lu=0 on the upper half-space \({\mathbb{R}} ^{n+1}_{+}, n\ge1\), with transversally independent coefficients. We use the first order formalism introduced by Auscher-Axelsson-McIntosh and further developed with a better understanding of the classes of solutions in the subsequent work of Auscher-Axelsson. The interesting fact is that we prove only half of the Rellich boundary inequality without knowing the other half.

Poincaré Problem for Nonlinear Elliptic Equations of Second Order in Unbounded Domains

In [1], I. N. Vekua propose the Poincaré problem for some second order elliptic equations, but it can not be solved. In [2], the authors discussed the boundary value problem for nonlinear elliptic equations of second order in some bounded domains. In this article, the Poincaré boundary value problem for general nonlinear elliptic equations of second order in unbounded multiply connected domains have been completely investigated. We first provide the formulation of the above boundary value problem and corresponding modified well posed-ness. Next we obtain the representation theorem and a priori estimates of solutions for the modified problem. Finally by the above estimates of solutions and the Schauder fixed-point theorem, the solvability results of the above Poincaré problem for the nonlinear elliptic equations of second order can be obtained. The above problem possesses many applications in mechanics and physics and so on.

Left omega algebras and regular equations

Abstract Left omega algebras, where one of the usual star induction axioms is absent, are studied in the context of recursive regular equations. Abstract conditions for explicitly defining the omega operation are presented. They are used for developing abstract side conditions on Arden’s rule that are necessary for solving such equations. The definability and solvability results are refined to concrete models, to languages, traces and relations. It turns out, for instance, that the omega operation captures precisely the empty word property in regular languages and wellfoundedness in relational models. The approach also leads to simple new relative completeness results for left omega algebras, and for Salomaa’s axioms for regular expressions. Since automated theorem proving and counterexample search within the theorem proving environment Isabelle/HOL are instrumental for this investigation, it is also an exercise in formalised mathematics.

SCATTERING OF PLANE ELASTIC WAVES BY THREE-DIMENSIONAL DIFFRACTION GRATINGS

The reflection and transmission of a time-harmonic plane wave in an isotropic elastic medium by a three-dimensional diffraction grating is investigated. If the diffractive structure involves an impenetrable surface, we study the first, second, third and fourth kind boundary value problems for the Navier equation in an unbounded domain by the variational approach. A radiation condition based on the Rayleigh expansion of the quasi-periodic solutions is presented. Existence of solutions in Sobolev spaces is established if the grating profile is a two-dimensional Lipschitz surface, while uniqueness is proved only for small frequencies or for all frequencies excluding a discrete set. Similar solvability results are obtained for multilayered transmission gratings in the case of an incident pressure wave. Moreover, by a periodic Rellich identity, uniqueness of the solution to the first kind (Dirichlet) boundary value problem is established for all frequencies under the assumption that the impenetrable surface is given by the graph of a Lipschitz function.

Half eigenvalues and the Fučík spectrum of multi-point, boundary value problems

We consider the nonlinear boundary value problem consisting of the equation (1) − u ″ = f ( u ) + h , a.e. on ( − 1 , 1 ) , where h ∈ L 1 ( − 1 , 1 ) , together with the multi-point, Dirichlet-type boundary conditions (2) u ( ± 1 ) = ∑ i = 1 m ± α i ± u ( η i ± ) , where m ± ⩾ 1 are integers, α ± = ( α 1 ± , … , α m ± ) ∈ [ 0 , 1 ) m ± , η ± ∈ ( − 1 , 1 ) m ± , and we suppose that ∑ i = 1 m ± α i ± < 1 . We also suppose that f : R → R is continuous, and 0 < f ± ∞ : = lim s → ± ∞ f ( s ) s < ∞ (we assume that these limits exist). We allow f ∞ ≠ f − ∞ — such a nonlinearity f is said to be jumping . Related to (1) is the equation (3) − u ″ = λ ( a u + − b u − ) , on ( − 1 , 1 ) , where λ , a , b > 0 , and u ± ( x ) = max { ± u ( x ) , 0 } for x ∈ [ − 1 , 1 ] . The problem (2) , (3) is ‘positively-homogeneous’ and jumping. Regarding a , b as fixed, values of λ = λ ( a , b ) for which (2) , (3) has a non-trivial solution u will be called half-eigenvalues , while the corresponding solutions u will be called half-eigenfunctions . We show that a sequence of half-eigenvalues exists, the corresponding half-eigenfunctions having specified nodal properties, and we obtain certain spectral and degree theoretic properties of the set of half-eigenvalues. These properties lead to solvability and non-solvability results for the problem (1) , (2) . The set of half-eigenvalues is closely related to the ‘Fučík spectrum’ of the problem, which we briefly describe. Equivalent solvability and non-solvability results for (1) , (2) are obtained from either the half-eigenvalue or the Fučík spectrum approach.

The [formula omitted] Dirichlet problem for second-order, non-divergence form operators: solvability and perturbation results

We establish Dahlbergʼs perturbation theorem for non-divergence form operators L=A∇2. If L0 and L1 are two operators on a Lipschitz domain such that the Lp Dirichlet problem for the operator L0 is solvable for some p∈(1,∞) and the coefficients of the two operators are sufficiently close in the sense of Carleson measure, then the Lp Dirichlet problem for the operator L1 is solvable for the same p. This is a refinement of the A∞ version of this result proved by Rios (2003) in [10]. As a consequence we also improve a result from Dindoš et al. (2007) [4] for the Lp solvability of non-divergence form operators (Theorem 3.2) by substantially weakening the condition required on the coefficients of the operator. The improved condition is exactly the same one as is required for divergence form operators L=divA∇.

Uncontrollable dissipative dynamical systems

Abstract The classical definition of dissipativity for Linear Time Invariant differential (LTI) systems requires non-negativity of the net power absorbed over all system trajectories of compact support. This definition forces one to deal with only controllable systems. The more general definition uses the existence of an internal storage function as the required condition for dissipativity. While this definition allows considering uncontrollable systems also, the existential aspect of the definition raises questions about what systems admit a storage function. In this paper we provide conditions on an LTI system under which a storage function exists. Further, we also prove that this storage function is a state function , meaning the storage of energy requires no more variables than those that have to satisfy constraints for impulse-free concatenability of system trajectories. While such results have been proved for controllable systems or for a special class of uncontrollable systems, we use techniques from indefinite linear algebra to strengthen past results. The proofs of the main results above requires new results in Algebraic Riccati Equation (ARE) solvability in the context of uncontrollable state space systems: these ARE solvability results are also stated and proved in this paper.