Nonlinear Elliptic Boundary Value Problems Research Articles

Multiple Positive Solutions For a Class of Nonlinear Elliptic Equations

Abstract This paper deals with a class of nonlinear elliptic Dirichlet boundary value problems where the combined effects of a sublinear and a superlinear term allow us to establish some existence and multiplicity results.

Maximum principles for a class of nonlinear second-order elliptic boundary value problems in divergence form

For a class of nonlinear elliptic boundary value problems in divergence form, we construct some general elliptic inequalities for appropriate combinations of and , where are the solutions of our problems. From these inequalities, we derive, using Hopf's maximum principles, some maximum principles for the appropriate combinations of and , and we list a few examples of problems to which these maximum principles may be applied.

Existence of solutions for a nonlinear elliptic Dirichlet boundary value problem with an inverse square potential

Via the linking theorem, the existence of nontrivial solutions for a nonlinear elliptic Dirichlet boundary value problem with an inverse square potential is proved.

Global Bifurcation Theory of Deep-Water Waves with Vorticity

The classical deep-water wave problem is to find a periodic traveling wave with a free surface of infinite depth. The main result is the construction of a global connected set of rotational solutions for a general class of vorticities. Each nontrivial solution on the continuum has a wave profile symmetric around the crests and monotone between crest and trough. The problem is formulated as a nonlinear elliptic boundary value problem in an unbounded domain with a parameter. The analysis is based on generalized degree theory and the global theory of bifurcation. The unboundedness of the domain renders consideration of approximate problems with stronger compactness properties.

Homogenization of a Nonlinear Elliptic Boundary Value Problem Modeling Galvanic Currents

We study a nonlinear elliptic boundary value problem arising from electrochemistry in the study of heterogeneous electrode surfaces. The boundary condition is of exponential type (Butler--Volmer) and has a periodic structure. We find a limiting or effective problem as the period approaches zero, along with a first order correction. We establish convergence estimates and provide numerical experiments.

A new fourth order discretization for singularly perturbed two dimensional non-linear elliptic boundary value problems

In this article, we derive a new difference method of O( h 4), so called, arithmetic average discretization for the solution of two dimensional non-linear singularly perturbed elliptic partial differential equation of the form ε( u xx + u yy ) = f( x, y, u, u x , u y ), 0 < x, y < 1, subject to appropriate Dirichlet boundary conditions where ε > 0 is a small parameter .We also derive new methods of O( h 4) for the estimates of (∂ u/∂ n), which are quite often of interest in many physical problems. In all cases, we require only 9-grid points and a single computational cell. The main advantage of the proposed methods is that the methods are directly applicable to singular problems. We do not require any special technique or modification to solve singular problems. Numerical results are provided to demonstrate the usefulness of the methods discussed.

Degenerate nonlinear higher-order elliptic problems in domains with fine-grained boundary

We study the convergence of the solutions of the Dirichlet problem associated to a degenerate nonlinear higher-order elliptic equations in divergence form in variables domains, to a limit solution of the same type problem in a fixed domain, following the methods of the asymptotic expansion developed by Skrypnik [Methods for Analysis of Nonlinear Elliptic Boundary Value Problems, AMS, Providence, RI, 1994] modified to weighted higher-order case.

Maximum Principles for a Class of Nonlinear Elliptic Boundary Value Problems

The Hopf’s maximum principles are utilized to obtain maximum principles for functions defined on solutions of nonlinear elliptic equations in divergence form (g(u)u, i ), i +f(x, u, q) = 0 (q = |∇u|2), subject to Dirichlet, Neumann or Robin boundary conditions. The principles derived may be used to deduce bounds on the gradient q.

On a nonlocal model of image segmentation

We understand an image as binary grey ‘alloy’ of a black and a white component and use a nonlocal phase separation model to describe image segmentation. The model consists in a degenerate nonlinear parabolic equation with a nonlocal drift term additionally to the familiar Perona-Malik model. We formulate conditions for the model parameters to guarantee global existence of a unique solution that tends exponentially in time to a unique steady state. This steady state is solution of a nonlocal nonlinear elliptic boundary value problem and allows a variational characterization. Numerical examples demonstrate the properties of the model.

A robust linearisation scheme for a nonlinear elliptic boundary value problem: Error estimates

AbstractWe consider a nonlinear second-order elliptic boundary value problem in a bounded domain Ω ⊂ RN with mixed boundary conditions. The solution is found via linearisation. We design a robust and efficient approximation scheme. Error estimates for the linearisation algorithm are derived in L2(Ω), H1(Ω) and L∞(Ω) spaces under the minimal regularity assumptions of the exact solution.

Existence theorems of multiple critical points and applications

In this paper, using the method of invariant sets of descending flow, the multiplicity of critical points of a functional is discussed. A main theorem, which is called chain of rings theorem, is obtained. The theoretical results are applied to nonlinear elliptic boundary value problems and at least seven or nine solutions are obtained.

Quasilinearization Method for Nonlinear Elliptic Boundary-Value Problems

The quasilinearization method is developed for strong solutions of semilinear and nonlinear elliptic boundary-value problems. We obtain two monotone, Lp-convergent sequences of approximate solutions. The order of convergence is two. The tools are some results on the abstract quasilinearization method and from weakly–near operators theory.

On nonlocal existence results for elliptic equations with convex–concave nonlinearities

In this paper we study the family of nonlinear elliptic Dirichlet boundary value problems with p-Laplacian and with concave–convex nonlinearity which depend on real parameter λ . We introduce nonlocal intervals ( λ i , λ i + 1 ) such that the characteristic points λ i , λ i + 1 (a priori bifurcation values) expressed in terms of exact variational principles. In these intervals, new results on the existence of positive solutions, multiple positive solutions and existence of multiple disjoint sets with infinitely many solutions are proved.

MULTIPLICITY OF POSITIVE SOLUTIONS UNDER NONLINEAR BOUNDARY CONDITIONS FOR DIFFUSIVE LOGISTIC EQUATIONS

AbstractIn this paper we consider the existence and multiplicity of positive solutions of a nonlinear elliptic boundary-value problem with nonlinear boundary conditions which arises in population dynamics. While bifurcation problems from lines of trivial solutions are studied, the existence of bifurcation positive solutions from infinity is discussed. The former will be caught by the reduction to a bifurcation equation following the Lyapunov and Schmidt procedure. The latter will be based on a variational argument depending on the corresponding constrained minimization problem.AMS 2000 Mathematics subject classification: Primary 35J65; 35J20; 35P30; 35B32; 92D25

Magnetic Field Confinement in the Solar Corona. I. Force‐free Magnetic Fields

Axisymmetric force-free magnetic fields external to a unit sphere are studied as solutions to boundary value problems in an unbounded domain posed by the equilibrium equations. It is well known from virial considerations that stringent global constraints apply for a force-free field to be confined in equilibrium against expansion into the unbounded space. This property as a basic mechanism for solar coronal mass ejections is explored by examining several sequences of axisymmetric force-free fields of an increasing total azimuthal flux with a power-law distribution over the poloidal field. Particular attention is paid to the formation of an azimuthal rope of twisted magnetic field embedded within the global field, and to the energy storage properties associated with such a structure. These sequences of solutions demonstrate (1) the formation of self-similar regions in the far global field where details of the inner boundary conditions are mathematically irrelevant, and (2) the possibility that there is a maximum to the amount of azimuthal magnetic flux confined by a poloidal field of a fixed flux anchored rigidly to the inner boundary. The nonlinear elliptic boundary value problems we treat are mathematically interesting and challenging, requiring a specially designed solver, which is described in the Appendix.

Nontrivial solutions of a semilinear elliptic problem via variational methods

Using variational methods, we investigate the existence of nontrivial solutions of a nonlinear elliptic boundary value problem at resonance under generalised Ahmad-Lazer-Paul conditions. Some new results are obtained and some results in the literature are improved.

Positive solutions of nonlinear elliptic singular boundary value problems in a ball

This paper deals with existence of positive solutions of non-linear elliptic singular boundary value problems in a ball. It is shown that results of Grandall et al. [1] and [2] follow as special cases of our results proved in this article.

SIMULATIONS OF THE POISSON TYPE PARTIAL DIFFERENTIAL EQUATIONS WHERE THE NODES ARE BOUND BY THE KNIGHT'S MOVES

International Journal of Computational Engineering ScienceVol. 05, No. 01, pp. 25-39 (2004) No AccessSIMULATIONS OF THE POISSON TYPE PARTIAL DIFFERENTIAL EQUATIONS WHERE THE NODES ARE BOUND BY THE KNIGHT'S MOVESRIO HIROWATI SHARIFFUDIN and ITHNIN ABDUL JALILRIO HIROWATI SHARIFFUDINInstitute of Mathematical Sciences, Faculty of Science, University of Malaya, 50603 Kuala Lumpur, Malaysia Search for more papers by this author and ITHNIN ABDUL JALILDepartment of Physics, Faculty of Science, University of Malaya, 50603 Kuala Lumpur, Malaysia Search for more papers by this author https://doi.org/10.1142/S1465876304002241Cited by:0 PreviousNext AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail AbstractIt is very much desired to have subsystems resulting from finite difference modeling of partial differential equations where the matrices are block diagonal as the work required to solve such subsystems is reduced greatly. The simulation nodes in the red/black order and bound by the usual five-point finite difference molecule have their coefficient matrix with such pattern. In this paper, we attempt to have such subsystems for the coefficient matrix for the nodes bound by the nine-point molecule. This can be accomplished by the knight's move. A comparison of the performances between this model and the usual nine-point model is given. We note that the coefficient matrix favors the acceleration of the basic iterative methods.Keywords:Hamiltonian Circuited SimulationsPoisson's EquationsConjugate Gradient SimulationsKnight's Move Model References F. Harary , Graph Theory ( Addison Wesley , New York , 1971 ) . Google ScholarM. R. Hestenes and E. Steifel, Journal of Research of the Nat. Bur. of Stds. 49, 409 (1952). Crossref, Google ScholarD. F. Shanno, Math. of Operations Research 3919780, 244 (1977). Google Scholar E. M. L. Beale , A Derivation of Conjugate Gradients in Numerical Methods for Monlinear Optimization , ed. F. A. Lootsman ( Academic Press , London , 1972 ) . Google ScholarM. J. D. Powell, Math. Prog. 12, 241 (1977). Crossref, Google ScholarE. Polak and G. Riebiere, Revue Francaise Inform. Rech. Operation 16-R1, 35 (1965). Google ScholarA. G. Buckley, Math. Prog. 15, 200 (1978). Crossref, Google ScholarD. Le, Math. Prog. 32, 41 (1985). Crossref, Google Scholar J. W. Daniel , The conjugate Gradient Method for Linear and Nonlinear Operator Equations doctoral thesis: Stanford University . Google ScholarJ. K. Reid, On the method of Conjugate Gradients for The Solution Of Large Sparse Systems of Linear Equations, Proc. Conf. Large Sets of Linear Equations pp. 231–254. Google Scholar R. Bartels and J. W. Daniel , A Conjugate Approach to Nonlinear Elliptic Boundary Value Problem in Irregular Regions , Proc. Conf. Numerical Solution of Differential Equations ( Dundee , 1974 ) . Google Scholar Axelson, O., 1974. On Preconditioning and Convergence Acceleration in Sparse Matrix Problems. CERN 74-10: European Organization for Nuclear research, Data Handling Division . Google Scholar O'Leary D.P.,1975. Hybrid Conjugate Gradient Algorithms, doctoral thesis: Stanford University Pato Alto, California . Google Scholar R. Chandra , Conjugate Gradient Methods for Partial Differential Equations , Research Report 129 ( Connecticut, Department of Comp. Sc., University of New Haven , 1978 ) . Google ScholarP. Concus and G. H. Golub, SIAM J. Num. Anal. 10, 1103 (1973). Crossref, Google Scholar P. Concus and G. H. Golub , A generalized Conjugate Gradient Method For Nonsymmetric Systems of Linear Equations ( Comp. Sc. Dept., Stanford University , 1976 ) . Crossref, Google ScholarD. J. Evans, J.I.M.A. 4, 295 (1968). Google ScholarJ. A. Meijerink and H. A. van der Vorst, Math. Comp. 31, 148 (1977). Google ScholarR. Hirowati Shariffudin and A. R. Abdullah, International Journal of Computer Mathematics 76, (2001). Google ScholarR. Hirowati Shariffudin and A. R. Abdullah, Applied Mathematics Letters 14, 423 (2001). Google Scholar FiguresReferencesRelatedDetails Recommended Vol. 05, No. 01 Metrics History KeywordsHamiltonian Circuited SimulationsPoisson's EquationsConjugate Gradient SimulationsKnight's Move ModelPDF download

Solutions of some nonlinear elliptic problems with perturbation terms of arbitrary growth

In thispaper, existence and multiplicity of nontrivial solutions areobtained for some nonlinear elliptic boundary value problems withperturbation terms of arbitrary growth. Results are obtained viavariational arguments.

Solvability of nonlinear elliptic boundary value problem via its associated linear problem

We investigate the existence of solutions of a nonlinear elliptic boundary value problem at resonance. Under the condition that the associated linear boundary value problem has no sign-changing solution or nontrivial solution and some other additional conditions, we prove the existence of solutions or nontrivial (even multiple) solutions to the nonlinear problem. Thus the existence of solutions can be obtained when the nonlinearity may cross any finite number of eigenvalues of the linear problem.