Prove Existence Results Research Articles

Existence of solutions for some noncoercive elliptic problems involving derivatives of nonlinear terms

+u= f in Ω,u= 0 on ∂Ω,have been studied in [6], [3], [7] and [4] proving existence results. Inthis note we prove that the same results hold even in the presence of aterm in divergence form, that is, for problem (7).We are going to prove the following theorems, according to thegrowth of Φ.Theorem 1. Assume (2), (4), (8) and (9) and that there exists apositiveCsuchthat(12) |Φ(t)| ≤ C|t|

Optimal control of some simplified models of tumour growth

In this article we study some optimal control problems for a system of PDEs that describes the growth of a spherical tumour influenced by the mechanical action of chemicals. We make two different choices of the cost functional. We prove existence results, deduce the associated optimality systems and present iterative algorithms for the computation of the solution.

Monotone convergence theorems for Henstock–Kurzweil integrable functions and applications

In this paper we prove monotone convergence theorems for Henstock–Kurzweil integrable functions from a compact real interval to an ordered Banach space. These theorems are then applied to prove existence results for solutions of a discontinuous functional integral equation containing Henstock–Kurzweil integrable functions.

Design of a Composite Membrane with Patches

This paper is concerned with minimization and maximization problems of eigenvalues. The principal eigenvalue of a differential operator is minimized or maximized over a set which is formed by intersecting a rearrangement class with an affine subspace of finite co-dimension. A solution represents an optimal design of a 2-dimensional composite membrane Ω, fixed at the boundary, built out of two different materials, where certain prescribed regions (patches) in Ω are occupied by both materials. We prove existence results, and present some features of optimal solutions. The special case of one patch is treated in detail.

Periodic solutions of the N-body problem with Lennard-Jones-type potentials

In this article we consider a system of N identical particles interacting through a potential of Lennard-Jones (LJ) type. We consider subsets of loop space that satisfy certain basic conditions including the notion of ‘tiedness’ introduced by Gordon in [W. Gordon, Conservative dynamical systems involving strong force, Trans. Amer. Math. Soc. 204 (1975), pp. 113–135]. We then show, by means of critical point theory, that this system admits periodic solutions in every homotopy class of these subsets of loop space. More precisely we show that every homotopy class contains at least two periodic solutions for sufficiently large periods. One of these solutions is a local minimum and the other is a mountain-pass critical point of the action functional. We also prove that, given a homotopy class of one of these subsets, there do not exist any periodic solutions in it for sufficiently small periods. Our results have wide applicability. For example, one can consider the space of choreographies and prove existence results for choreographic solutions. Our existence proof relies upon an assumption that global minimizers of standard strong force potentials on suitable spaces are non-degenerate up to some symmetries.

Radial solutions for some nonlinear problems involving mean curvature operators in Euclidean and Minkowski spaces

In this paper, using the Schauder fixed point theorem, we prove existence results of radial solutions for Dirichlet problems in the unit ball and in an annular domain, associated to mean curvature operators in Euclidean and Minkowski spaces.

On Abstract Variational Inequalities in Viscoplasticity with Frictional Contact

In this paper, we study quasistatic abstract variational inequalities with time-dependent constraints. We prove existence results and present an approximation method valid for nonsmooth constraints. Then, we apply our results to the approximation of the quasistatic evolution of an elastic body in bilateral contact with a rigid foundation. The contact involves viscous friction of the Tresca or Coulomb type. We prove existence results for approximate problems and give a full asymptotic analysis, proving strong or weak convergence results. Our work is motivated by the numerical study in the paper [Delost, M.: Quasistatic Problem with Frictional Contact: Comparison between Numerical Methods and Asymptotic Analysis Related to Semi Discrete and Fully Discrete Approximations. University of Nice, Nice (2007, to appear)] and explains the choice of the approximation made in it.

On the isoperimetric problem in Euclidean space with density

We study the isoperimetric problem for Euclidean space endowed with a continuous density. In dimension one, we characterize isoperimetric regions for a unimodal density. In higher dimensions, we prove existence results and we derive stability conditions, which lead to the conjecture that for a radial log-convex density, balls about the origin are isoperimetric regions. Finally, we prove this conjecture and the uniqueness of minimizers for the density exp\((|x|^2)\) by using symmetrization techniques.

Solutions of vectorial Hamilton–Jacobi equations and minimizers of nonquasiconvex functionals

We provide a unified approach to prove existence results for the Dirichlet problem for Hamilton–Jacobi equations and for the minimum problem for nonquasiconvex functionals of the Calculus of Variations with affine boundary data. The idea relies on the definition of integro-extremal solutions introduced in the study of nonconvex scalar variational problem.

Non-periodic finite-element formulation of orbital-free density functional theory

We propose an approach to perform orbital-free density functional theory calculations in a non-periodic setting using the finite-element method. We consider this a step towards constructing a seamless multi-scale approach for studying defects like vacancies, dislocations and cracks that require quantum mechanical resolution at the core and are sensitive to long range continuum stresses. In this paper, we describe a local real-space variational formulation for orbital-free density functional theory, including the electrostatic terms and prove existence results. We prove the convergence of the finite-element approximation including numerical quadratures for our variational formulation. Finally, we demonstrate our method using examples.

Symmetry of solutions to optimization problems related to partial differential equations

We investigate maxima and minima of some functionals associated with solutions to Dirichlet problems for elliptic equations. We prove existence results and, under suitable restrictions on the data, we show that any maximal configuration satisfies a special system of two equations. Next, we use the moving-plane method to find symmetry results for solutions of a system. We apply these results in our discussion of symmetry for the maximal configurations of the previous problem.

Existence results for nonlocal and nonsmooth hemivariational inequalities

We consider an elliptic hemivariational inequality with nonlocal nonlinearities. Assuming only certain growth conditions on the data, we are able to prove existence results for the problem under consideration. In particular, no continuity assumptions are imposed on the nonlocal term. The proofs rely on a combined use of recent results due to the authors on hemivariational inequalities and operator equations in partially ordered sets.

Diffusive approximations for a class of nonlinear parabolic equations

We study a class of discrete velocity type approximations to nonlinear parabolic equations with source. After proving existence results and estimates on the solution to the relaxation system, we pass into the limit towards a weak solution, which is the unique entropy solution if the coefficients of the parabolic equation are constant.

A tour of the theory of absolutely minimizing functions

These notes are intended to be a rather complete and self-contained exposition of the theory of absolutely minimizing Lipschitz extensions, presented in detail and in a form accessible to readers without any prior knowledge of the subject. In particular, we improve known results regarding existence via arguments that are simpler than those that can be found in the literature. We present a proof of the main known uniqueness result which is largely self-contained and does not rely on the theory of viscosity solutions. A unifying idea in our approach is the use of cone functions. This elementary geometric device renders the theory versatile and transparent. A number of tools and issues routinely encountered in the theory of elliptic partial differential equations are illustrated here in an especially clean manner, free from burdensome technicalities - indeed, usually free from partial differential equations themselves. These include a priori continuity estimates, the Harnack inequality, Perron’s method for proving existence results, uniqueness and regularity questions, and some basic tools of viscosity solution theory. We believe that our presentation provides a unified summary of the existing theory as well as new results of interest to experts and researchers and, at the same time, a source which can be used for introducing students to some significant analytical tools.

P-Laplacian Systems on Resonance

We assume certain (adapted to the case of a system) Landesman–Lazer-type conditions, in order to prove existence results for a p-Laplacian system on resonance.

A Characterization of Approximate Solutions of Multiobjective Stochastic Optimal Control Problems

We introduce several concepts of approximate solutions of multiobjective optimization problems, prove existence results and an k -minimum principle for multiobjective stochastic optimal control problems.

Optimal design via variational principles: the one-dimensional case

By reformulating a typical optimal design problem in a variational format, we are able to prove existence results in a rather general framework allowing non-linear state equations and costs functional depending on derivatives of states. In this work, we restrict attention to the one-dimensional situation.

Variational methods for boundary value problems

The variational formulation of boundary value problems is valuable in providing remarkably easy computational algorithms as well as an alternative framework with which to prove existence results. Boundary conditions impose constraints which can be annoying from a computational point of view. The question is then posed: what is the most general boundary value problem which can be posed in variational form with the boundary conditions appearing naturally? Special cases of two-point problems in one-dimension and some higher dimensional problems are addressed. There is a deep connection with self-adjointness for the linear case. Further cases under which a Lagrangian may or may not exist are explained.

Existence and multiplicity results for quasilinear elliptic differential systems

We use a nonsmooth critical point theory to prove existence results for a variational system of quasilinear elliptic equations in both the sublinear and superlinear cases. We extend a technique of Bartsch to obtain multiplicity results when the system is invariant under the action of a compact Lie group. The problem is rather different from its scalar version, because a suitable condition on the coefficients of the system seems to be necessary in order to prove the convergence of the Palais-Smale sequences. Such condition is in some sense a restriction to the "distance" between the quasilinear operator and a semilinear one.

Holomorphic cylinders with lagrangian boundaries and periodic orbits of hamiltonian systems

This paper outlines how to use holomorphic cylinders with Lagrangian boundaries to prove existence results for periodic orbits of Hamiltonian systems. We describe the case of the cotangent bundle of the Klein bottle, where these results lead to a new obstruction to the existence of Lagrangian embeddings of the Klein bottle into C 2.