Elliptic Problems Research Articles

Infinitely Many Solutions for a Class of Quasi-linear Elliptic Problem

Infinitely Many Solutions for a Class of Quasi-linear Elliptic Problem

Base of exponent representation matters -- more efficient reduction of discrete logarithm problem and elliptic curve discrete logarithm problem to the QUBO problem

This paper presents further improvements in the transformation of the Discrete Logarithm Problem (DLP) and Elliptic Curve Discrete Logarithm Problem (ECDLP) over prime fields to the Quadratic Unconstrained Binary Optimization (QUBO) problem. This is significant from a cryptanalysis standpoint, as QUBO problems may be solved using quantum annealers, and the fewer variables the resulting QUBO problem has, the less time is expected to obtain a solution.The main idea presented in the paper is allowing the representation of the exponent in different bases than the typically used base 2 (binary representation). It is shown that in such cases, the reduction of the discrete logarithm problem over the prime field \(\mathbb{F}_p\) to the QUBO problem may be obtained using approximately \(1.89n^2\) logical variables for \(n\) being the bitlength of prime \(p\), instead of the \(2n^2\) which was previously the best-known reduction method. The paper provides a practical example using the given method to solve the discrete logarithm problem over the prime field \(\mathbb{F}_{47}\). Similarly, for the elliptic curve discrete logarithm problem over the prime field \(\mathbb{F}_p\), allowing the representation of the exponent in different bases than typically used base two results in a lower number of required logical variables for \(n\) being the bitlength of prime \(p\), from \(3n^3\) to \(\frac{6n^3}{\log_{2}\left(\frac{3}{4}n\right)}\) logical variables, in the case of Edwards curves.

Regularization errors introduced by the one-fluid formulation in the solution of two-phase elliptic problems

This manuscript discusses the structure of the errors in the solution of elliptic problems introduced by the regularization of the fluid properties discontinuity in a small region of finite size. By using a multiscale approach, the problem of the error calculation is splitted into an outer problem, where the regularization region is replaced by a sharp interface, and an inner local one dimensional problem that eventually imposes effective jump conditions across the interface for the outer problem. Except in some particular cases, the use of regularization techniques introduces first order errors in the solution imposing an error flux jump that is proportional to the tangential Laplacian of the averaged solution and an error jump that is proportional to the normal flux, both multiplied by a prefactor that depends on the averaging rule used. In general, the optimal averaging procedure is shown to depend on the structure of the problem at hand. The errors introduced by standard arithmetic and harmonic averages are obtained for various analytical and numerical examples which are used to discuss the nature of the errors introduced and the importance of first order errors in the solution. The influence of the ratio between the regularization thickness and the grid size is also investigated in numerical implementations of the one-fluid model.

Convergence analysis of enhanced Phragmén–Lindelöf methods for solving elliptic Dirichlet problems

Convergence analysis of enhanced Phragmén–Lindelöf methods for solving elliptic Dirichlet problems

A Finite Difference Method for Two Dimensional Elliptic Interface Problems with Imperfect Contact

A Finite Difference Method for Two Dimensional Elliptic Interface Problems with Imperfect Contact

A Parallel Iterative Procedure for Weak Galerkin Methods for Second Order Elliptic Problems

A Parallel Iterative Procedure for Weak Galerkin Methods for Second Order Elliptic Problems

Efficient Authentication Steps for Vehicle Transmission in Ad-Hoc Networks

Vehicular ad-hoc networks (VANETs) are underactive development, thanks in part to recent advances in wireless communication and networking technologies. The most fundamental part of VANET is to enable message authentications between vehicles and roadside units. Message authentication using proxy vehicles has been proposed to reduce the computational overhead of roadside units significantly. In this message authentication scheme, a proxy vehicle that verifies multiple messages at the same time improves roadside units’ efficiency. In this paper first, we show that the only proxy-based authentication scheme (PBAS) presented for this goal by Liu et al. cannot guarantee message authenticity and also it is not resistant to impersonation and modification attacks and false acceptance of batched invalid signatures. Next, we propose a new identity-based message authentication scheme using proxy vehicles (ID-MAP). Then guarantee that it can satisfy the message authentication requirements, existential unforgeability of underlying signatures against adaptability chosen message, and identity attack is proved under the Elliptic Curve discrete logarithm problem (ECDLP) in the random oracle model. It should be highlighted that ID MAP not only is more efficient than PBAS since it is pairing-free and identity based and also it does not use map-to-point hash functions, also, it satisfies the security and privacy requirements of VANETs. Furthermore, analysis shows that the required time to verify 3000 messages in ID-MAP is reduced by 76% compared to that PBAS.

High multiplicity of positive solutions in a superlinear problem of Moore–Nehari type

In this paper we consider a superlinear one-dimensional elliptic boundary value problem that generalizes the one studied by Moore and Nehari in Moore and Nehari (1959). Specifically, we deal with piecewise-constant weight functions in front of the nonlinearity with an arbitrary number κ≥1 of vanishing regions. We study, from an analytic and numerical point of view, the number of positive solutions, depending on the value of a parameter λ and on κ.Our main results are twofold. On the one hand, we study analytically the behavior of the solutions, as λ↓−∞, in the regions where the weight vanishes. Our result leads us to conjecture the existence of 2κ+1−1 solutions for sufficiently negative λ. On the other hand, we support such a conjecture with the results of numerical simulations which also shed light on the structure of the global bifurcation diagrams in λ and the profiles of positive solutions.Finally, we give additional numerical results suggesting that high multiplicity also holds true for a much larger class of weights, even arbitrarily close to situations where there is uniqueness of positive solutions.

Existence of solutions to elliptic equations on compact Riemannian manifolds

The aim of this paper is to investigate the existence of weak solutions of a nonlinear elliptic problem with Dirichlet boundary value condition, in the framework of Sobolev spaces on compact Riemannian manifolds

An elliptic problem in dimension N with a varying drift term bounded in L N

An elliptic problem in dimension N with a varying drift term bounded in L N

Generalization of PINNs for elliptic interface problems

In this letter, we investigate the statistical limits of deep learning for learning solutions of elliptic interface problems from randomly generated data by employing Physics-informed Neural Networks (PINNs). We prove a lower bound and an upper bound for the generalization error of PINNs for solving elliptic interface equations with the Dirichlet boundary condition. In particular, the upper bound on the generalization error of PINNs is obtained by computing the fixed points of the local Rademacher complexity. We show that variations in volume across distinct regions exert influence on the requisite sample complexity. This unveils an underlying principle within the sampling strategy: the sample size of the data is contingent upon the Lebesgue measure of the domain, coupled with the smoothness and dimensionality characteristics of the interface problem. This letter sheds light on the network architecture design for solving interface problems using PINNs.

Combined effect of Poynting-Robertson (P-R) drag, oblateness and radiation on the triangular points in the elliptic restricted three-body problem

This study investigates the motion of a test particle around triangular equilibrium points in the elliptic restricted three-body problem (ER3BP) under the influence of the two oblate and radiating primaries having Poynting-Robertson (P-R) drag. It is observed that the position of triangular points of the problem is affected by oblateness, radiation pressure, eccentricity, semi-major axis and Poynting-Robertson (P-R) drag. The stability of these points is demonstrated analytically by the Routh-Hurwitz criterion. It is seen that they are unstable under the combined effect of involved parameters. The effect of these parameters on the position of triangular points is examined numerically using the binary systems, 61 Cygni and Archird. The results obtained by these binary systems can be used to broaden the scope of interest in astronomy, astrophysics, space science and celestial mechanics in general.

On a localization-in-frequency approach for a class of elliptic problems with singular boundary data

We consider a class of nonhomogeneous elliptic equations in the half-space with critical singular boundary potentials and nonlinear fractional derivative terms. The forcing terms are considered on the boundary and can be taken as singular measure. Employing a functional setting and approach based on localization-in-frequency and Littlewood–Paley decomposition, we obtain results on solvability, regularity, and symmetry of solutions.

Optimal Approximation of Unique Continuation

AbstractWe consider numerical approximations of ill-posed elliptic problems with conditional stability. The notion of optimal error estimates is defined including both convergence with respect to discretisation and perturbations in data. The rate of convergence is determined by the conditional stability of the underlying continuous problem and the polynomial order of the approximation space. A proof is given that no approximation can converge at a better rate than that given by the definition without increasing the sensitivity to perturbations, thus justifying the concept. A recently introduced class of primal-dual finite element methods with weakly consistent regularisation is recalled and the associated error estimates are shown to be optimal in the sense of this definition.

Variational Worn Stones

We introduce an evolution model à la Firey for a convex stone which tumbles on a beach and undertakes an erosion process depending on some variational energy, such as torsional rigidity, a principal Dirichlet Laplacian eigenvalue, or Newtonian capacity. Relying on the assumption of the existence of a solution to the corresponding parabolic flow, we prove that the stone tends to become asymptotically spherical. Indeed, we identify an ultimate shape of these flows with a smooth convex body whose ground state satisfies an additional boundary condition, and we prove symmetry results for the corresponding overdetermined elliptic problems. Moreover, we extend the analysis to arbitrary convex bodies: we introduce new notions of cone variational measures and we prove that, if such a measure is absolutely continuous with constant density, the underlying body is a ball.

Regularity results for solutions to elliptic obstacle problems in limit cases

We prove the Lewy–Stampacchia’s inequality for elliptic variational inequalities with obstacle involving Leray–Lions type operator whose simpler model case is given by the following u∈W01,N(Ω)↦-ΔNu-divB(x)|u|N-2u\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} u \\in W^{1,N}_0(\\Omega )\\mapsto -\\Delta _N u-\ ext {div}\\left( B (x) |u|^{N-2}u \\right) \\end{aligned}$$\\end{document}where Ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega $$\\end{document} is a smooth bounded domain of RN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^N$$\\end{document} with N⩾2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N\\geqslant 2$$\\end{document}, ΔNu\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta _N u$$\\end{document} denotes the classical N–Laplacian operator and the coefficient B:Ω→RN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B:\\Omega \\rightarrow \\mathbb {R}^N$$\\end{document} belongs to a suitable Lorentz–Zygmund space. For this kind of obstacle problems, we also provide regularity results and amongst them we give sufficient conditions to get boundedness of solutions.

Multiplicity of normalized solutions to biharmonic Schrödinger equation with mixed nonlinearities

This paper is concerned with the existence of multiple normalized solutions to the elliptic problems { Δ 2 u = λu + h ( ϵx ) | u | q − 2 u + | u | p − 2 u in R N , ∫ R N u 2 d x = c , where c, 0 $ ]]> ϵ > 0 , N ≥ 5 , 2 < q < 2 + 8 N < p ≤ 4 ∗ := 2 N N − 4 , λ ∈ R is a Lagrangian multiplier and h : R N → [ 0 , ∞ ) is a continuous function. We find some 0 $ ]]> c 0 > 0 and present a class of reasonable assumptions on h to guarantee that the numbers of normalized solutions are at least the numbers of global maximus points of h when ε is small enough and 0 < c ≤ c 0 .

Existence of Solutions for p(x)-Laplacian Elliptic BVPs on a Variable Sobolev Space Via Fixed Point Theorems

In this paper, we prove some existence theorems for elliptic boundary value problems within the p(x)-Laplacian on a variable Sobolev space. For this purpose, the main problem is transformed into a fixed point problem and then fixed point arguments such as Schaefer’s and Schauder’s theorems are used. Our approach involves fewer stringent assumptions on the nonlinearity function than the prior findings. An interesting example is presented to examine the validity of the theoretical findings.

Weak Galerkin methods for elliptic interface problems on curved polygonal partitions

This paper presents a new weak Galerkin (WG) method for elliptic interface problems on general curved polygonal partitions. The method’s key innovation lies in its ability to transform the complex interface jump condition into a more manageable Dirichlet boundary condition, simplifying the theoretical analysis significantly. The numerical scheme is designed by using locally constructed weak gradient on the curved polygonal partitions. We establish error estimates of optimal order for the numerical approximation in both discrete H1 and L2 norms. Additionally, we present various numerical results that serve to illustrate the robust numerical performance of the proposed WG interface method.

Positive solution for a singular and nonlocal problem of the N-Kirchhoff type

We investigate the existence of a positive solution for a singular elliptic problem of the N-Kirchhoff type involving a nonlocal operator. The nonlinearity considered in the equation consists of two terms: one has a subcritical, critical or supercritical exponential growth governed by the Trudinger–Moser inequality, and the other one presents a singularity at the origin. We also show a result of nonexistence of solutions in dimension 2. Finally, we study the asymptotic behavior of the solutions with respect to the parameter.