Minimization Argument Research Articles

The simplest minimal subtraction for massive scalar field theory

The simplest minimal subtraction method for massive λϕ4 scalar field theory is presented. We utilize the one-particle irreducible vertex parts framework to deal only with the primitive divergent ones that can be renormalized multiplicatively. We give a unified description for spacetime metric tensor with either Minkowski or Euclidean signature. The partial-p operation in the remaining diagrams of the two-point vertex part eliminates its overlapping divergences. We show how the parametric dissociation transform effectively removes the external momentum dependence of the coefficient of the squared bare mass after performing the partial-p operation in the two-point vertex part diagrams. The resemblance of this method with a minimal subtraction scheme in the massless theory is pointed out. We derive the Callan-Symanzik equations using minimal subtraction arguments and discuss the scaling limit in the ultraviolet region. We apply the method to determine critical exponents for an O(N) internal symmetry at least up to two-loop order with a flat Euclidean metric and find perfect agreement with all previous results in the literature.

Sign-changing solutions for a class of fractional Choquard equation with the Sobolev critical exponent in [formula omitted]

In this paper, we study the following fractional Choquard equation with critical exponent(−Δ)σu+u=(Iα⁎|u|p)|u|p−2u+|u|2σ⁎−2uinR3, where σ∈(34,1), α∈(2σ,3) and α+2σ3−2σ<p<2α,σ⁎:=3+α3−2σ, 2α,σ⁎ is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality, 2σ⁎:=63−2σ is the fractional Sobolev critical exponent and the operator (−Δ)σ stands for the fractional Laplacian of order σ. Based on the above assumptions, we establish the existence of positive ground state solutions, and if p∈(α+2σ3−2σ,2α3−2σ), we also have the corresponding regular property. Subsequently, by introducing some other additional hypotheses on σ, α, p and with the help of quantitative deformation lemma, we employ constrained minimization arguments on the sign-changing Nehari manifold to obtain the existence of ground state sign-changing solutions.

Fractional Sobolev space: Study of Kirchhoff-Schrödinger systems with singular nonlinearity

This study extensively investigates a specific category of Kirchhoff-Schrödinger systems in fractional Sobolev space with Dirichlet boundary conditions. The main focus is on exploring the existence and multiplicity of non-negative solutions. The non-linearity of the problem generally exhibits singularity. By employing minimization arguments involving the Nehari manifold and a variational approach, we establish the existence and multiplicity of positive solutions for our problem with respect to the parameters \(\eta\) and \(\zeta\) in suitable fractional Sobolev spaces. Our key findings are novel and contribute significantly to the literature on coupled systems of Kirchhoff-Schrödinger system with Dirichlet boundary conditions.

On a class of generalized capillarity system involving fractional ψ#x02010;Hilfer derivative with p(·)‐Laplacian operator

This research delves into a comprehensive investigation of a class of ‐Hilfer generalized fractional nonlinear differential system originated from a capillarity phenomena with Dirichlet boundary conditions, focusing on issues of existence and multiplicity of nonnegative solutions. The nonlinearity of the problem, in general, does not satisfy the Ambrosetti–Rabinowitz type condition. We use minimization arguments of Nehari manifold together with variational approach to show the existence and multiplicity of positive solutions of our problem with respect to the parameter in appropriate fractional ‐Hilfer spaces. Our main result is novel, and its investigation will enhance the scope of the literature on coupled systems of fractional ‐Hilfer generalized capillary phenomena.

Defense of Freedom of Abortion: An Analysis Based on the Case of Dobbs vs. Jackson Womens Health Organization

This paper delves into the great complexity of the right to abortion. The paper defends the freedom of abortion through perspectives of womens health, the premise that fetuses are human beings, and rights-based theory with reference to arguments by prominent philosophers. The paper discusses the famous thought experiment of the violinist analogy, claim rights, and liberty rights by Judith Jarvis Thomson to argue why abortion should be morally permissible. Moreover, the paper dissects objections to these arguments, revealing their inherent limitations by stating that traveling to other states for abortion is unjust. The paper also uses the minimal decent Samaritan argument to address possible objections. The paper strongly argues that the right to abortion is a legal and moral imperative to fulfill. The paper aims to justify the right to abortion and that people should recognize the current threat to the right to abortion with the case of Dobbes vs. Jackson Women's Health Organization.

Normalized ground state of a mixed dispersion nonlinear Schrodinger equation with combined power-type nonlinearities

We study the existence of normalized ground state solutions to a mixed dispersion fourth-order nonlinear Schrodinger equation with combined power-type nonlinearities. By analyzing the subadditivity of the ground state energy with respect to the prescribed mass, we employ a constrained minimization method to establish the existence of ground state that corresponds to a local minimum of the associated functional. Under certain conditions, by studying the monotonicity of ground state energy as the mass varies, we apply the constrained minimization arguments on the Nehari-Pohozaev manifold to prove the existence of normalized ground state solutions.&#x0D; For more information see https://ejde.math.txstate.edu/Volumes/2024/29/abstr.html

Sign-changing solution for a generalized Kirchhoff problem in the fractional Orlicz-Sobolev space with nonsmooth nonlinearity

In this paper, we study a nonlocal generalized Kirchhoff problem driven by the fractional Orlicz g-Laplace operator and involving a nonsmooth nonlinearity. Although this problem has a double nonlocality, we prove the existence of a least-energy nodal (or sign-changing) solution. Note that the tools used are minimization argument and quantitative deformation lemma.

European payment order as a means of cross-border debt collection in the European Union

This article examines the European Payment Order (EPO). The importance of this instrument as a component of the principle of the EU internal market is emphasized. It has been argued that as in any trade or other commercial relationship, there is a risk of arrears, which creates problems for creditors. Problems can be further complicated by the fact that the debtor is located outside the state in which the creditor is located, and therefore it is a different state, with different laws, approaches to dispute resolution, etc. The main stages of the EPO procedure are considered. It is emphasized that the procedure looks simple in general, but the main drawback is immediately noticeable - for a dispute it is enough to simply submit an application without even providing sufficient grounds for the dispute. Thus, it allows the debtor to delay the case rather easily, transfer it to the national court (although it provides that the debtor must react within 30 days, which may not always happen in practice, especially if the debtor does not use the services of a qualified legal advisor). Also important is the fact that the EPO procedure is used in all EU member states, except for Denmark. This means that its application is truly universal and can be applied to almost any debtor in the European Union. It is indicated that EU Regulation No. 1896/2006 of December 12, 2006 applies to civil and commercial matters in cross-border cases, and also provides for exceptions when it cannot be applied. It was concluded that the main drawback of the procedure is that it would be more expedient if the debtor was obliged to provide at least minimal arguments, reasons for filing such an objection. This would allow to avoid groundless objections aimed solely at prolonging the resolution of the debt issue. In addition, it would force debtors to take the objection procedure more seriously and encourage them to seek help from qualified legal counsel. Since the services of such a legal advisor can be quite expensive, the debtor could weigh the expediency of such - for example, whether it makes sense to spend money and resources on such services, if the grounds for the claim are truly unconditional and the chances of a challenge already in court are zero. Instead, the debtor can very easily deny the application without any additional reasons. It was concluded that, at the same time, after Ukraine becomes a member of the European Union, it will be useful to join this procedure and it will be necessary to develop the main national features of the EPO application.

Generalized Quasilinear Elliptic Equations in {mathbb {R}}^N

In this paper, we aim at establishing a new existence result for the quasilinear elliptic equation -div(a(x,u,∇u))+At(x,u,∇u)+V(x)|u|p-2u=g(x,u)inRN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} - \ extrm{div} (a(x,u,\ abla u)) + A_t(x,u,\ abla u) + V(x) {\\vert u \\vert }^{p-2} u= g(x,u) \\quad \\quad \\hbox { in }{{\\mathbb {R}}}^{N} \\end{aligned}$$\\end{document}with p>1, Nge 2 and V:{mathbb {R}}^Nrightarrow {mathbb {R}} suitable measurable positive function. Here, we suppose A: {mathbb {R}}^N times {mathbb {R}}times {mathbb {R}}^N rightarrow {mathbb {R}} is a given {C}^{1}-Carathéodory function which grows as |xi |^p, with A_t(x,t,xi ) = frac{partial A}{partial t}(x,t,xi ), a(x,t,xi ) = nabla _xi A(x,t,xi ), V:{mathbb {R}}^Nrightarrow {mathbb {R}} is a suitable measurable function and g:{mathbb {R}}^N times {mathbb {R}}rightarrow {mathbb {R}} is a given Carathéodory function which grows as |xi |^q with 1<q<p. Since the coefficient of the principal part depends on the solution itself, under suitable assumptions on A(x,t,xi ), V(x) and g(x, t), we study the interaction of two different norms in a suitable Banach space with the aim of obtaining a good variational approach. Thus, a minimization argument on bounded sets can be used to state the existence of a nontrivial weak bounded solution on an arbitrary bounded domain. Then, one nontrivial bounded solution of the given equation can be found by passing to the limit on a sequence of solutions on bounded domains. Finally, under slightly stronger hypotheses, we can able to find a positive solution of the problem.

Positive and nodal ground state solutions for a critical Schrödinger-Poisson system with indefinite potentials

We consider the Schrödinger-Poisson system{−Δu+V(x)u+K(x)ϕu=a(x)|u|p−2u+|u|4u,x∈R3,−Δϕ=K(x)u2,x∈R3, where 4<p<6 and the potentials V, a are allowed to change their signs. Under some reasonable assumptions on V, K and a, we apply the constraint minimization argument to establish the existence of positive ground state solutions and ground state nodal solutions.

The global well‐posedness for nonlinear Schrödinger equation with the delta potential in the energy space

In this paper, we concern with the global well‐posedness for nonlinear Schrödinger equation with a Dirac delta potential ( ‐NLS): Firstly, we give a new variational characterization of the solitary waves of ( ‐NLS) by making use of the symmetric decreasing rearrangement and constrained minimization argument. Secondly, we construct an invariant set , via the variational characterization of the solitary waves. Finally, we show the solutions, with initial data in , exist globally.

Weak solvability via bipotentials and approximation results for a class of bilateral frictional contact problems

A contact model with Tresca’s friction law is considered, the behavior of the material being modeled by the subdifferential of a convex constitutive map. Using a separable bipotential defined by means of the constitutive map and its Fenchel conjugate we deliver a two-field weak formulation of the model under consideration. This formulation allows us to deliver existence and uniqueness results based on minimization arguments. Subsequently, we study some properties of the solution and the data dependence. An alternative one-field formulation, in displacements, is also discussed focusing on the connections between formulations. Finally, we consider a regularization of the Tresca law and we deliver convergence results.

Least-energy nodal solutions of nonlinear equations with fractional Orlicz–Sobolev spaces

In this paper, we study the existence of least-energy nodal (sign-changing) weak solutions for a class of fractional Orlicz equations given by ( − △ g ) α u + g ( u ) = K ( x ) f ( u ) , in R N , where N ⩾ 3 , ( − △ g ) α is the fractional Orlicz g-Laplace operator, while f ∈ C 1 ( R ) and K is a positive and continuous function. Under a suitable conditions on f and K, we prove a compact embeddings result for weighted fractional Orlicz–Sobolev spaces. Next, by a minimization argument on Nehari manifold and a quantitative deformation lemma, we show the existence of at least one nodal (sign-changing) weak solution.

Existence of ground state solutions of elliptic system in Fractional Orlicz-Sobolev Spaces

We employing a minimization arguments on appropriate Nehari manifolds, we obtain ground state solutionsfor a non-local elliptic system driven by the fractional a(.)-Laplacian operator, with Dirichlet boundaryconditions type.

Existence of solution for magnetic Schrödinger equation with the Neumann boundary condition

We consider the nonlinear Schrödinger equation with magnetic field and the Neumann boundary condition: where Ω is a boundary domain in with a boundary, ν is the outward normal vector field at , , , ( is given by (3)), is a magnetic vector potential. When the exponent is subcritical, we can obtain solutions by Nehari manifold. When the exponent is critical, , we can obtain solutions by constrained minimization arguments.

Minimal bad sequences are necessary for a uniform Kruskal theorem

Kruskal's theorem on trees is a classical result of combinatorics with important applications in computer science. The minimal bad sequence argument of Nash-Williams yields a proof that is very elegant but not as elementary as possible, by previous results from mathematical logic. In the present paper, we formulate a uniform Kruskal theorem, which extends the original theorem from trees to general recursive data types. Our main result shows that the minimal bad sequence argument does yield the most elementary proof of this uniform Kruskal theorem. So in the uniform case, the elegant proof coincides with the most elementary one. In more precise and technical terms, we work in the framework of reverse mathematics, where we show that the uniform Kruskal theorem is equivalent to Π11-comprehension, over RCA0 extended by the chain-antichain principle. As a by-product of our investigation, we obtain uniform proofs of several Kruskal-type independence results.

Positive Solutions for a Class of Quasilinear Schr&amp;amp;#246;dinger Equations with Nonlocal Term

This paper is considered the existence of positive solutions for a class of generalized quasilinear Schrödinger equations with nonlocal term in RN which have appeared from plasma physics, as well as high-power ultrashort laser in matter. We use a charge of variables and obtain the existence of solutions via minimization argument.

Minimization arguments in analysis of variational–hemivariational inequalities

In this paper, an alternative approach is provided in the well-posedness analysis of elliptic variational–hemivariational inequalities in real Hilbert spaces. This includes the unique solvability and continuous dependence of the solution on the data. In most of the existing literature on elliptic variational–hemivariational inequalities, well-posedness results are obtained by using arguments of surjectivity for pseudomonotone multivalued operators, combined with additional compactness and pseudomonotonicity properties. In contrast, following (Han in Nonlinear Anal B Real World Appl 54:103114, 2020; Han in Numer Funct Anal Optim 42:371–395, 2021), the approach adopted in this paper is based on the fixed point structure of the problems, combined with minimization principles for elliptic variational–hemivariational inequalities. Consequently, only elementary results of functional analysis are needed in the approach, which makes the theory of elliptic variational–hemivariational inequalities more accessible to applied mathematicians and engineers. The theoretical results are illustrated on a representative example from contact mechanics.

DNA Mutations via Chern\u2013Simons Currents

We test the validity of a possible schematization of DNA structure and dynamics based on the Chern–Simons theory, that is a topological field theory mostly considered in the context of effective gravity theories. By means of the expectation value of the Wilson Loop, derived from this analogue gravity approach, we find the point-like curvature of genomic strings in KRAS human gene and COVID-19 sequences, correlating this curvature with the genetic mutations. The point-like curvature profile, obtained by means of the Chern–Simons currents, can be used to infer the position of the given mutations within the genetic string. Generally, mutations take place in the highest Chern–Simons current gradient locations and subsequent mutated sequences appear to have a smoother curvature than the initial ones, in agreement with a free energy minimization argument.

Multiplicity results for a system involving the p(x)- Laplacian operator

In this manuscript, it is considered existence and multiplicity of solutions for a system involving the -Laplacian operator which are related to several applications such as electrorheological fluids, image processing, robotics and space technology. A first solution for the system considered is obtained, under general conditions, by combining sub-supersolutions with a minimization argument in convex sets. Additionally, considering an Ambrosetti-Rabinowitz type condition and applying the Mountain Pass Theorem it is proved the existence of a second solution for the problem. The hypotheses allow to consider an wide class of nonlinearities and the results in this paper improve and complement some recent contributions.