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Mathematics
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Advances in Elliptic Equations and Their Applications
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Advances in Elliptic Equations and Their Applications

A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: closed (31 December 2024) | Viewed by 920

Special Issue Editors

Dr. João Vítor Da Silva

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Guest Editor
Departamento de Matemática, Instituto de Matemática, Estatística e Computação Científica-IMECC, Universidade Estadual de Campinas (Unicamp), Cidade Universitária Zeferino Vaz, Campinas 13083-591, SP, Brazil
Interests: regularity theory in elliptic/parabolic PDEs; nonlinear PDEs of degenerate type; free boundary problems; geometric measure theory and calculus of variations; nonlinear models driven by infinity-Laplacian
Prof. Dr. Analía C. Silva

E-Mail Website
Guest Editor
Departamento de Matemática, FCFMyN, Universidad Nacional de San Luis and Instituto de Matemática Aplicada San Luis (IMASL), CONICET-UNSL, San Luis, Argentina
Interests: nonlinear PDEs with nonstandard growth; models with lack of compactness; nonlocal PDEs and calculus of variations and related topics
Prof. Dr. Hernán A. Vivas

E-Mail Website
Guest Editor
Centro Marplatense de Investigaciones Matemáticas (UNMDP-Argentina), Conicet De´an Funes 3350, Mar del Plata 7600, Argentina
Interests: regularity theory in elliptic PDEs; nonlinear PDEs; integrodifferential PDEs with nonstandard growth; free boundary problems; calculus of variations and related topics

Special Issue Information

Dear Colleagues, 

It is our pleasure to announce the launch of a new Special Issue of Mathematics. In this current Special Issue titled "Advances in Elliptic Equations and Their Applications", we look forward to producing significant contributions in the modern area of the theory of PDEs, which covers topics such as elliptic PDEs, integrodifferential PDEs, calculus of variations, geometric measure theory, free boundary problems, and a range of further topics from mathematical analysis to its related issues. Such topics represent modern and relevant subjects of investigation in the current scopes of nonlinear PDEs and their related questions in the framework of pure and applied sciences. We also hope to deal with the applications of PDEs in the scenarios of applied sciences, which cover applications of PDEs in mathematical physics and also include the context of geometric analysis. We would like to stress that both analytic and numerical methods are welcome in this current Special Issue.

Dr. João Vítor Da Silva
Prof. Dr. Analía C. Silva
Prof. Dr. Hernán A. Vivas
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Regularity Theory in Elliptic PDEs
  • Calculus of Variations and Related Topics
  • Nonlinear Partial Differential Equations
  • Models with Nonstandard Growth
  • Local Models and Integrodifferential PDEs
  • Existence and Multiplicity of Solutions
  • Free Boundary Problems and Related Issues

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Published Papers (2 papers)

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Research

67 pages, 689 KiB  
Article
New Applications of Elliptic Functions and Integrals in GPS Inter-Satellite Communications with Account of General Relativity Theory
by Bogdan Dimitrov
Mathematics 2025, 13(8), 1286; https://doi.org/10.3390/math13081286 - 14 Apr 2025
Viewed by 212
Abstract
During the last 15–20 years, the experimental methods for autonomous navigation and inter-satellite links have been developing rapidly in order to ensure navigation control and data processing without commands from Earth stations. Inter-satellite links are related to relative ranging between the satellites from [...] Read more.
During the last 15–20 years, the experimental methods for autonomous navigation and inter-satellite links have been developing rapidly in order to ensure navigation control and data processing without commands from Earth stations. Inter-satellite links are related to relative ranging between the satellites from one constellation or different constellations and measuring the distances between them with the precision of at least 1 μm micrometer (=10−6 m), which should account for the bending of the light (radio or laser) signals due to the action of the Earth’s gravitational field. Thus, the theoretical calculation of the propagation time of a signal should be described in the framework of general relativity theory and the s.c. null cone equation. This review paper summarizes the latest achievements in calculating the propagation time of a signal, emitted by a GPS satellite, moving along a plane elliptical orbit or a space-oriented orbit, described by the full set of six Kepler parameters. It has been proved that for the case of plane elliptical orbit, the propagation time is expressed by a sum of elliptic integrals of the first, the second and the third kind, while for the second case (assuming that only the true anomaly angle is the dynamical parameter), the propagation time is expressed by a sum of elliptic integrals of the second and of the fourth order. For both cases, it has been proved that the propagation time represents a real-valued expression and not an imaginary one, as it should be. For the typical parameters of a GPS orbit, numerical calculations for the first case give acceptable values of the propagation time and, especially, the Shapiro delay term of the order of nanoseconds, thus confirming that this is a propagation time for the signal and not for the time of motion of the satellite. Theoretical arguments, related to general relativity and differential geometry have also been presented in favor of this conclusion. A new analytical method has been developed for transforming an elliptic integral in the Legendre form into an integral in the Weierstrass form. Two different representations have been found, one of them based on the method of four-dimensional uniformization, exposed in the monograph of Whittaker and Watson. The result of this approach is a new formulae for the Weierstrass invariants, depending in a complicated manner on the modulus parameter q of the elliptic integral in the Legendre form. Full article
(This article belongs to the Special Issue Advances in Elliptic Equations and Their Applications)
8 pages, 213 KiB  
Article
Regularizing Effects for Some Nonlocal Elliptic Problems
by Ida de Bonis
Mathematics 2025, 13(5), 687; https://doi.org/10.3390/math13050687 - 20 Feb 2025
Viewed by 215
Abstract
We study the regularizing effect of the zero order term in some elliptic problems, which present in the principal part a nonlocal operator, i.e., the fractional Laplacian operator. We prove the existence of bounded energy solutions, even if the data are assumed to [...] Read more.
We study the regularizing effect of the zero order term in some elliptic problems, which present in the principal part a nonlocal operator, i.e., the fractional Laplacian operator. We prove the existence of bounded energy solutions, even if the data are assumed to be very irregular. Full article
(This article belongs to the Special Issue Advances in Elliptic Equations and Their Applications)
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