This article mainly uses two methods of solving the conservation laws of two partial differential equations and a system of equations. The first method is to construct the conservation law directly and the second method is to apply the Ibragimov method to solve the conservation laws of the target equation systems, which are constructed based on the symmetric rows of the target equation system. In this paper, we select two equations and an equation system, and we try to apply these two methods to the combined KdV-MKdV equation, the Klein-Gordon equation and the generalized coupled KdV equation, and simply verify them. The combined KdV-MKdV equation describes the wave propagation of bound particles, sound waves and thermal pulses. The Klein-Gordon equation describes the nonlinear sine-KG equation that simulates the motion of the Josephson junction, the rigid pendulum connected to the stretched wire, and the dislocations in the crystal. And the coupled KdV equation has also attracted a lot of research due to its importance in theoretical physics and many scientific applications. In the last part of the article, we try to briefly analyze the Hamiltonian structures and adjoint symmetries of the target equations, and calculate their linear soliton solutions.
{"title":"Conservation laws analysis of nonlinear partial differential equations and their linear soliton solutions and Hamiltonian structures","authors":"Long Ju, Jian Zhou, Yufeng Zhang","doi":"10.3934/cam.2023002","DOIUrl":"https://doi.org/10.3934/cam.2023002","url":null,"abstract":"This article mainly uses two methods of solving the conservation laws of two partial differential equations and a system of equations. The first method is to construct the conservation law directly and the second method is to apply the Ibragimov method to solve the conservation laws of the target equation systems, which are constructed based on the symmetric rows of the target equation system. In this paper, we select two equations and an equation system, and we try to apply these two methods to the combined KdV-MKdV equation, the Klein-Gordon equation and the generalized coupled KdV equation, and simply verify them. The combined KdV-MKdV equation describes the wave propagation of bound particles, sound waves and thermal pulses. The Klein-Gordon equation describes the nonlinear sine-KG equation that simulates the motion of the Josephson junction, the rigid pendulum connected to the stretched wire, and the dislocations in the crystal. And the coupled KdV equation has also attracted a lot of research due to its importance in theoretical physics and many scientific applications. In the last part of the article, we try to briefly analyze the Hamiltonian structures and adjoint symmetries of the target equations, and calculate their linear soliton solutions.","PeriodicalId":233941,"journal":{"name":"Communications in Analysis and Mechanics","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131768991","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The dissipative (2 + 1)-dimensional AKNS equation is considered in this paper. First, the Lie symmetry analysis method is applied to the dissipative (2 + 1)-dimensional AKNS and six point symmetries are obtained. Symmetry reductions are performed by utilizing these obtained point symmetries and four differential equations are derived, including a fourth-order ordinary differential equation and three partial differential equations. Thereafter, the direct integration approach and the $ (G'/G^{2})- $expansion method are employed to solve the ordinary differential respectively. As a result, a periodic solution in terms of the Weierstrass elliptic function is obtained via the the direct integration approach, while six kinds of including the hyperbolic function types and the hyperbolic function types are derived via the $ (G'/G^{2})- $expansion method. The corresponding graphical representation of the obtained solutions are presented by choosing suitable parametric values. Finally, the multiplier technique and the classical Noether's theorem are employed to derive conserved vectors for the dissipative (2 + 1)-dimensional AKNS respectively. Consequently, eight local conservation laws for the dissipative (2 + 1)-dimensional AKNS equation are presented by utilizing the multiplier technique and five local conservation laws are derived by invoking Noether's theorem.
{"title":"Lie symmetry analysis, particular solutions and conservation laws for the dissipative (2 + 1)- dimensional AKNS equation","authors":"Sixing Tao","doi":"10.3934/cam.2023024","DOIUrl":"https://doi.org/10.3934/cam.2023024","url":null,"abstract":"The dissipative (2 + 1)-dimensional AKNS equation is considered in this paper. First, the Lie symmetry analysis method is applied to the dissipative (2 + 1)-dimensional AKNS and six point symmetries are obtained. Symmetry reductions are performed by utilizing these obtained point symmetries and four differential equations are derived, including a fourth-order ordinary differential equation and three partial differential equations. Thereafter, the direct integration approach and the $ (G'/G^{2})- $expansion method are employed to solve the ordinary differential respectively. As a result, a periodic solution in terms of the Weierstrass elliptic function is obtained via the the direct integration approach, while six kinds of including the hyperbolic function types and the hyperbolic function types are derived via the $ (G'/G^{2})- $expansion method. The corresponding graphical representation of the obtained solutions are presented by choosing suitable parametric values. Finally, the multiplier technique and the classical Noether's theorem are employed to derive conserved vectors for the dissipative (2 + 1)-dimensional AKNS respectively. Consequently, eight local conservation laws for the dissipative (2 + 1)-dimensional AKNS equation are presented by utilizing the multiplier technique and five local conservation laws are derived by invoking Noether's theorem.","PeriodicalId":233941,"journal":{"name":"Communications in Analysis and Mechanics","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115995442","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this note we give a short proof of Arnold's conjecture for the zero section of a cotangent bundle of a closed manifold. The proof is based on some basic properties of Lagrangian spectral invariants from Floer theory.
{"title":"A short proof of cuplength estimates on Lagrangian intersections","authors":"Wenmin Gong","doi":"10.3934/cam.2023003","DOIUrl":"https://doi.org/10.3934/cam.2023003","url":null,"abstract":"<abstract><p>In this note we give a short proof of Arnold's conjecture for the zero section of a cotangent bundle of a closed manifold. The proof is based on some basic properties of Lagrangian spectral invariants from Floer theory.</p></abstract>","PeriodicalId":233941,"journal":{"name":"Communications in Analysis and Mechanics","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133254838","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper focuses on the spectral analysis of equations that describe the oscillations of a heavy pendulum partially filled with a homogeneous incompressible viscoelastic fluid. The constitutive equation of the fluid follows the simpler Oldroyd model. By examining the eigenvalues of the linear operator that describes the dynamics of the coupled system, it was demonstrated that under appropriate assumptions the equilibrium configuration remains stable in the linear approximation. Moreover, when the viscosity coefficient is sufficiently large the spectrum comprises three branches of eigenvalues with potential cluster points at $ 0 $, $ beta $ and $ infty $ where $ beta $ represents the viscoelastic parameter of the fluid. These three branches of eigenvalues correspond to frequencies associated with various types of waves.
{"title":"Analysis of small oscillations of a pendulum partially filled with a viscoelastic fluid","authors":"H. Essaouini, P. Capodanno","doi":"10.3934/cam.2023019","DOIUrl":"https://doi.org/10.3934/cam.2023019","url":null,"abstract":"This paper focuses on the spectral analysis of equations that describe the oscillations of a heavy pendulum partially filled with a homogeneous incompressible viscoelastic fluid. The constitutive equation of the fluid follows the simpler Oldroyd model. By examining the eigenvalues of the linear operator that describes the dynamics of the coupled system, it was demonstrated that under appropriate assumptions the equilibrium configuration remains stable in the linear approximation. Moreover, when the viscosity coefficient is sufficiently large the spectrum comprises three branches of eigenvalues with potential cluster points at $ 0 $, $ beta $ and $ infty $ where $ beta $ represents the viscoelastic parameter of the fluid. These three branches of eigenvalues correspond to frequencies associated with various types of waves.","PeriodicalId":233941,"journal":{"name":"Communications in Analysis and Mechanics","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128321702","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we introduce the Laguerre bounded variation space and the Laguerre perimeter, thereby investigating their properties. Moreover, we prove the isoperimetric inequality and the Sobolev inequality in the Laguerre setting. As applications, we derive the mean curvature for the Laguerre perimeter.
{"title":"Laguerre BV spaces, Laguerre perimeter and their applications","authors":"Heming Wang, Yu Liu","doi":"10.3934/cam.2023011","DOIUrl":"https://doi.org/10.3934/cam.2023011","url":null,"abstract":"In this paper, we introduce the Laguerre bounded variation space and the Laguerre perimeter, thereby investigating their properties. Moreover, we prove the isoperimetric inequality and the Sobolev inequality in the Laguerre setting. As applications, we derive the mean curvature for the Laguerre perimeter.","PeriodicalId":233941,"journal":{"name":"Communications in Analysis and Mechanics","volume":"53 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126456016","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this work, our main concern is to study the existence and multiplicity of solutions for the following sub-elliptic system with Hardy type potentials and multiple critical exponents on Carnot group begin{document}$ begin{equation*} left{begin{aligned} &-Delta_{mathbb{G}}u = frac{psi^{alpha}|u|^{2^*(alpha)-2}u}{d(z)^{alpha}}+ frac{p_{1}}{2^*(gamma)}frac{psi^{gamma}|u|^{p_{1}-2}u|v|^{p_{2}}}{d(z, z_{0})^{gamma}} +lambda h(z)frac{psi^{sigma}|u|^{q-2}u}{d(z)^{sigma}} , , & text{in } , , Omega, &-Delta_{mathbb{G}}v = frac{psi^{beta}|v|^{2^*(beta)-2}v}{d(z)^{beta}}+ frac{p_{2}}{2^*(gamma)}frac{psi^{gamma}|u|^{p_{1}}|v|^{p_{2}-2}v}{d(z, z_{0})^{gamma}} +lambda h(z)frac{psi^{sigma}|v|^{q-2}v}{d(z)^{sigma}}, , &text{in } , , Omega, &quad u = v = 0, , &text{on } , , partialOmega, end{aligned}right. end{equation*} $end{document} where $ -Delta_{mathbb{G}} $ is a sub-Laplacian on Carnot group $ mathbb{G} $, $ alpha, beta, gamma, sigmain [0, 2) $, $ d $ is the $ Delta_{mathbb{G}} $-natural gauge, $ psi = |nabla_{mathbb{G}}d| $ and $ nabla_{mathbb{G}} $ is the horizontal gradient associated to $ Delta_{mathbb{G}} $. The positive parameters $ lambda $, $ q $ satisfy $ 0 < lambda < infty $, $ 1 < q < 2 $, and $ p_{1} $, $ p_{2} > 1 $ with $ p_{1}+p_{2} = 2^*(gamma) $, here $ 2^*(alpha): = frac{2(Q-alpha)}{Q-2} $, $ 2^*(beta): = frac{2(Q-beta)}{Q-2} $ and $ 2^*(gamma) = frac{2(Q-gamma)}{Q-2} $ are the critical Hardy-Sobolev exponents, $ Q $ is the homogeneous dimension of the space $ mathbb{G} $. By means of variational methods and the mountain-pass theorem of Ambrosetti and Rabonowitz, we study the existence of multiple solutions to the sub-elliptic system.
In this work, our main concern is to study the existence and multiplicity of solutions for the following sub-elliptic system with Hardy type potentials and multiple critical exponents on Carnot group begin{document}$ begin{equation*} left{begin{aligned} &-Delta_{mathbb{G}}u = frac{psi^{alpha}|u|^{2^*(alpha)-2}u}{d(z)^{alpha}}+ frac{p_{1}}{2^*(gamma)}frac{psi^{gamma}|u|^{p_{1}-2}u|v|^{p_{2}}}{d(z, z_{0})^{gamma}} +lambda h(z)frac{psi^{sigma}|u|^{q-2}u}{d(z)^{sigma}} , , & text{in } , , Omega, &-Delta_{mathbb{G}}v = frac{psi^{beta}|v|^{2^*(beta)-2}v}{d(z)^{beta}}+ frac{p_{2}}{2^*(gamma)}frac{psi^{gamma}|u|^{p_{1}}|v|^{p_{2}-2}v}{d(z, z_{0})^{gamma}} +lambda h(z)frac{psi^{sigma}|v|^{q-2}v}{d(z)^{sigma}}, , &text{in } , , Omega, &quad u = v = 0, , &text{on } , , partialOmega, end{aligned}right. end{equation*} $end{document} where $ -Delta_{mathbb{G}} $ is a sub-Laplacian on Carnot group $ mathbb{G} $, $ alpha, beta, gamma, sigmain [0, 2) $, $ d $ is the $ Delta_{mathbb{G}} $-natural gauge, $ psi = |nabla_{mathbb{G}}d| $ and $ nabla_{mathbb{G}} $ is the horizontal gradient associated to $ Delta_{mathbb{G}} $. The positive parameters $ lambda $, $ q $ satisfy $ 0 < lambda < infty $, $ 1 < q < 2 $, and $ p_{1} $, $ p_{2} > 1 $ with $ p_{1}+p_{2} = 2^*(gamma) $, here $ 2^*(alpha): = frac{2(Q-alpha)}{Q-2} $, $ 2^*(beta): = frac{2(Q-beta)}{Q-2} $ and $ 2^*(gamma) = frac{2(Q-gamma)}{Q-2} $ are the critical Hardy-Sobolev exponents, $ Q $ is the homogeneous dimension of the space $ mathbb{G} $. By means of variational methods and the mountain-pass theorem of Ambrosetti and Rabonowitz, we study the existence of multiple solutions to the sub-elliptic system.
{"title":"On criticality coupled sub-Laplacian systems with Hardy type potentials on Stratified Lie groups","authors":"Jinguo Zhang, Shuhai Zhu","doi":"10.3934/cam.2023005","DOIUrl":"https://doi.org/10.3934/cam.2023005","url":null,"abstract":"In this work, our main concern is to study the existence and multiplicity of solutions for the following sub-elliptic system with Hardy type potentials and multiple critical exponents on Carnot group\u0000\u0000 begin{document}$ begin{equation*} left{begin{aligned} &-Delta_{mathbb{G}}u = frac{psi^{alpha}|u|^{2^*(alpha)-2}u}{d(z)^{alpha}}+ frac{p_{1}}{2^*(gamma)}frac{psi^{gamma}|u|^{p_{1}-2}u|v|^{p_{2}}}{d(z, z_{0})^{gamma}} +lambda h(z)frac{psi^{sigma}|u|^{q-2}u}{d(z)^{sigma}} , , & text{in } , , Omega, &-Delta_{mathbb{G}}v = frac{psi^{beta}|v|^{2^*(beta)-2}v}{d(z)^{beta}}+ frac{p_{2}}{2^*(gamma)}frac{psi^{gamma}|u|^{p_{1}}|v|^{p_{2}-2}v}{d(z, z_{0})^{gamma}} +lambda h(z)frac{psi^{sigma}|v|^{q-2}v}{d(z)^{sigma}}, , &text{in } , , Omega, &quad u = v = 0, , &text{on } , , partialOmega, end{aligned}right. end{equation*} $end{document} \u0000where $ -Delta_{mathbb{G}} $ is a sub-Laplacian on Carnot group $ mathbb{G} $, $ alpha, beta, gamma, sigmain [0, 2) $, $ d $ is the $ Delta_{mathbb{G}} $-natural gauge, $ psi = |nabla_{mathbb{G}}d| $ and $ nabla_{mathbb{G}} $ is the horizontal gradient associated to $ Delta_{mathbb{G}} $. The positive parameters $ lambda $, $ q $ satisfy $ 0 < lambda < infty $, $ 1 < q < 2 $, and $ p_{1} $, $ p_{2} > 1 $ with $ p_{1}+p_{2} = 2^*(gamma) $, here $ 2^*(alpha): = frac{2(Q-alpha)}{Q-2} $, $ 2^*(beta): = frac{2(Q-beta)}{Q-2} $ and $ 2^*(gamma) = frac{2(Q-gamma)}{Q-2} $ are the critical Hardy-Sobolev exponents, $ Q $ is the homogeneous dimension of the space $ mathbb{G} $. By means of variational methods and the mountain-pass theorem of Ambrosetti and Rabonowitz, we study the existence of multiple solutions to the sub-elliptic system.","PeriodicalId":233941,"journal":{"name":"Communications in Analysis and Mechanics","volume":"188 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121527746","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article we revisit the global existence result of the wave-Klein-Gordon model of the system of the self-gravitating massive field. Our new observation is that, by applying the conformal energy estimates on hyperboloids, we obtain mildly increasing energy estimate up to the top order for the Klein-Gordon component, which clarify the question on the hierarchy of the energy bounds of the Klein-Gordon component in our previous work. Furthermore, a uniform-in-time energy estimate is established for the wave component up to the top order, as well as a scattering result. These improvements indicate that the partial conformal symmetry of the Einstein-massive scalar system will play an important role in the global analysis.
{"title":"Conformal-type energy estimates on hyperboloids and the wave-Klein-Gordon model of self-gravitating massive fields","authors":"Senhao Duan, Yue Ma, Weidong Zhang","doi":"10.3934/cam.2023007","DOIUrl":"https://doi.org/10.3934/cam.2023007","url":null,"abstract":"In this article we revisit the global existence result of the wave-Klein-Gordon model of the system of the self-gravitating massive field. Our new observation is that, by applying the conformal energy estimates on hyperboloids, we obtain mildly increasing energy estimate up to the top order for the Klein-Gordon component, which clarify the question on the hierarchy of the energy bounds of the Klein-Gordon component in our previous work. Furthermore, a uniform-in-time energy estimate is established for the wave component up to the top order, as well as a scattering result. These improvements indicate that the partial conformal symmetry of the Einstein-massive scalar system will play an important role in the global analysis.","PeriodicalId":233941,"journal":{"name":"Communications in Analysis and Mechanics","volume":"83 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126068696","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper is concerned with a nonlinear plate equation modeling the oscillations of suspension bridges. Under mixed boundary conditions consisting of simply supported and free boundary conditions, we obtain the global well-posedness of solutions in suitable function spaces. In addition, we use the perturbed energy method to prove the existence of a bounded absorbing set and establish a stabilizability estimate. Then, we derive the existence of a global attractor by verifying the asymptotic smoothness of the corresponding dissipative dynamical system.
{"title":"Global attractors for a nonlinear plate equation modeling the oscillations of suspension bridges","authors":"Yang Liu","doi":"10.3934/cam.2023021","DOIUrl":"https://doi.org/10.3934/cam.2023021","url":null,"abstract":"This paper is concerned with a nonlinear plate equation modeling the oscillations of suspension bridges. Under mixed boundary conditions consisting of simply supported and free boundary conditions, we obtain the global well-posedness of solutions in suitable function spaces. In addition, we use the perturbed energy method to prove the existence of a bounded absorbing set and establish a stabilizability estimate. Then, we derive the existence of a global attractor by verifying the asymptotic smoothness of the corresponding dissipative dynamical system.","PeriodicalId":233941,"journal":{"name":"Communications in Analysis and Mechanics","volume":"44 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132184792","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
R. Pinčák, A. Pigazzini, Saeid Jafari, Özge Korkmaz, C. Özel, E. Bartoš
The aim of this paper is to use a special type of Einstein warped product manifolds recently introduced, the so-called PNDP-manifolds, for the differential geometric study, by focusing on some aspects related to dark field in financial market such as the concept of dark volatility. This volatility is not fixed in any relevant economic parameter, a sort of negative dimension, a ghost field, that greatly influences the behavior of real market. Since the PNDP-manifold has a "virtual" dimension, we want to use it in order to show how the Global Market is influenced by dark volatility, and in this regard we also provide an example, by considering the classical exponential models as possible solutions to our approach. We show how dark volatility, combined with specific conditions, leads to the collapse of a forward price.
{"title":"A possible interpretation of financial markets affected by dark volatility","authors":"R. Pinčák, A. Pigazzini, Saeid Jafari, Özge Korkmaz, C. Özel, E. Bartoš","doi":"10.3934/cam.2023006","DOIUrl":"https://doi.org/10.3934/cam.2023006","url":null,"abstract":"The aim of this paper is to use a special type of Einstein warped product manifolds recently introduced, the so-called PNDP-manifolds, for the differential geometric study, by focusing on some aspects related to dark field in financial market such as the concept of dark volatility. This volatility is not fixed in any relevant economic parameter, a sort of negative dimension, a ghost field, that greatly influences the behavior of real market. Since the PNDP-manifold has a \"virtual\" dimension, we want to use it in order to show how the Global Market is influenced by dark volatility, and in this regard we also provide an example, by considering the classical exponential models as possible solutions to our approach. We show how dark volatility, combined with specific conditions, leads to the collapse of a forward price.","PeriodicalId":233941,"journal":{"name":"Communications in Analysis and Mechanics","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129374399","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this manuscript, existence and multiplicity results are obtained for a problem involving an anisotropic $ (p, q) $-Laplacian-type operator by means of sub-supersolutions and variational techniques. This problem arises in various applications such as in the study of the enhancement of images, the spread of epidemic disease and in the dynamic of fluids. Under a general condition, the existence of a solution is proved, and the multiplicity of solutions is obtained by considering an additional natural hypothesis.
{"title":"Solutions for a class of problems driven by an anisotropic $ (p, q) $-Laplacian type operator","authors":"Leandro Tavares","doi":"10.3934/cam.2023026","DOIUrl":"https://doi.org/10.3934/cam.2023026","url":null,"abstract":"In this manuscript, existence and multiplicity results are obtained for a problem involving an anisotropic $ (p, q) $-Laplacian-type operator by means of sub-supersolutions and variational techniques. This problem arises in various applications such as in the study of the enhancement of images, the spread of epidemic disease and in the dynamic of fluids. Under a general condition, the existence of a solution is proved, and the multiplicity of solutions is obtained by considering an additional natural hypothesis.","PeriodicalId":233941,"journal":{"name":"Communications in Analysis and Mechanics","volume":"107 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122119423","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：