We consider the singular Dirichlet problem for the Monge-Ampère type equation■=0,whereΩis a strictly convex and bounded smooth domain in■is positive and strictly decreasing in(0,∞)with■is positive inΩ.We ob...We consider the singular Dirichlet problem for the Monge-Ampère type equation■=0,whereΩis a strictly convex and bounded smooth domain in■is positive and strictly decreasing in(0,∞)with■is positive inΩ.We obtain the existence,nonexistence and global asymptotic behavior of the convex solution to such a problem for more general b and g.Our approach is based on the Karamata regular variation theory and the construction of suitable sub-and super-solutions.展开更多
20世纪以来,随着科学技术的发展,出现了许多新型的偏微分方程,如椭圆型方程、双曲线型方程和抛物线方程等,这些偏微分方程求解问题都可以转化为求相应常微分方程的解或研究解的性质的问题。Monge-Ampère方程是一类完全非线性偏微...20世纪以来,随着科学技术的发展,出现了许多新型的偏微分方程,如椭圆型方程、双曲线型方程和抛物线方程等,这些偏微分方程求解问题都可以转化为求相应常微分方程的解或研究解的性质的问题。Monge-Ampère方程是一类完全非线性偏微分方程,起源于几何学,在微分几何,流体力学,最优化问题等领域有广泛应用。近年来,关于Monge-Ampère方程的研究已取得了很大的突破,众多学者运用不同的方法去讨论这类方程解的存在性,多解性及其唯一性,例如,不动点定理、上下解方法、单调迭代方法、变分理论、分歧理论等。基于Monge-Ampère方程模型的实际应用背景,本文主要讨论一类Monge-Ampère方程的Dirichlet问题,运用Krasnosel’skii-Precup不动点定理得到了其径向解存在的充分条件。Since the 20th century, with the development of science and technology, there are many new types of partial differential equations, such as elliptic equations, hyperbolic equations and parabolic equations, etc. These partial differential equations can be transformed into the solution of the corresponding ordinary differential equations or the study of the nature of the solution of the problem. Monge-Ampère equations are a class of fully nonlinear partial differential equations that originated in geometry and have wide applications in differential geometry, fluid mechanics, optimization problems, and other fields. In recent years, the research on Monge-Ampère equations has made great breakthroughs, and many scholars have used different methods to discuss the existence, multisolvability and uniqueness of the solutions of these equations, such as the fixed point theorem, the upper and lower solution method, the monotone iteration method, the theory of variations, the theory of divergence, etc. Based on the practical application background of the Monge-Ampère equation model, this paper focuses on the Dirichlet problem for a class of Monge-Ampère equations, and the sufficient condition for the existence of radial solutions of Monge-Ampère equations is obtained by applying the Krasnosel’skii-Precup fixed point theorem.展开更多
In this short note we revisit the convex integration approach to constructing very weak solutions to the 2D Monge-Ampére equation with H?lder-continuous ?rst derivatives of exponent β < 1/5. Our approach is b...In this short note we revisit the convex integration approach to constructing very weak solutions to the 2D Monge-Ampére equation with H?lder-continuous ?rst derivatives of exponent β < 1/5. Our approach is based on combining the approach of Lewicka and Pakzad(2017) with a new diagonalization procedure which avoids the use of conformal coordinates, which was introduced by De Lellis et al.(2018) for the isometric immersion problem.展开更多
基金supported by Shandong Provincial NSF(ZR2022MA020).
文摘We consider the singular Dirichlet problem for the Monge-Ampère type equation■=0,whereΩis a strictly convex and bounded smooth domain in■is positive and strictly decreasing in(0,∞)with■is positive inΩ.We obtain the existence,nonexistence and global asymptotic behavior of the convex solution to such a problem for more general b and g.Our approach is based on the Karamata regular variation theory and the construction of suitable sub-and super-solutions.
文摘20世纪以来,随着科学技术的发展,出现了许多新型的偏微分方程,如椭圆型方程、双曲线型方程和抛物线方程等,这些偏微分方程求解问题都可以转化为求相应常微分方程的解或研究解的性质的问题。Monge-Ampère方程是一类完全非线性偏微分方程,起源于几何学,在微分几何,流体力学,最优化问题等领域有广泛应用。近年来,关于Monge-Ampère方程的研究已取得了很大的突破,众多学者运用不同的方法去讨论这类方程解的存在性,多解性及其唯一性,例如,不动点定理、上下解方法、单调迭代方法、变分理论、分歧理论等。基于Monge-Ampère方程模型的实际应用背景,本文主要讨论一类Monge-Ampère方程的Dirichlet问题,运用Krasnosel’skii-Precup不动点定理得到了其径向解存在的充分条件。Since the 20th century, with the development of science and technology, there are many new types of partial differential equations, such as elliptic equations, hyperbolic equations and parabolic equations, etc. These partial differential equations can be transformed into the solution of the corresponding ordinary differential equations or the study of the nature of the solution of the problem. Monge-Ampère equations are a class of fully nonlinear partial differential equations that originated in geometry and have wide applications in differential geometry, fluid mechanics, optimization problems, and other fields. In recent years, the research on Monge-Ampère equations has made great breakthroughs, and many scholars have used different methods to discuss the existence, multisolvability and uniqueness of the solutions of these equations, such as the fixed point theorem, the upper and lower solution method, the monotone iteration method, the theory of variations, the theory of divergence, etc. Based on the practical application background of the Monge-Ampère equation model, this paper focuses on the Dirichlet problem for a class of Monge-Ampère equations, and the sufficient condition for the existence of radial solutions of Monge-Ampère equations is obtained by applying the Krasnosel’skii-Precup fixed point theorem.
基金support of the European Research Council Grant Agreement (Grant No. 724298)
文摘In this short note we revisit the convex integration approach to constructing very weak solutions to the 2D Monge-Ampére equation with H?lder-continuous ?rst derivatives of exponent β < 1/5. Our approach is based on combining the approach of Lewicka and Pakzad(2017) with a new diagonalization procedure which avoids the use of conformal coordinates, which was introduced by De Lellis et al.(2018) for the isometric immersion problem.
