Research of the structure of unbounded viscosity solutions to semilinear degenerate elliptic equations in R^N

R^N中半线性简并椭圆方程无界粘性解的结构研究

• 批准号: 16540151
• 负责人: MARUO Kenji
• 金额: $ 1.47万
• 依托单位: Kobe University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2004
• 资助国家: 日本
• 起止时间: 2004 至 2006
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-16540151/
• 关键词: Semilinear Elliptic Differential Equations; Structure of Solutions; Viscosity Solutions; 解析学; 関数方程式; 半線形楕円型方程式; 解の漸近挙動; 球対称解; 非球対称解の非存在; 非球対称解の存在; 粘性解の解の構造

Consider a following semilinear degenerate partial differential equation:-g(|x|)Δu + u|u|^p - f(|x|) = 0 ∈ R^N (1)where g is a nonnegative plynominal of degree ell > 2 in a neighborhood of a point at infinity and f is also a plynominal in a neighborhood of a point at infinity. Moreover, assume g holds bounded zero points. Hence, the differential equation is a degenerate tyle. We don't impose the boundary condition to solutions of (1) in the neighborhood of the point infinity. Then, It is possible to exist many continuous viscosity solutions.Our purpose of this research was to analyze the structure of the set of many continuous viscosity solutions of (1). The results of our research are as follows. We the first showed that an inequality of relations of N and k is a necessary and sufficient condition to decide whether radically symmetric solutions are infinite or single where k is a coefficient of a maxima order of f. Moreover, we proved that a set of many radically symmetric solutions is homeomorphic to R^1. The secondly, under the assumptions N = 2 and that lower order terms of polynominal of f, g do not exist, we found the condition to judge whether there exists non radically symmetric solution or not. If non-radically symmetric solution exists we also showed how many non-radically symmetric solutions there were. That is, We showed that the number of solutions was different depending on the value that related to l, p, k.

考虑以下半线性偏二方方程：-g（| x |）ΔU + u | u | u | u | u | u | u | u | u | u | u | u | u | u | u | 't将边界条件施加到点无穷大的邻居引擎盖中的解决方案。 ）其次，在n = 2的假设下，f的多项式术语不存在，我们发现了是否存在非根本对称的溶液，我们也表明了如何采用的soluti soluti ons。我们显示的解决方案的数量不同，具体取决于与L，P，K的值。

项目成果

Non-existence of non radially symmetric viscosity solutions to semilinear degerate elliptic equations with radially symmetric cofficents in R^2.

R^2 中具有径向对称坐标的半线性退化椭圆方程不存在非径向对称粘度解。

• 发表时间: 2004
• 期刊: Adv. in Math. Sci. and Appl. 14
• 作者: Hiroki Sato;C.Li;M.Oichi;Sin-Ei Takahasi;Kenji Maruo
• 通讯作者: Kenji Maruo

Asymptotic behavior of unbounded radially symmetric solutions of semilinear degerate elliptic equations.

半线性退化椭圆方程的无界径向对称解的渐近行为。

• 发表时间: 2006
• 期刊: Adv. Math. Sci. Appl. 16
• 作者: Yoshiyuki Hino;Satoru Murakami;Kenji Maruo
• 通讯作者: Kenji Maruo

Non-existence of non radically symmetric viscosity solutions to semilinear degerate elliptic equations with radically symmetric coefficients in R2.

R2 中具有激进对称系数的半线性退化椭圆方程不存在非激进对称粘度解。

• 发表时间: 2004
• 期刊: Adv. in Math. Sci. and Appl. 14
• 作者: Kenji;Maruo
• 通讯作者: Maruo

Non-existence of non-radially symmetric solutions to semilinear degenerate elliptic equations with radially symmetric coefficients in R^2

R^2 中具有径向对称系数的半线性简并椭圆方程不存在非径向对称解

• 发表时间: 2004
• 期刊: Advances in Mahematical Sciences and Applications Vol.14,No.2
• 作者: Hora;A.;K.Yamaguchi;H.Matsunaga;Akira Koyama;K.Maruo
• 通讯作者: K.Maruo

Method of the distance function to the Bence-Merriman-Osher algorithm for motion by mean curvature

Bence-Merriman-Osher 平均曲率运动算法的距离函数方法

• 发表时间: 2005
• 期刊: Communications on Pure and Applied Analysis 4・2
• 作者: Yoko GOTO;Katsuyuki ISHII;Takayoshi OGAWA
• 通讯作者: Takayoshi OGAWA