Symmetry of solutions for elliptic partial differential equations

• Project/Area Number: 24540179
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Saga University
• Principal Investigator: Kajikiya Ryuji 佐賀大学, 工学(系)研究科(研究院), 教授 (10183261)
• Project Period (FY): 2012-04-01 – 2016-03-31
• Project Status: Completed (Fiscal Year 2015)
• Budget Amount: ¥4,940,000 (Direct Cost: ¥3,800,000、Indirect Cost: ¥1,140,000); Fiscal Year 2015: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2014: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2013: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2012: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
• Keywords: 楕円型偏微分方程式 / 変分法 / 最小エネルギー解 / 解の対称性 / 群不変性 / 劣線形熱方程式 / 定常解の安定性 / 準線形放物型偏微分方程式 / p-Laplace方程式 / 非線形ノイマン問題 / 分岐 / p-Laplace 方程式 / least energy solution / elliptic equation

Outline of Final Research Achievements

We study elliptic partial differential equations in symmetric domains. Let H and G be closed subgroups of the orthogonal group such that H is a closed subgroup of G and the domain is G invariant. Then we show the existence of a positive solution which is H invariant but G non-invariant under suitable assumptions of H, G and the coefficient function of the equation. Such a solution is obtained as a least energy solution. Here a least energy solution is a solution which is a minimizer of the Rayleigh quotient. Our theorem ensures the existence of various solutions which has a weak symmetry but does not have a strong symmetry.

Research Products

• [Journal Article] Multiple positive solutions of the Emden-Fowler equation in hollow thin symmetric domains. (2015)
  • Author(s): R. Kajikiya
  • Journal Title: Calculus of Variations and Partial Differential Equations Volume: 52 Issue: 3-4 Pages: 681-704
  • DOI: 10.1007/s00526-014-0729-6
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] A priori estimate for the first eigenvalue of the p-Laplacian. (2015)
  • Author(s): R. Kajikiya
  • Journal Title: Differential and Integral Equations Volume: 28 Pages: 1011-1028
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Partially symmetric solutions of the generalized Henon equation in symmetric domains. (2015)
  • Author(s): R. Kajikiya
  • Journal Title: Topological Methods in Nonlinear Analysis Volume: 46 Pages: 191-221
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Stability of stationary solutions for semilinear heat equations with concave nonlinearity. (2015)
  • Author(s): G. Akagi and R. Kajikiya
  • Journal Title: Communications in Contemporary Mathematics Volume: 17 Issue: 06 Pages: 1-29
  • DOI: 10.1142/s0219199715500017
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Multiple bifurcations of sign-changing solutions for one-dimensional p-Laplace equation with a critical weight. (2014)
  • Author(s): R. Kajikiya and Y.-H. Lee
  • Journal Title: Advances in Differential Equations Volume: 19 Pages: 283-316
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Two sequences of solutions for indefinite superlinear-sublinear elliptic equations with nonlinear boundary conditions. (2014)
  • Author(s): R. Kajikiya and D. Naimen
  • Journal Title: Commun. Pure and Appl. Anal. Volume: 13 Issue: 4 Pages: 1593-1612
  • DOI: 10.3934/cpaa.2014.13.1593
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Positive solutions of the p-Laplace Emden-Fowler equation in hollow thin symmetric domains. (2014)
  • Author(s): R. Kajikiya
  • Journal Title: Math. Bohem. Volume: 139 Pages: 145-154
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Three positive solutions of the one-dimensional generalized Henon equation. (2014)
  • Author(s): R. Kajikiya
  • Journal Title: Results in Mathematics Volume: 66 Issue: 3-4 Pages: 427-459
  • DOI: 10.1007/s00025-014-0385-3
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Symmetry and stability of asymptotic profiles for fast diffusion equations in annuli. (2014)
  • Author(s): G. Akagi and R. Kajikiya
  • Journal Title: Annales de l'Institut Henri Poincare. Analyse Non Lineaire Volume: 31 Issue: 6 Pages: 1155-1173
  • DOI: 10.1016/j.anihpc.2013.08.006
  • NAID: 120005372387
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Multiple solutions of sublinear elliptic equations with small perturbations. (2013)
  • Author(s): Ryuji Kajikiya
  • Journal Title: J. Math. Anal. Appl., Volume: 398 Issue: 2 Pages: 857-866
  • DOI: 10.1016/j.jmaa.2012.09.045
  • Peer Reviewed
• [Journal Article] Positive solutions of semilinear elliptic equations with small perturbations (2013)
  • Author(s): Ryuji Kajikiya
  • Journal Title: Proc. Amer. Math. Soc. Volume: 141 Issue: 4 Pages: 1335-1342
  • DOI: 10.1090/s0002-9939-2012-11569-2
  • Peer Reviewed
• [Journal Article] Non-even positive solutions of the Emden-Fowler equations with sign-changing weights (2013)
  • Author(s): Ryuji Kajikiya
  • Journal Title: Proc. Roy. Soc. Edinburgh Sect. A Volume: 143 Issue: 3 Pages: 631-642
  • DOI: 10.1017/s0308210511001594
  • Peer Reviewed
• [Journal Article] Least energy solutions of the Emden-Fowler equation in hollow thin symmetric domains (2013)
  • Author(s): Ryuji Kajikiya
  • Journal Title: J. Math. Anal. Appl., Volume: 406 Issue: 1 Pages: 277-286
  • DOI: 10.1016/j.jmaa.2013.04.068
  • Peer Reviewed
• [Journal Article] Stability analysis of asymptotic profiles for sign-changing solutions to fast diffusion equations (2013)
  • Author(s): Goro Akagi and Ryuji Kajikiya
  • Journal Title: Manuscripta Math. Volume: 141 Issue: 3-4 Pages: 559-587
  • DOI: 10.1007/s00229-012-0583-9
  • NAID: 120005455953
  • Peer Reviewed
• [Journal Article] Least energy solutions and group invariant solutions of the Henon equation. (2013)
  • Author(s): Ryuji Kajikiya
  • Journal Title: Advanced Nonlinear Studies Volume: 13 Pages: 721-738
  • Peer Reviewed
• [Journal Article] Sobolev norm estimates of solutions for the sublinear Emden-Fowler equation (2013)
  • Author(s): Ryuji Kajikiya
  • Journal Title: Opuscula Mathematica Volume: 33 Issue: 4 Pages: 713-723
  • DOI: 10.7494/opmath.2013.33.4.713
  • Peer Reviewed
• [Journal Article] Two positive solutions for one-dimensional p-Laplacian with a singular weight. (2013)
  • Author(s): R. Kajikiya, Y.-H. Lee and I. Sim
  • Journal Title: Topological Methods in Nonlinear Analysis Volume: 42 Pages: 427-448
  • Peer Reviewed
• [Journal Article] 1. Non-radial least energy solutions of the generalized Henon equation (2012)
  • Author(s): R. Kajikiya
  • Journal Title: J. Differential Equations Volume: 252 Issue: 2 Pages: 1987-2003
  • DOI: 10.1016/j.jde.2011.08.032
  • Peer Reviewed
• [Journal Article] Non-even least energy solutions of the Emden-Fowler equation (2012)
  • Author(s): R. Kajikiya
  • Journal Title: Proc. Amer. Math. Soc. Volume: 140 Issue: 4 Pages: 1353-1362
  • DOI: 10.1090/s0002-9939-2011-11172-9
  • Peer Reviewed
• [Journal Article] Mountain pass theorem in ordered Banach spaces and its applications to semilinear elliptic equations (2012)
  • Author(s): R. Kajikiya
  • Journal Title: Nonlinear Differential Equations and Applications Volume: 19 Issue: 2 Pages: 159-175
  • DOI: 10.1007/s00030-011-0122-5
  • Peer Reviewed
• [Journal Article] Least energy solutions of the generalized Henon equation in reflectionally symmetric or point symmetric domains (2012)
  • Author(s): R. Kajikiya
  • Journal Title: J. Differential Equations Volume: 253 Issue: 5 Pages: 1621-1646
  • DOI: 10.1016/j.jde.2012.05.004
  • Peer Reviewed
• [Journal Article] Non-even positive solutions of the one dimensional p-Laplace Emden-Fowler equation (2012)
  • Author(s): R. Kajikiya
  • Journal Title: Applied Mathematics Letters Volume: 25 Issue: 11 Pages: 1891-1895
  • DOI: 10.1016/j.aml.2012.02.057
  • Peer Reviewed
• [Presentation] Nonradial positive solutions of the p-Laplace Emden-Fowler equation. (2016)
  • Author(s): 梶木屋龍治
  • Organizer: 日本数学会
  • Place of Presentation: 筑波大学
  • Year and Date: 2016-03-16
• [Presentation] Symmetric solutions of semilinear elliptic equations. (2015)
  • Author(s): 梶木屋龍治
  • Organizer: 3rd Chile-Japan Workshop on Nonlinear PDEs
  • Place of Presentation: 大阪大学
  • Year and Date: 2015-12-08
  • Int'l Joint Research
• [Presentation] A priori estimate for the first eigenvalue of the p-Laplacian. (2015)
  • Author(s): 梶木屋龍治
  • Organizer: 日本数学会
  • Place of Presentation: 京都産業大学
  • Year and Date: 2015-09-13
• [Presentation] Symmetric mountain pass lemma and sublinear elliptic equations. (2015)
  • Author(s): 梶木屋龍治
  • Organizer: Equadiff2015
  • Place of Presentation: Universite Claude Bernard Lyon
  • Year and Date: 2015-07-06
  • Int'l Joint Research
• [Presentation] Infinitely many solutions for sublinear elliptic equations. (2015)
  • Author(s): 梶木屋龍治
  • Organizer: 2015 International Workshop on Nonlinear PDE and Applications.
  • Place of Presentation: 釜山大学
  • Year and Date: 2015-06-11
  • Int'l Joint Research
• [Presentation] Symmetric mountain pass lemma and sublinear elliptic equations. (2015)
  • Author(s): 梶木屋龍治
  • Organizer: 日本数学会
  • Place of Presentation: 明治大学
  • Year and Date: 2015-03-21 – 2015-03-24
• [Presentation] Symmetric mountain pass lemma and sublinear elliptic equations (2015)
  • Author(s): 梶木屋龍治
  • Organizer: 常微分方程式ワークショップ 松山 2015
  • Place of Presentation: 愛媛大学
  • Year and Date: 2015-03-10
• [Presentation] 半線形楕円型方程式の群不変解の存在 (2014)
  • Author(s): 梶木屋龍治
  • Organizer: 熊本大学応用解析セミナー
  • Place of Presentation: 熊本大学
  • Year and Date: 2014-10-04
  • Invited
• [Presentation] Bifurcation of positive solutions for the one dimensional (p,q)-Laplace equation. (2014)
  • Author(s): 田中 視英子, 田中 敏, 梶木屋龍治
  • Organizer: 日本数学会
  • Place of Presentation: 広島大学
  • Year and Date: 2014-09-25 – 2014-09-28
• [Presentation] 1次元(p,q)ラプラス方程式の正値解の分岐 (2014)
  • Author(s): 梶木屋龍治
  • Organizer: 熊本大学における微分方程式セミナー
  • Place of Presentation: 熊本大学
  • Year and Date: 2014-09-09 – 2014-09-10
• [Presentation] Group invariant positive solutions of the generalized Henon equation. (2014)
  • Author(s): R. Kajikiya
  • Organizer: The 10th AIMS Conference on Dynamical Systems, Differential Equations and Applications.
  • Place of Presentation: スペイン, マドリッド
  • Year and Date: 2014-07-07 – 2014-07-11
• [Presentation] 1次元 Henon 方程式の３つの正値解の存在 (2014)
  • Author(s): 梶木屋龍治
  • Organizer: 振動理論ワークショップ -- 金沢 2014
  • Place of Presentation: 金沢大学サテライト・プラザ
• [Presentation] 一般化された Henon 方程式の３つの正値解の存在 (2014)
  • Author(s): 梶木屋龍治
  • Organizer: 日本数学会
  • Place of Presentation: 学習院大学
• [Presentation] Positive solutions of the Emden-Fowler equation in hollow thin symmetric domains. (2014)
  • Author(s): 梶木屋龍治
  • Organizer: 2014 International Workshop on Nonlinear PDE and Applications
  • Place of Presentation: Pusan National University, Pusan, Korea
  • Invited
• [Presentation] 穴の空いた対称領域におけるEmden-Fowler方程式の正値解の多重存在 (2013)
  • Author(s): 梶木屋龍治
  • Organizer: 大阪市立大学 「南大阪応用数学セミナー」
  • Place of Presentation: 大阪市立大学
• [Presentation] Positive solutions of the p-Laplace Emden-Fowler equation in hollow thin symmetric domains. (2013)
  • Author(s): 梶木屋龍治
  • Organizer: Equadiff13.
  • Place of Presentation: Charles University, Prague, Czech Republic
• [Presentation] Positive symmetric solutions of the generalized Henon equation (2013)
  • Author(s): 梶木屋龍治
  • Organizer: 岡山理科大学における微分方程式セミナー
  • Place of Presentation: 岡山理科大学
• [Presentation] 優線形項と劣線形項を持つ楕円型方程式の2種類の解の列の存在について (2013)
  • Author(s): 内免大輔, 梶木屋龍治
  • Organizer: 日本数学会
  • Place of Presentation: 愛媛大学
• [Presentation] Partially symmetric solutions of the generalized Henon equation (2013)
  • Author(s): 梶木屋龍治
  • Organizer: 日本数学会
  • Place of Presentation: 愛媛大学
• [Presentation] 劣線形放物型方程式の定常解の安定性について (2013)
  • Author(s): 赤木剛朗, 梶木屋龍治
  • Organizer: 日本数学会
  • Place of Presentation: 愛媛大学
• [Presentation] Three positive solutions of the generalized Henon equation (2013)
  • Author(s): 梶木屋龍治
  • Organizer: 岡山理科大学 「半田山微分方程式セミナー」
  • Place of Presentation: 岡山理科大学
• [Presentation] 一般化された１次元 Henon 方程式の３つの正値解の存在 (2013)
  • Author(s): 梶木屋龍治
  • Organizer: 横浜国立大学 「横浜数学セミナー」
  • Place of Presentation: 横浜国立大学
• [Presentation] 穴の空いた対称領域におけるEmden-Fowler方程式の正値解の非対称性 (2013)
  • Author(s): 梶木屋龍治
  • Organizer: 日本数学会
  • Place of Presentation: 京都大学
• [Presentation] 穴の空いた対称領域におけるEmden-Fowler方程式の正値解の多重存在 (2013)
  • Author(s): 梶木屋龍治
  • Organizer: 日本数学会
  • Place of Presentation: 京都大学
• [Presentation] １次元 pラプラス方程式の解の多重分岐 (2012)
  • Author(s): 梶木屋龍治
  • Organizer: 日本数学会
  • Place of Presentation: 九州大学
• [Presentation] 点対称及び面対称領域におけるHenon 方程式の最小エネルギー解 (2012)
  • Author(s): 梶木屋龍治
  • Organizer: 日本数学会
  • Place of Presentation: 九州大学
• [Presentation] 一般の対称領域における Henon 方程式の解の非対称性 (2012)
  • Author(s): 梶木屋龍治
  • Organizer: 日本数学会
  • Place of Presentation: 九州大学
• [Presentation] Symmetry and stability of asymptotic profiles for fast diffusion equations (2012)
  • Author(s): 赤木 剛朗
  • Organizer: 日本数学会
  • Place of Presentation: 九州大学
• [Presentation] Symmetry breaking of least energy solutions of Emden-Fowler equations (2012)
  • Author(s): 赤木 剛朗
  • Organizer: 日本数学会
  • Place of Presentation: 九州大学