Asymmetric solutions of elliptic partial differential equations
| Project/Area Number |
16K05236
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Multi-year Fund |
|---|
| Section | 一般 |
|---|
| Research Field |
Mathematical analysis
|
|---|
| Research Institution | Saga University |
|---|
Principal Investigator |
|
|---|
| Project Period (FY) |
2016-04-01 – 2022-03-31
|
| Project Status |
Completed (Fiscal Year 2021)
|
| Budget Amount *help |
¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000)
Fiscal Year 2019: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2018: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2017: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2016: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
|
|---|
| Keywords | 楕円型偏微分方程式 / 変分法 / 境界値問題 / 対称解 / ２点境界値問題 / 常微分方程式 / 非対称解 / 半線形楕円型偏微分方程式 / nodal solution / 正値解 / 球対称解 / 放物型偏微分方程式 / 定常解の安定性 / 最小エネルギー解 / p ラプラス方程式 / (p,q) ラプラス方程式 / 定常解 / 無限に多くの解 / 関数方程式 / 非線形偏微分方程式 / 非線形楕円型偏微分方程式 / 解の非対称性 |
|---|
| Outline of Final Research Achievements |
We study the p-Laplace equation on the Dirichlet boundary condition. Under a very weak condition on the nonlinear term, we prove the existence of a sequence of solutions which converges to the zero solution. For some elliptic equations with parameters, we decide whether the zero solution is an accumulation point or an isolated point in the set of all solutions.
We study the p-Laplace Emden-Fowler equation with a radial and sign-changing weight in the unit ball under the Dirichlet boundary condition. We show that if the weight function is negative in the unit ball except for a small neighborhood of the boundary and positive at somewhere in this neighborhood, then no least energy solution is radially symmetric. Here a least energy solution is defined by the minimizer of the Rayleigh quotient.
|
| Academic Significance and Societal Importance of the Research Achievements |
この研究は, 数学や物理学, 工学などにおいて非常に重要な偏微分方程式である楕円型偏微分方程式の解の性質を調べるものである. 解を無限に多くもつ楕円型偏微分方程式の研究は, 今までにも行われてきた. 本研究では, 極めて弱い仮定のもとに解の無限個の存在を証明している.
また, 方程式が対称性をもつにもかかわらずに, 解が非対称になることが起きる. このような対称性の崩れは, 解の分岐理論にも密接に関係していて, 非対称解の研究は, 楕円型偏微分方程式論において極めて重要である.
|
Report
(7 results)
Research Products
(47 results)
