[Paper Review] Morse theory methods for quasi-linear elliptic systems of higher order

Guangcun Lu|arXiv (Cornell University)|Feb 22, 2017

Advanced Mathematical Modeling in Engineering46 references3 citations

TL;DR

This paper develops a new local Morse theory for non-twice continuously differentiable functionals on Hilbert spaces, introducing a generalized Gromoll-Meyer splitting theorem and a weaker Marino-Prodi perturbation result. These tools enable the generalization of critical point and bifurcation theorems, which are then applied to study quasi-linear elliptic systems of higher order via variational methods under Hypothesis $\mathfrak{F}_{p,N}$, extending prior results beyond $C^2$ functionals.

ABSTRACT

We develop the local Morse theory for a class of non-twice continuously differentiable functionals on Hilbert spaces, including a new generalization of the Gromoll-Meyer's splitting theorem and a weaker Marino-Prodi perturbation type result. With them some critical point theorems and famous bifurcation theorems are generalized. Then we show that these are applicable to studies of quasi-linear elliptic equations and systems of higher order given by multi-dimensional variational problems as in (1.3).

Motivation & Objective

• To extend local Morse theory to a class of non-twice continuously differentiable functionals on Hilbert spaces, overcoming limitations of classical Gromoll-Meyer theory.
• To establish a new splitting theorem and a weaker Marino-Prodi-type perturbation result for such functionals.
• To generalize critical point and bifurcation theorems for potential operators in infinite-dimensional settings.
• To apply the developed theory to quasi-linear elliptic systems of higher order arising from variational problems.
• To verify the applicability of the new framework under Hypothesis $\mathfrak{F}_{p,N}$ for $p \geq 2$ and $m$-th order systems.

Proposed method

• Introduce a generalized Gromoll-Meyer splitting theorem for $C^1$-functionals that are not necessarily $C^2$, based on critical point structure and local splitting near non-degenerate critical points.
• Develop a parameterized splitting and shifting theorem for families of potential operators, enabling bifurcation analysis in non-smooth settings.
• Use a new implicit function theorem for families of potential operators to handle perturbations in the absence of full $C^2$ regularity.
• Establish a weaker version of the Marino-Prodi perturbation result applicable to non-$C^2$ functionals, ensuring robustness in bifurcation analysis.
• Apply the theory to functionals defined on Sobolev spaces $W_0^{m,p}(\Omega)$, verifying the $(PS)$ and $(C)$-conditions under Hypothesis $\mathfrak{F}_{p,N}$.
• Derive Morse inequalities and bifurcation results for higher-order quasi-linear elliptic systems using the generalized critical point theory.

Experimental results

Research questions

• RQ1 Can Morse theory be extended to non-$C^2$ functionals on Hilbert spaces, particularly in the context of higher-order elliptic systems?
• RQ2 What is the appropriate generalization of the Gromoll-Meyer splitting theorem for $C^1$-functionals that are not twice continuously differentiable?
• RQ3 How can bifurcation theorems be generalized when the functional lacks $C^2$ smoothness?
• RQ4 Under what conditions does the $(PS)$ or $(C)$-condition hold for functionals associated with higher-order quasi-linear elliptic systems?
• RQ5 Can the new theory be applied to variational problems of the form (1.3) under Hypothesis $\mathfrak{F}_{p,N}$ for $p \geq 2$?

Key findings

• A new generalized Gromoll-Meyer splitting theorem is established for $C^1$-functionals that are not necessarily $C^2$, enabling local Morse theory in non-smooth settings.
• A weaker version of the Marino-Prodi perturbation result is derived, allowing bifurcation analysis under reduced regularity assumptions.
• The $(PS)$ and $(C)$-conditions are verified for the functional $\mathcal{F}$ under Hypothesis $\mathfrak{F}_{p,N}$, ensuring compactness for critical point sequences.
• Morse inequalities are proven for the functional $\mathcal{F}$, relating critical groups to topological invariants of the underlying space.
• Bifurcation theorems are generalized for quasi-linear elliptic systems of higher order, extending results of Chow-Lauterbach and Rabinowitz to non-$C^2$ settings.
• The theory is successfully applied to multi-dimensional variational problems of the form (1.3), demonstrating its utility in studying higher-order elliptic systems.