The main result of this thesis is the proof of higher-order Poincar´e inequalities for Baouendi-Grushin vector fields on non-smooth domains. The finding is successfully used in the study of the higher-order Fisher-KPP equation. I would like to thank my thesis supervisor, Associate Professor Durvudkhan Suragan, who guided me throughout the entire creation process, for his fruitful advice during the work.
Notations
Preliminaries
Although in this treatise we are dealing with a non-classical case, assertion plays a crucial role here. In fact, we simply follow the restrictions on the set set by Ilyin and Poznyak. Another classical result of vector fields, which will play an essential role in this thesis, is Green's first identity.
Historical review
It is essential to note that the author does not mention the Baouendi-Grushin operator itself. The hypoelliptic result is obtained by extending the author's discoveries for the case when the operator (1.4) is degenerate on the boundary of the domain [Bao67]. Despite the chronological pioneering of Baouendi in this discovery, Grushin rightly deserves his merits in the study of the operator.
In 2009, Dario Daniele Monticelli and Kevin Ray Payne achieve the most groundbreaking result of the theory of the Baouendi-Grushin operator to date: existence and positivity of the main (first) eigenvalue and ae While the boundary is still the type of Tricomi, the coefficient was further generalized. A good insight into why this happened is given in [PHNS17]: ..The multidimensional case, however, is quite different and there is no general understanding of the situation.
Celebrated inequalities
The greatest success in studying Rellich-type inequalities for Baouendi-Grushin vector fields is the Lp Rellich inequality of Kombe and Yener [KY17]. The Caffarelli-Kohn-Nirenberg inequality has been successfully translated into Baouendi-Grushin vector fields by Manli Song and Wenjuan Li. Although the previous setting is the most common at this time, the constant C is still unknown.
Rather, in a narrower context, the Caffarelli-Kohn-Nirenberg inequalities for Baouendi-Grushin vector fields are established with a sharp constant. It is also worth mentioning the works [MPP19], [AL11], [LRY19] as a contribution to the study of famous inequalities involving Baouendi-Grushin vector fields. In [MPP19] the authors obtained the L2-Poincare inequality for Baouendi-Grushin vector fields with a non-explicit constant, while in [AL11] and [LRY19] the authors showed refined versions of the Hardy inequality for Baouendi-Grushin vector fields.
Pillar works
It is also assumed that the initial data u0 ∈C1(V) is a non-negative and non-trivial function with u0 = 0 on∂V. This is the most important article for the current work as the main result of the entire thesis is fully inspired by their findings. The authors complement the result of the previous work [SY21]: one can consider Theorem 2.1 as an inductive extension of Theorem 1.1.
Here ee1 is the ground state of the (minus) Dirichlet Laplacian in V and eλ1 is the corresponding eigenvalue. And the last article is [KST21] by Ardak Kashkynbayev, Durvudkhan Suragan and Berikbol Torebek, which inspired the Applications chapter of this thesis. The results of the chapter are the higher-order Baouendi-Grushin analogues of the theorems from the paper.
Structure
We will consider type-K domains with respect to yj axes, which means that the previous statement only holds for axes y1, .., yk. Theorem (Green's first identity for Baouendi-Grushin vector fields on type-K domains with respect to yj axes).
Integrability criterion for degenerate domains
Representation formulae
The higher-order Poincar´ e inequalities
Proof of the Gauss-Ostrogradsky divergence theorem for Baouendi-Grushin vector fields on K-type domains with respect to the yj axes. Due to the projectability property of V, for each zj ∈ Vyj there is one segment — the intersection of V and the line that crosses zj and is parallel to the yj axis. Since the set V is intersected at most twice by lines parallel to the axes y1, .., yk, we continue ˆ.
The negative before the integral over the lower part of ∂V changes the direction of the outward circulation. Proof of Green's first identity for Baouendi-Grushin vector fields on type-K domains with respect to yj axes. Again we denote the orthogonal projection on the hyperplane without yj-axis as Vyj = Proj.
For a fixed zj ∈ Vyj, we consider the line passing through zj and parallel to the yj axis.
Proof of Integrability criterion for degenerate domains
Proof of Remainders
Since all functions are integrable according to the reasoning of the integrability criterion for degenerate domains (the integrands can be considered as O(|∇γ2ru|2) functions), we have. As an application, we consider the higher-order Fisher-KPP equation on Baouendi-Grushin vector fields. We discuss the boundedness of global solutions, the asymptotic behavior of global solutions, and the exploding of solutions.
Moreover, we extend the obtained results to the higher-order time-fractional Fisher-KPP equation on Baouendi-Grushin vector fields. 4.2) Here we consider a subelliptic version of the Fisher-KPP equation with the Baouendi-Grushin sub-Laplace-∆γ of a strange power in the space variables: 4.5) Despite existence being the first priority when studying math problems, let's leave this topic out of scope.
We prove the boundedness of global solutions when the initial data are continuous and bounded between 0 and 1. To our knowledge, these results are new as we have not been able to find them in the existing literature. We believe it is of fundamental interest to the theory of higher-order subelliptic operators since the sub-Laplacian serves as a model operator.
We extend our arguments from the first part of this chapter to a problem involving the initial limits for the higher-order time-fractional Fisher-KPP equation.
Main results
Boundedness of global solutions
Large-time behavior of global solutions
Blow-up of solutions
Time-fractional extension
Boundedness of global solutions
Then repeat the same procedure as in the proof of the theorem Boundedness of global solutions and use an estimate [AAK17]. The case u61 is proved in a similar way as in the proof of the theorem Boundedness of global solutions.
Large-time behavior of global solutions
Blow-up of solutions
In this work we have proven the higher order Poincar´e inequalities for Baouendi-Grushin vector fields. The fundamental contribution to the proof undoubtedly belongs to Tohru Ozawa and Durvudkhan Suragan [OS20]. To be precise, we moved on to the case where the domain has the degeneracy {x = 0}: you have to treat this carefully because an intersection with the hyperlane opens the way to a possible division by zero.
Another result to appreciate is the classical vector calculus theorems for Baouendi-Grushin vector fields. I am extremely grateful for the development of these theorems to Vladimir Aleksandrovich Ilyin and Eduard Grigoryevich Pozniak, who provided thorough proofs of analogous theorems in the usual case for a special class of domains and showed a way to extend the result for a common type of domains [IP82, ch. As far as we have searched, the classical vector field theorems for Baouendi-Grushin vector fields have not been proved or even correctly formulated.
Thus, the performed tests can be considered as an educational contribution to the theory of Baouendi-Grushin vector fields. Torebek [KST21]: all the results they obtained for the Fisher-KPP equation in the Heisenberg group translate completely to the case of Baouendi-Grushin vector fields. We have successfully extended their findings by considering the highest-order Baouendi-Grushin operator of an odd power.
Gellerstedt.Sur un probl`eme aux Limites pour une Equation Lin´eaire aux D´eriv´ees Partielles du Second Ordre de Type Mixte. Strong inequalities for Landau Hamiltonian and for Baouendi-Grushin operator with aharonov-bohm type magnetic field. Poincar´e inequalities for Sobolev spaces with matrix value weights and applications to degenerate partial differential equations.
On the existence and uniqueness of a generalized solution of the Protter problem for equations of the (3+1)-D Keldysh type.
