Grushin Operator Research Articles

Liouville results for stable solutions of weighted elliptic equations involving the Grushin operator

Liouville results for stable solutions of weighted elliptic equations involving the Grushin operator

On some operator-valued Fourier pseudo-multipliers associated to Grushin operators

This is a continuation of our work [2,3] where we have initiated the study of sparse domination and quantitative weighted estimates for Grushin pseudo-multipliers. In this article, we further extend this analysis to study analogous estimates for a family of operator-valued Fourier pseudo-multipliers associated to Grushin operators G=−Δx′−|x′|2Δx″ on Rn1+n2.

Liouville-type theorems for double-phase problems involving the Grushin operator

In this paper, we are concerned with the double-phase problem involving the Grushin operator in the whole space \mathbb{R}^{N}=\mathbb{R}^{N_1}\times\mathbb{R}^{N_2} - \operatorname{div}_{G} (|\nabla_{G} u|^{p-2}\nabla_{G} u + w(z) |\nabla_{G} u|^{q-2}\nabla_{G} u) = f(z)|u|^{r-1}u, where \nabla_{G} is the Grushin gradient, \Delta_{G} is the Grushin operator, q\geq p \geq 2 , r>q-1 and w, f \in L^{1}_{\mathrm{loc}}(\mathbb{R}^{N}) are two nonnegative functions satisfying some growth conditions at infinity. Our purpose is to establish some Liouville-type theorems for stable weak solutions or for weak solutions which are stable outside a compact set of the equation above.

A Brézis–Nirenberg Type Problem for a Class of Degenerate Elliptic Problems Involving the Grushin Operator

A Brézis–Nirenberg Type Problem for a Class of Degenerate Elliptic Problems Involving the Grushin Operator

On L2-boundedness of pseudo-multipliers associated to the Grushin operator

In this article we define analogues of pseudo-differential operators associated to the joint functional calculus of the Grushin operator using their spectral resolution, and study Calderón–Vaillancourt-type theorems for these operators.

Existence of the solution for a class of the semilinear degenerate elliptic equation involving the Grushin operator in R2$\mathbb {R}^2$: The interaction between Grushin's critical exponent and exponential growth

AbstractIn this work, we study the existence of the solution for a class of semilinear degenerate elliptic equations in the whole space involving the Grushin operator and a nonlinearity that involves the Grushin's critical growth and the exponential growth. The existence of at least one nontrivial weak solution is done by combining the mountain‐pass theorem, Trudinger–Moser inequality, and a version of a result due to Lions for the exponential growth in in the corresponding weighted Sobolev space.

Sparse Bounds for Pseudo-multipliers Associated to Grushin Operators, I

In this article, we prove sharp quantitative weighted $$L^p$$ -estimates for Grushin pseudo-multipliers satisfying Hörmander’s condition as an application of pointwise domination of Grushin pseudo-multipliers by appropriate sparse operators.

On Stable Weak Solutions of the Weighted Static Choquard Equation Involving Grushin Operator

On Stable Weak Solutions of the Weighted Static Choquard Equation Involving Grushin Operator

Liouville type theorem for weak solutions of nonlinear system for Grushin operator

<abstract><p>In this paper, we prove Liouville type theorem of positive weak solution of nonlinear system for Grushin operator. We give some integral inequalities, which combine the method of moving plane with the integral inequality to get the result for nonlinear system.</p></abstract>

An optimal multiplier theorem for Grushin operators in the plane, I

Let \mathcal{L} = -\partial_x^2 - V(x) \partial_y^2 be the Grushin operator on \mathbb{R}^2 with coefficient V \colon \mathbb{R} \to [0,\infty) . Under the sole assumptions that V(-x) \simeq V(x) \simeq xV'(x) and x^2 |V''(x)| \lesssim V(x) , we prove a spectral multiplier theorem of Mihlin–Hörmander type for \mathcal{L} , whose smoothness requirement is optimal and independent of V . The assumption on the second derivative V'' can actually be weakened to a Hölder-type condition on V' . The proof hinges on the spectral analysis of one-dimensional Schrödinger operators, including universal estimates of eigenvalue gaps and matrix coefficients of the potential.

A Berestycki–Lions type result for a class of degenerate elliptic problems involving the Grushin operator

In this work we study the existence of nontrivial solution for the following class of semilinear degenerate elliptic equations \[ -\Delta_{\gamma} u + a(z)u = f(u) \quad \mbox{in}\ \mathbb{R}^{N}, \]where $\Delta _{\gamma }$ is known as the Grushin operator, $z:=(x,y)\in \mathbb {R}^{m}\times \mathbb {R}^{k}$ and $m+k=N\geqslant 3$, $f$ and $a$ are continuous function satisfying some technical conditions. In order to overcome some difficulties involving this type of operator, we have proved some compactness results that are crucial in the proof of our main results. For the case $a=1$, we have showed a Berestycki–Lions type result.

A p-specific spectral multiplier theorem with sharp regularity bound for Grushin operators

In a recent work, Chen and Ouhabaz proved a p-specific L^p-spectral multiplier theorem for the Grushin operator acting on {mathbb {R}}^{d_1}times {mathbb {R}}^{d_2} which is given by L=-∑j=1d1∂xj2-(∑j=1d1|xj|2)∑k=1d2∂yk2.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} L =-\\sum _{j=1}^{d_1} \\partial _{x_j}^2 - \\Bigg ( \\sum _{j=1}^{d_1} |x_j|^2\\Bigg ) \\sum _{k=1}^{d_2}\\partial _{y_k}^2. \\end{aligned}$$\\end{document}Their approach yields an L^p-spectral multiplier theorem within the range 1< ple min { 2d_1/(d_1+2), 2(d_2+1)/(d_2+3) } under a regularity condition on the multiplier which is sharp only when d_1ge d_2. In this paper, we improve on this result by proving L^p-boundedness under the expected sharp regularity condition s>(d_1+d_2)(1/p-1/2). Our approach avoids the usage of weighted restriction type estimates which played a key role in the work of Chen and Ouhabaz, and is rather based on a careful analysis of the underlying sub-Riemannian geometry and restriction type estimates where the multiplier is truncated along the spectrum of the Laplacian on {mathbb {R}}^{d_2}.

Correction to: An Optimal Multiplier Theorem for Grushin Operators in the Plane, II

Correction to: An Optimal Multiplier Theorem for Grushin Operators in the Plane, II

Probabilistic local well-posedness for the Schrödinger equation posed for the Grushin Laplacian

We study the local well-posedness of the nonlinear Schrödinger equation associated to the Grushin operator with random initial data. To the best of our knowledge, no well-posedness result is known in the Sobolev spaces Hk when k⩽32. In this article, we prove that there exists a large family of initial data such that, with respect to a suitable randomization in Hk, k∈(1,32], almost-sure local well-posedness holds. The proof relies on bilinear and trilinear estimates.

An Optimal Multiplier Theorem for Grushin Operators in the Plane, II

In a previous work we proved a spectral multiplier theorem of Mihlin--H\"ormander type for two-dimensional Grushin operators $-\partial_x^2 - V(x) \partial_y^2$, where $V$ is a doubling single-well potential, yielding the surprising result that the optimal smoothness requirement on the multiplier is independent of $V$. Here we refine this result, by replacing the $L^\infty$ Sobolev condition on the multiplier with a sharper $L^2$ condition. As a consequence, we obtain the sharp range of $L^1$ boundedness for the associated Bochner--Riesz means. The key new ingredient of the proof is a precise pointwise estimate in the transition region for eigenfunctions of one-dimensional Schr\"odinger operators with doubling single-well potentials.

Null Controllability of the Parabolic Spherical Grushin Equation

We investigate the null controllability property of the parabolic equation associated with the Grushin operator defined by the canonical almost-Riemannian structure on the 2-dimensional sphere 𝕊2. This is the natural generalization of the Grushin operator 𝒢 = ∂x2 + x2∂y2 on ℝ2 to this curved setting and presents a degeneracy at the equator of 𝕊2. We prove that the null controllability is verified in large time when the control acts as a source term distributed on a subset ω̅ = {(x1, x2, x3) ∈ 𝕊2 | α &lt; | x3 | &lt; β} for some 0 ≤ α &lt; β ≤ 1. More precisely, we show the existence of a positive time T* &gt; 0 such that the system is null controllable from ω̅ in any time T ≥ T*, and that the minimal time of control from ω̅ satisfies Tmin ≥ log(1/√(1 - α2)) . Here, the lower bound corresponds to the Agmon distance of ω̅ from the equator. These results are obtained by proving a suitable Carleman estimate using unitary transformations and Hardy-Poincaré type inequalities to show the positive null-controllability result. The negative statement is proved by exploiting an appropriate family of spherical harmonics, concentrating at the equator, to falsify the uniform observability inequality.

Null controllability and inverse source problem for stochastic Grushin equation with boundary degeneracy and singularity

In this paper, we consider a null controllability and an inverse source problem for stochastic Grushin equation with boundary degeneracy and singularity. We construct two special weight functions to establish two Carleman estimates for the whole stochastic Grushin operator with singular potential by a weighted identity method. One is for the backward stochastic Grushin equation with singular weight function. We then apply it to prove the null controllability for stochastic Grushin equation for any T and any degeneracy γ &gt; 0, when our control domain touches the degeneracy line {x = 0}. In order to study the inverse source problem of determining two kinds of sources simultaneously, we prove the other Carleman estimate, which is for the forward stochastic Grushin equation with regular weight function. Based on this Carleman estimate, we obtain the uniqueness of the inverse source problem.

ON THE TRANSITION DENSITY FUNCTION OF THE DIFFUSION PROCESS GENERATED BY THE GRUSHIN OPERATOR

The short-time asymptotic behavior of the transition density function of the diffusion process generated by the general Grushin operator will be investigated, by using its explicit expression in terms of expectation. Further the dependence on of the asymptotics will be seen.

Liouville type theorems for stable solutions of elliptic system involving the Grushin operator

&lt;p style='text-indent:20px;'&gt;We examine the following degenerate elliptic system:&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE1"&gt; \begin{document}$ -\Delta_{s} u \! = \! v^p, \quad -\Delta_{s} v\! = \! u^\theta, \;\; u, v&amp;gt;0 \;\;\mbox{in }\; \mathbb{R}^N = \mathbb{R}^{N_1}\times \mathbb{R}^{N_2}, \quad\mbox{where}\;\; s \geq 0\;\; \mbox{and} \;\;p, \theta \!&amp;gt;\!0. $\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;We prove that the system has no stable solution provided &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ p, \theta &amp;gt;0 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; and &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ N_s: = N_1+(1+s)N_2&amp;lt; 2 + \alpha + \beta, $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; where&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE2"&gt; \begin{document}$ \alpha = \frac{2(p+1)}{p\theta - 1} \quad\mbox{and} \quad \beta = \frac{2(\theta +1)}{p\theta - 1}. $\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;This result is an extension of some results in [&lt;xref ref-type="bibr" rid="b15"&gt;15&lt;/xref&gt;]. In particular, we establish a new integral estimate for &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$ u $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; and &lt;inline-formula&gt;&lt;tex-math id="M4"&gt;\begin{document}$ v $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; (see Proposition 1.1), which is crucial to deal with the case &lt;inline-formula&gt;&lt;tex-math id="M5"&gt;\begin{document}$ 0 &amp;lt; p &amp;lt; 1. $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;&lt;/p&gt;

Nonexistence of positive solutions to a system of elliptic inequalities involving the Grushin operator

We study the nonexistence of positive solutions to the degenerate system of elliptic inequalities where and is the Grushin operator. Recently, it has been established in Le et al. (Liouville-type theorems for sub-elliptic systems involving -Laplacian. Complex Var Ellip Equ. DOI:10.1080/17476933.2020.1816981) that the system has no positive classical solution in the case where the exponents satisfy one of the following conditions or , p, q>0 and , p, q>0, pq>1 and . Here, is the homogeneous dimension associated to the Grushin operator . The nonexistence of positive solutions is left open in the borderline case In this paper, we shall prove that the system has no positive classical solution in this case. In particular, when , this result agrees with the well-known one for the system involving the Laplace operator. Our approach is based on the development of the idea of Serrin and Zou in (Non-existence of positive solutions of Lane–Emden systems. Differ Integral Equ. 1996;9(4):635–653) and the spherical mean formula for the Grushin operator.