Related papers: A compactness result for an equation with Holderia…
In this paper, we are devoted to establishing the compactness and existence results of the solutions to the fractional Nirenberg problem for $n=3,$ $\sigma=1/2,$ when the prescribing $\sigma$-curvature function satisfies the…
In this note, we give the affirmative answer of the question in [18], which is a compactness result of the non-radial Sobolev spaces. As an application, we show the existence of an extremal function of the critical Hardy inequality under…
We prove a compactness theorem for K\"ahler metrics with various bounds on Ricci curvature and the $\mathcal I$ functional. We explore applications of our result to the continuity method and the Calabi flow.
We study approximations of compact linear multivariate operators defined over Hilbert spaces. We provide necessary and sufficient conditions on various notions of tractability. These conditions are mainly given in terms of sums of certain…
In this paper, the initial and boundary problem of the difference equation which is a discretization of the semi-linear heat equation. The difference equation derived by discretizing the semi-linear heat equation has solutions which show…
We consider the nonlinear Schr\"odinger equation on ${\mathbb R}^N $, $N\ge 1$, \begin{equation*} \partial _t u = i \Delta u + \lambda | u |^\alpha u \quad \mbox{on ${\mathbb R}^N $, $\alpha>0$,} \end{equation*} with $\lambda \in {\mathbb…
In this paper, we extend the analysis of the subcritical approximation of the Nirenberg problem on spheres recently conducted in \cite{MM19, MM}. Specifically, we delve into the scenario where the sequence of blowing up solutions exhibits a…
We consider the inhomogeneous nonlinear Schr\"odinger equation (INLS) in $\mathbb{R}^N$, $N \geq 1$, $$i \partial_t u + \Delta u + |x|^{-b} |u|^{p-1}u = 0,$$ with finite-variance initial data $u_0 \in H^1(\mathbb{R}^N)$. We extend the…
We study the possibility of finite-time blow-up for a two dimensional Broadwell model. In a set of rescaled variables, we prove that no self-similar blow-up solution exists, and derive some a priori bounds on the blow-up rate. In the final…
We investigate the 2D combustion model with Dirichlet boundary conditions and slip boundary conditions in bounded domains. The global existence of weak and strong solutions and the uniqueness of strong solutions are obtained provided the…
We are concerned with existence results for a critical problem of Brezis-Nirenberg Type involving an integro-differential operator. Our study includes the fractional Laplacian. Our approach still applies when adding small singular terms. It…
We obtain a Struwe type global compactness result for a class of nonlinear nonlocal problems involving the fractional $p-$Laplacian operator and nonlinearities at critical growth.
We establish the existence of compactly supported solutions of the inviscid incompressible 2D Boussinesq equation with $C^{1,\sqrt{\frac{4}{3}}-1-\varepsilon}\cap L^{2}$ force that develop a singularity in finite time. Importantly, the…
We study a Floer-theoretic approach to harmonic maps from the two-torus into non-flat K\"ahler manifolds. Building on the complex-regularized polysymplectic (CRPS) formalism of [BF24], which provides a Hamiltonian description of harmonic…
Given a compact Kahler manifold with an extremal metric (M,\omega), we give sufficient conditions on finite sets points p_1,...,p_n and weights a_1,...a_n for which the blow up of M at p_1,...,p_n has an extremal metric in the Kahler class…
Let $(M,g)$ be a 2-dimensional compact Riemannian manifold. In this paper, we use the method of blowing up analysis to prove several Moser-Trdinger type inequalities for vector bundle over $(M,g)$. We also derive an upper bound of such…
In this paper, we establish a Minkowski-type inequality for weak Lebesgue space, which allows us to obtain a characterization of relative compactness in these spaces. Furthermore, we are the first to investigate the compactness results of…
Our aim is to give a version of the Moser-Trudinger inequality in the setting of complex geometry. As a very particular case, our result already gives a new Moser-Trudinger inequality for functions in the Sobolev space $W^{1,2}$ of a domain…
On complete metric spaces that support doubling measures, we show that the validity of a Rademacher theorem for Lipschitz functions can be characterised by Keith's "Lip-lip" condition. Roughly speaking, this means that at almost every…
We use quantum harmonic analysis for densely defined operators to provide a simplified proof of the Berger-Coburn theorem for boundedness of Toeplitz operators. In addition, we revisit compactness and Schatten-class membership of densely…