arXiv subjects

Michael Ruzhansky

Publications and source records attributed to Michael Ruzhansky.

At least 19 recordsLinked to original sources

arXiv2026

Topological Effects on Bubbling in the Critical Dirichlet Problem on Hyperbolic Domains: $3\leq N \leq5$

Let $\Omega\Subset\mathbb H^N$, $N\in\{3,4,5\}$, be a bounded connected $C^2$ domain. We prove that the pure critical Dirichlet problem \[ -\Delta_{\mathbb H}u=u^{\frac{N+2}{N-2}} \quad\text{in }\Omega, \qquad u=0 \quad\text{on }\partial\Omega \] admits a positive solution whenever $H_d(\Omega;\mathbb F_2)\neq0$ for some $1\le d\le N-1$. This gives a hyperbolic Bahri-Coron theorem for $3\le N\le5$ under $C^2$ boundary regularity. Under conformal reduction, the hyperbolic geometry produces a positive potential and leads to dimension-dependent bubbling mechanisms. In dimension three, the required energy drop follows from the balance between diagonal corrections and pair interactions at fixed large multiplicity. In dimensions four and five, it is obtained at the matched scale through a normalized defect estimate and an all-pairs source-transfer bound. A unified Thom-barycenter construction converts these analytic estimates into the topological contradiction. Thus nontrivial domain topology forces existence despite critical loss of compactness and the additional geometric potential.

math.AP↗

arXiv2026

Mixed local-nonlocal quasilinear problems with mixed interpolated Hardy potential

This paper addresses the existence of nontrivial solutions to a class of mixed local-nonlocal problems involving a mixed interpolated Hardy potential. We first establish a concentration-compactness principle for mixed local and nonlocal operators. This result is combined with Ricceri's variational principle to obtain an existence result for quasilinear elliptic problems under different growth assumptions on the nonlinearity. Furthermore, we apply the classical mountain pass theorem to obtain a second existence result in the superlinear case.

math.AP↗

arXiv2026

Very weak solutions of the heat equation with anisotropically singular time-dependent diffusivity

We investigate the heat equation with a time-dependent, anisotropic, and potentially singular diffusivity tensor. Since weak (in the Sobolev sense) or distributional solutions may not exist in this setting, we employ the framework of very weak solutions to establish the existence and uniqueness of solutions to the heat equation with singular, anisotropic, time-dependent diffusivity.

math.AP↗

arXiv2026

On a fractional nonlinear Schr\"odinger equation with irregular coefficients. case: d<2s

In the case when $d<2s$, where $d$ is the space dimension and $s$ is the fractional power of the Laplacian, we study the well-posedness for a cubic nonlinear Schr\"odinger equation (CNLSE) generated by the fractional Laplacian and involving distributional, or less regular, coefficients. We formulate our problem in the setting of the concept of so-called very weak solutions and prove that it has a very weak solution. Moreover, we prove the uniqueness in some adequate sense as well as the compatibility of the very weak solution with the classical one when the latter exists. Our results cover the classical case when: $d=1, s=1$. A second task in this paper is to conduct some numerical experiments where interesting behaviours of the very weak solution are observed. The obtained result is the first example of the very weak well-posedness in the setting of nonlinear partial differential equations.

math.AP↗

arXiv2026

Compactness and Spectral Properties of Multiplier Operators in the Walsh System

We investigate compactness and spectral properties of multiplier operators associated with the Walsh system in the spaces $L^p[0,1]$, $1<p<\infty$. Building upon previously established criteria for boundedness of Walsh multipliers, we prove an exact compactness criterion in the $L^p\to L^p$ regime for all $1<p<\infty$(assuming boundedness of the multiplier), and also in the $L^p\to L^2$ regime for $2<p<\infty$. The key result states that compactness is equivalent to the condition $a_n\to 0$ for the multiplier symbol. We also examine in detail the point spectrum and derive strict spectral inclusions; in the Hilbert space case $p=2$ we obtain a complete description of the spectrum. For $p\neq 2$, we emphasize the limitations of transferring "diagonal" arguments and formulate results in a form that does not admit incorrect generalizations.

math.FA↗

arXiv2026

Heat Equation driven by mixed local-nonlocal operators with non-regular space-dependent coefficients

In this paper, we study the Cauchy problem for a heat equation governed by a mixed local--nonlocal diffusion operator with spatially irregular coefficients. We first establish classical well-posedness in an energy framework for bounded, measurable coefficients that satisfy uniform positivity, and we derive an a priori estimate ensuring uniqueness and continuous dependence on the initial data. We then extend the notion of solution to distributional coefficients and initial data by a Friedrichs-type regularisation procedure. Within this very weak framework, we establish the existence and uniqueness of solution nets and prove consistency with the classical weak solution whenever the coefficients are regular.

math.AP↗

arXiv2026

Kernel estimates and weak (1,1)-boundedness of pseudo-differential operators on compact Lie groups

Given a compact Lie group $G$ and its unitary dual $\widehat{G}$, we establish the weak (1,1) continuity for pseudo-differential operators in the global H\"ormander classes of order $-n(1-\rho)/2$ on $G\times \widehat{G}$. Our approach consists of proving suitable estimates for the kernel of such operators. Furthermore, we use these kernel estimates to give an alternative proof for the $H^1(G)$-$L^1(G)$-continuity of these classes now allowing the full range $0\leq\delta\leq\rho\leq1, \;\rho\neq0,\;\delta\neq1$. The conditions for the operators are formulated using the H\"ormander classes $S^m_{\rho,\delta}(G):=S^m_{\rho,\delta}(G\times \widehat{G})$ of symbols in the non-commutative phase space $G\times \widehat{G}$, which are extensions of the well-known $(\rho,\delta)$-classes in the Euclidean space. Our results are formulated in the complete range $0\leq \delta\leq \rho\leq 1,$ $\rho\neq0,\;$$\delta\neq 1$. As an application of this boundedness result we provide end-point a-priori $L^1$-estimates for the sub-Laplacian $\mathcal{L}_{sub}=X^2+Y^2,$ and for the heat type operator $T=Z-X^2-Y^2$ on $SU(2)\cong \mathbb{S}^3$ that cannot be obtained by application of the standard pseudo-differential calculus due to H\"ormander. More precisely, we prove that if one considers the subelliptic problem, \begin{equation}\label{IVP:abstract} \begin{cases}Tu=f ,& \text{ } \\u,f\in \mathscr{D}'(SU(2)):=(C^\infty(SU(2)))', & \text{ } \end{cases} \end{equation} then, for $f\in W^{1,-\frac{1}{4}}(SU(2)),$ one has that $u\in L^{1,\infty}(SU(2)).$

math.AP↗

arXiv2026

Reproducing Kernel Hilbert Spaces and entropy Kolmogorov numbers on compact Lie Groups

On a compact Lie group $G$, we consider the reproducing kernel Hilbert space $\mathcal{H}_K$ associated with the integral kernel $K$ of a left-invariant, positive, symmetric, trace class integral operator on $L^2(G)$. We present lower and upper asymptotic estimates for the entropy Kolmogorov numbers (also called covering numbers) for the embedding of $\mathcal{H}_K$ into the space $C(G)$ of continuous functions on $G$.

math.FA↗

arXiv2026

Lieb-Thirring inequalities for the Dirac operator on spheres

In this paper, we obtain bounds for the best constants in two inequalities which can be seen as analogues of the Lieb-Thirring inequality, but with the Dirac operator, on the $n-$sphere. We then apply these results in order to improve the known upper bounds on the classical Lieb-Thirring constant on the $n$-sphere for $n\geq 5$.

math.SP↗

arXiv2026

Fujita exponents on quantum Euclidean spaces

We study the well-posedness of a non-linear heat equation with power nonlinearity with positive initial data on quantum Euclidean spaces. We prove a noncommutative analogue of the classical Fujita theorem by identifying the critical exponent separating finite-time blow-up from global existence for small initial data. Moreover, we establish a fundamental inequality in general semifinite von Neumann algebras that is of independent interest and plays a crucial role in the study of global existence and local well-posedness of solutions of nonlinear equations in noncommutative setting.

math.AP↗

arXiv2025

Normalized Solutions and Semiclassical Concentration for Upper-Critical Fractional Choquard Equations

We study a fractional Choquard equation with an upper-critical Hartree term, an $L^2$-supercritical Hartree perturbation, and a semiclassical potential under a prescribed $L^2$-mass constraint. The potential is bounded and nonnegative, has a nonempty zero set, and has a positive lower limit at infinity. For every prescribed mass and all sufficiently small semiclassical parameters, we prove the existence of a pair of normalized solutions $\pm u_\varepsilon$ with a negative Lagrange multiplier. The proof combines a strict energy bound below the critical one-bubble level, compactness modulo translations for the autonomous ground-state set, a simultaneous cutoff of both Hartree terms, and a localized constrained mountain-pass argument. Moreover, suitable translates of $u_\varepsilon$ converge strongly in $H^s(\mathbb{R}^N)$ to a positive autonomous ground state, and the corresponding concentration points approach the zero set of the potential as $\varepsilon\to0$.

math.AP↗

arXiv2025

Nonlinear Dirac equations on noncompact quantum graphs with potentials: Multiplicity and Concentration

In this paper, we study the existence and multiplicity of solutions to the following class of nonlinear Dirac equations (NLDE) on noncompact quantum graphs: \[ -i\,\varepsilon c\,\sigma_1\,\partial_x u + m c^2 \sigma_3 u + V(x)\,u = f(|u|)\,u, \quad x\in \mathcal{G}, \tag{P} \] where $V:\mathcal{G}\to\mathbb{R}$ and $f:\mathbb{R}\to\mathbb{R}$ are continuous, $\varepsilon>0$ is a semiclassical parameter, $m>0$ denotes the mass, and $c>0$ the speed of light. Here $\sigma_1,\sigma_3$ are Pauli matrices, and $\mathcal{G}$ is a noncompact quantum graph. We prove that when $\varepsilon$ is sufficiently small, the number of solutions to $(P)$ is at least the number of global minima of $V$. Moreover, these solutions exhibit semiclassical concentration: as $\varepsilon\to0$, their concentration points approach the set of global minima of $V$.

math.AP↗

arXiv2025

Numerical Approaches for Identifying the Time-Dependent Potential Coefficient in the Diffusion Equation

We address the inverse problem of identifying a time-dependent potential coefficient in a one-dimensional diffusion equation subject to Dirichlet boundary conditions and a nonlocal integral overdetermination constraint reflecting spatially averaged measurements. After establishing well-posedness for the forward problem and deriving an a priori estimate that ensures uniqueness and continuous dependence on the data, we prove existence and uniqueness for the inverse problem. To compute numerically the unknown coefficient, we propose and compare three numerical methods: an integration-based scheme, a Newton-Raphson iterative solver, and a physics-informed neural network (PINN). Numerical experiments on both exact and noisy data demonstrate the accuracy, robustness, and efficiency of each approach.

math.NA↗

arXiv2025

Fujita exponent for heat equation with H\"{o}rmander vector fields

In this paper, we show global existence and non-existence results for the heat equation with some of the squares of smooth vector fields on $\Rn$ satisfying H\"{o}rmander's rank condition with a non-linearity of the form $f(u)$, where $f$ is a suitable function and $u$ is the solution. In particular, when $f(u)=u^p$, we calculate the critical Fujita exponent. We also give necessary conditions for blow-up or, alternatively, a sufficient condition for the existence of positive global solutions for time-dependent nonlinearities of the type $\varphi(t)f(u)$.

math.AP↗

arXiv2025

Spectral analysis, maximum principles and shape optimization for nonlinear superposition operators of mixed fractional order

The main objective of this paper is to investigate the spectral properties, maximum principles, and shape optimization problems for a broad class of nonlinear ``superposition operators" defined as continuous superpositions of operators of mixed fractional order, modulated by a signed finite Borel measure on the unit interval. This framework encompasses, as particular cases, mixed local and nonlocal operators such as $-\Delta_p+(-\Delta_p)^s$, finite (possibly infinite) sums of fractional $p$-Laplacians with different orders, as well as operators involving fractional Laplacians with ``wrong" signs. The main findings, obtained through variational techniques, concern the spectral analysis of the Dirichlet eigenvalue problem associated with general superposition operators with special emphasis on various properties of the first eigenvalue and its corresponding eigenfunction. We establish weak and strong maximum principles for positive superposition operators by introducing an appropriate notion of the {\it nonlocal tail} for this class of superposition operators and deriving a logarithmic estimate, both of which are of independent interest. Utilizing these newly developed tools, we further investigate the spectral properties of such superposition operators and prove that the first eigenvalue is isolated and simple. Moreover, we show that the eigenfunctions corresponding to positive eigenvalues are globally bounded and that they change sign when associated with higher eigenvalues. In addition, we demonstrate that the second eigenvalue is well-defined and provide the mountain pass characterization. Finally, we address shape optimization problems, in particular, the Faber--Krahn inequality associated with the principal frequency associated with the superposition operators.

math.AP↗

arXiv2025

Nonrelativistic limit of bound-state solutions for nonlinear Dirac equation on noncompact quantum graphs

In this paper, we investigate the nonrelativistic limit and qualitative properties of bound-state solutions for the nonlinear Dirac equation (NLDE) defined on noncompact quantum graphs: \[ -i c \frac{d}{d x} \sigma_1 \psi+m c^2 \sigma_3 \psi-\omega \psi=g(|\psi|) \psi, \quad \text { in } \mathcal{G} \] where $ g : \mathbb{R}\rightarrow\mathbb{R} $ is a continuous nonlinear function, $ c>0 $ represents the speed of light, $ m>0 $ is the particle's mass, $ \omega\in\mathbb{R} $ is related to the frequency, $ \sigma_1 $ and $ \sigma_3 $ denote the Pauli matrices, and $\mathcal{G}$ is a noncompact quantum graph. We establish the existence of bound-state solutions to the NLDE on $\mathcal{G}$, and prove that these solutions converge toward the corresponding bound-state solutions of a nonlinear Schr\"odinger equation (NLS) in the nonrelativistic limit (i.e., as the speed of light $ c \to \infty $) for particles of small mass. Furthermore, we prove uniform boundedness and exponential decay properties of the NLDE solutions, uniformly in $ c $, thereby offering insight into their asymptotic behavior.

math.AP↗

arXiv2025

Semi-sparsity Generalization for Variational Mesh Denoising

In this paper, we propose a new variational framework for 3D surface denoising over triangulated meshes, which is inspired by the success of semi-sparse regularization in image processing. Differing from the uniformly sampled image data, mesh surfaces are typically represented by irregular, non-uniform structures, which thus complicate the direct application of the standard formulation and pose challenges in both model design and numerical implementation. To bridge this gap, we first introduce the discrete approximations of higher-order differential operators over triangle meshes and then develop a semi-sparsity regularized minimization model for mesh denoising. This new model is efficiently solved by using a multi-block alternating direction method of multipliers (ADMM) and achieves high-quality simultaneous fitting performance -- preserving sharp features while promoting piecewise-polynomial smoothing surfaces. To verify its effectiveness, we also present a series of experimental results on both synthetic and real scanning data, showcasing the competitive and superior results compared to state-of-the-art methods, both visually and quantitatively.

cs.CG↗

arXiv2025

Functional calculus for Safarov pseudo-differential operators

Given a smooth, closed Riemannian manifold $(M,g)$ equipped with a linear connection $\nabla$ (not necessarily metric), we develop the holomorphic functional calculus for operators belonging to the global pseudo-differential classes $\Psi_{\rho, \delta}^m\left(\Omega^\kappa, \nabla, \tau\right)$ introduced by Safarov. As a consequence of our main result, we establish a Szeg\"o type-theorem, derive asymptotic expansion of the heat kernel trace, and calculate some associated spectral $\zeta$-functions.

math.AP↗