Tootfinder

'Negotiation can begin where the front line currently stands' — Zelensky says ahead of meeting with Trump: https://benborges.xyz/2025/08/17/negotiation-can-begin-where-the.html

A file I came across on my iPad. No idea where it came from. #randomThoughts
Thoughts to ponder:
Why isn’t the number 11 pronounced onety-one?
If 4 out of 5 people SUFFER from diarrhea, does that mean that one out of five enjoys it?
Why do croutons come in airtight packages? Aren't they just stale bread to begin with?
If a pig loses its voice, is it disgrunt…

Navigating High-Dimensional Backstage: A Guide for Exploring Literature for the Reliable Use of Dimensionality Reduction
Hyeon Jeon, Hyunwook Lee, Yun-Hsin Kuo, Taehyun Yang, Daniel Archambault, Sungahn Ko, Takanori Fujiwara, Kwan-Liu Ma, Jinwook Seo
https://arxiv.org/abs/2506.14820

Navigating High-Dimensional Backstage: A Guide for Exploring Literature for the Reliable Use of Dimensionality Reduction
Visual analytics using dimensionality reduction (DR) can easily be unreliable for various reasons, e.g., inherent distortions in representing the original data. The literature has thus proposed a wide range of methodologies to make DR-based visual analytics reliable. However, the diversity and extensiveness of the literature can leave novice analysts and researchers uncertain about where to begin and proceed. To address this problem, we propose a guide for reading papers for reliable visual ana…

Stochastic Processes and Diffusion Equations
Helder Rojas
https://arxiv.org/abs/2506.13913 https://arxiv.org/pdf/2506.13913

Stochastic Processes and Diffusion Equations
In these lecture notes, we explore the mathematical preliminaries and foundational concepts that connect stochastic processes with partial differential equations. We begin by investigating Brownian motion, which serves as a model for random fluctuations and is deeply connected to the heat equation. This connection forms the basis for understanding diffusion phenomena, where the probability distribution of Brownian motion evolves according to the heat equation over time. To extend this…

Normalized solutions of coupled Sobolev critical Schrodinger equations with mass subcritical couplings
Zhang Jianjun, Zhong Xuexiu, Zhou Jinfang
https://arxiv.org/abs/2507.13163

Normalized solutions of coupled Sobolev critical Schrodinger equations with mass subcritical couplings
We are concerned with qualitative properties of positive solutions to the following coupled Sobolev critical Schrödinger equations $$ \begin{cases} -Δu+λ_1 u=μ_1|u|^{2^*-2}u+να|u|^{α-2}|v|^βu ~\hbox{in}~ \R^N,\\ -Δv+λ_2 v=μ_2|v|^{2^*-2}v+νβ|u|^α|v|^{β-2}v ~\hbox{in}~ \R^N \end{cases} $$ subject to the mass constraints $\int_{\mathbb{R}^N}|u|^2 \ud x=a^2$ and $\int_{\mathbb{R}^N}|v|^2 \ud x=b^2$, where…

Asymptotic behavior of solutions to the Dirac system with respect to a spectral parameter
Alexander Gomilko, {\L}ukasz Rzepnicki
https://arxiv.org/abs/2507.12147

Asymptotic behavior of solutions to the Dirac system with respect to a spectral parameter
We consider the Dirac system of ordinary differential equations \[ Y'(x) + \begin{bmatrix} 0 & σ_1(x) \\ σ_2(x) & 0 \end{bmatrix} Y(x) = iμ\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} Y(x), \quad Y(x) = \begin{bmatrix} y_1(x) \\ y_2(x) \end{bmatrix}, \] where $x \in [0,1]$, $μ\in \mathbb{C}$ is a spectral parameter, and $σ_j \in L^p[0,1],$ $j = 1,2,$ for $p \in [1,2).$ We study the asymptotic behavior of the system's fundamental solutions as $|μ| \to \infty$ in the half-plane $…

@… while I agree with most of your points I do need to note that LLM is amazing as a booster when I don't even know what I don't know. Like jumping to some new framework with no clue where to even begin with the docs.
One example: I needed a throwaway MS Word script to sanitize a bunch of documents (translations with same formatting problems)…

Where Things Stand
Policy bill: After releasing a 940-page version of their sprawling domestic policy bill overnight, Senate Republicans on Saturday were seeking the votes to pass it, a key step in advancing President Trump’s agenda. They cut deals with holdouts in their ranks that they hoped would be enough to speed the bill through the chamber as early as this weekend.

Final cost: If Senate Republicans begin debate on the sweeping bill on Saturday, they will do so wit…

Trump Administration Live Updates: Republicans Hunt for Votes to Bring Policy Bill to the Floor
Senate Republicans were trying to find enough votes to pass their domestic policy bill, a key step in advancing President Trump’s agenda. Three Republicans have voted against opening debate.

Rankin-Selberg integrals for $\mathrm{GSpin}$ groups with application to the global Gan-Gross-Prasad conjecture
Pan Yan
https://arxiv.org/abs/2508.09066 https://

Rankin-Selberg integrals for $\mathrm{GSpin}$ groups with application to the global Gan-Gross-Prasad conjecture
We construct Rankin-Selberg integrals using Bessel models for a product of tensor product partial $L$-functions \begin{equation*} L^S(s,π\timesτ_1) L^S(s,π\timesτ_2)\cdots L^S(s,π\timesτ_r) \end{equation*} where $π$ is an irreducible cuspidal automorphic representation of a quasi-split $\mathrm{GSpin}$ group, and $τ_1, \cdots, τ_r$ are irreducible unitary cuspidal automorphic representations of $\mathrm{GL}_{k_1}, \cdots, \mathrm{GL}_{k_r}$ respectively. As an application, we prove one…

Extension of a complete monotonicity theorem with applications
Zhen-Hang Yang
https://arxiv.org/abs/2507.10954 https://arxiv.org/pdf/…

Extension of a complete monotonicity theorem with applications
Let $F_{p}(x) =L( t^{p}f(t)) =\int_{0}^{\infty }t^{p}f(t) e^{-xt}dt$ converge on $(0,\infty)$ for $p\in \mathbb{N}_{0}=\mathbb{N}\cup{0}$, where $f(t)$ is positive on $(0,\infty)$. In a recent paper [Z.-H. Yang, A complete monotonicity theorem related to Fink's inequality with applications, \emph{J. Math. Anal. Appl.} \textbf{551} (2025), no. 1, Paper no. 129600], the author proved the sufficient conditions for the function \begin{equation*} x\mapsto \prod_{j=1}^{n}F_{p_{j}}(x) -λ_{n}\prod_{j=…

AFC West Preseason Position Rankings: Where Does Geno Smith Land? https://raiderramble.com/2025/08/07/afc-west-preseason-position-rankings-where-does-geno-smith-land/

I'm currently reading the second book of the DAEMON (German: Darknet, English: Freedom), a techno thriller from Daniel Suarez.
Pretty cool I think.
But what's even cooler: I just checked his homepage ( https://daniel-suarez.com/daemon10thsynopsis.html ) and saw a Ma…

Global in-time rough large data solution to complex-valued semilinear damped evolution equations
Wenhui Chen, Michael Reissig
https://arxiv.org/abs/2507.08272

Global in-time rough large data solution to complex-valued semilinear damped evolution equations
We study the semilinear Cauchy problem for complex-valued damped evolution equations \begin{align*} \partial_t^2u+(-Δ)^σu+(-Δ)^δ\partial_tu=u^p,\ \ u(0,x)=u_0(x),\ \partial_tu(0,x)=u_1(x), \end{align*} with $δ\in[0,σ]$, $σ\in\mathbb{R}_+$ and $p\in\mathbb{N}_+\backslash\{1\}$, where the initial data belong to the rough space $E^α_s$ endowed with the norm \begin{align*} \|f\|_{E^α_s}=\big\|\langleξ\rangle^s\,2^{α|ξ|}\widehat{f}(ξ)\big\|_{L^2}\ \ \mb…

P\'olya's conjecture up to $\epsilon$-loss and quantitative estimates for the remainder of Weyl's law
Renjin Jiang, Fanghua Lin
https://arxiv.org/abs/2507.04307

Pólya's conjecture up to $ε$-loss and quantitative estimates for the remainder of Weyl's law
Let $Ω\subset\mathbb{R}^n$ be a bounded Lipschitz domain. For any $ε\in (0,1)$ we show that for any Dirichlet eigenvalue $λ_k(Ω)>Λ(ε,Ω)$, it holds \begin{align*} k&\le (1+ε)\frac{|Ω|ω(n)}{(2π)^n}λ_k(Ω)^{n/2}, \end{align*} where $Λ(ε,Ω)$ is given explicitly. This reduces the $ε$-loss version of Pólya's conjecture to a computational problem. This estimate is based on a uniform estimate on the remainder of the Weyl law on Lipschitz domains, which appears to be the first…

I left a comment last week on Smashing’s “Design Patterns For AI Interfaces” (which is still in moderation):
https://www.smashingmagazine.com/2025/07/design-patterns-ai-interfaces/
Essentially, if your UI doesn’t also try to prevent harm then it might …

Design Patterns For AI Interfaces — Smashing Magazine
Designing a new AI feature? Where do you even begin? Here’s a simple, practical overview with useful design patterns for better AI experiences.

Growth of generalized greatest common divisors along orbits of self-rational maps on projective varieties
Yohsuke Matsuzawa
https://arxiv.org/abs/2507.05027

Growth of generalized greatest common divisors along orbits of self-rational maps on projective varieties
Consider a dominant rational self-map $f$ on a smooth projective variety $X$ defined over $\overline{\mathbb{Q}}$. We prove that \begin{align} \lim_{n \to \infty} \frac{h_{Y}(f^{n}(x))}{h_{H}(f^{n}(x)) } = 0, \end{align} where $h_{Y}$ is a height associated with a closed subscheme $Y \subset X$ of codimension $c$, $h_{H}$ is any ample height on $X$, and $x \in X(\overline{\mathbb{Q}})$ is a point with well-defined orbit, under the following assumptions: (1) either $f$ is a morphism, or $Y$ is p…

Approximate Proportionality in Online Fair Division
Davin Choo, Winston Fu, Derek Khu, Tzeh Yuan Neoh, Tze-Yang Poon, Nicholas Teh
https://arxiv.org/abs/2508.03253 https://

Approximate Proportionality in Online Fair Division
We study the online fair division problem, where indivisible goods arrive sequentially and must be allocated immediately and irrevocably to agents. Prior work has established strong impossibility results for approximating classic fairness notions, such as envy-freeness and maximin share fairness, in this setting. In contrast, we focus on proportionality up to one good (PROP1), a natural relaxation of proportionality whose approximability remains unresolved. We begin by showing that three natura…

Learning and Testing Inverse Statistical Problems For Interacting Systems Undergoing Phase Transition
Stefano Bae, Dario Bocchi, Luca Maria Del Bono, Luca Leuzzi
https://arxiv.org/abs/2507.02574

Learning and Testing Inverse Statistical Problems For Interacting Systems Undergoing Phase Transition
Inverse problems arise in situations where data is available, but the underlying model is not. It can therefore be necessary to infer the parameters of the latter starting from the former. Statistical mechanics offers a toolbox of techniques to address this challenge. In this work, we illustrate three of the main methods: the Maximum Likelihood, Maximum Pseudo-Likelihood, and Mean-Field approaches. We begin with a thorough theoretical introduction to these methods, followed by their application…

Resonating Kagome Dimer coverings in Rydberg atom arrays
Xicheng Wang, Erich J Mueller
https://arxiv.org/abs/2506.21255 https://arxiv…

Resonating Kagome Dimer coverings in Rydberg atom arrays
Motivated by experiments on Rydberg atom arrays, we explore the properties of uniform quantum superpositions of kagome dimer configurations and construct an efficient algorithm for experimentally producing them. We begin by considering the thin cylinder limit, where these states have simple descriptions. We then develop a matrix product representation of the states on arbitrary cylinders, which leads to a natural protocol to efficiently grow them. We explain how our approach can be adapted to o…

Bifurcation from periodic solutions of central force problems in the three-dimensional space
Alberto Boscaggin, Guglielmo Feltrin, Duccio Papini
https://arxiv.org/abs/2506.05842

Bifurcation from periodic solutions of central force problems in the three-dimensional space
The paper deals with electromagnetic perturbations of a central force problem of the form \begin{equation*} \dfrac{\mathrm{d}}{\mathrm{d}t} \bigl( φ(\dot{x}) \bigr) = V'(|x|) \dfrac{x}{|x|} + E_{\varepsilon}(t,x)+\dot{x} \wedge B_{\varepsilon}(t,x), \qquad x \in \mathbb{R}^3 \setminus \{0\}, \end{equation*} where $V \colon (0,+\infty) \to \mathbb{R}$ is a smooth function, $E_\varepsilon$ and $B_\varepsilon$ are respectively the electric field and the magnetic field, smooth …

Untriangular factorization of holomorpic symplectic matrices
Gaofeng Huang, Frank Kutzschebauch, Phan Quoc Bao Tran
https://arxiv.org/abs/2507.18963 https://

Untriangular factorization of holomorpic symplectic matrices
We prove that every holomorphic symplectic matrix can be factorized as a product of holomorphic unitriangular matrices with respect to the symplectic form $ \left[\begin{array}{ccc} 0 & L_n \\ -L_n & 0\end{array}\right]$ where $L$ is the $n \times n$ matrix with $1$ along the skew-diagonal. Also we prove that holomorphic unitriangular matrices with respect to this symplectic form are products of not more than $7$ holomorphic unitriangular matrices with respect to the standard symplectic form $\…

Higher differentiability for solutions to stationary $p$-Stokes systems under sub-quadratic growth conditions
Anna Cavagnoli
https://arxiv.org/abs/2507.03499

Higher differentiability for solutions to stationary $p$-Stokes systems under sub-quadratic growth conditions
We consider weak solutions $(u,π):\mathbb{R}^n\supsetΩ\to\ \mathbb{R}^n\times\ \mathbb{R}$ to stationary $p$-Stokes systems of the type \[ \begin{cases} -\mathrm{div} (a(\mathcal{E} u))+\nablaπ=f \\ \mathrm{div}(u)=0, \end{cases} \] in $Ω,$ where the function $a(ξ)$ satisfies $p$-growth conditions in $ξ$. By $\mathcal{E} u$ we denote the symmetric part of the gradient $Du$. In this setting, we establish results on the fractional higher differentiability of both the symmetric part of the…

Toward Khintchine's theorem with a moving target: extra divergence or finitely centered target
Gilbert Michaud, Felipe A. Ram\'irez
https://arxiv.org/abs/2506.04187

Toward Khintchine's theorem with a moving target: extra divergence or finitely centered target
Sz{ü}sz's inhomogeneous version (1958) of Khintchine's theorem (1924) gives conditions on $ψ:\mathbb{N}\to\mathbb{R}_{\geq 0}$ under which for almost every real number $α$ there exist infinitely many rationals $p/q$ such that \begin{equation*} \lvertα- \frac{p+γ}{q}\rvert < \frac{ψ(q)}{q}, \end{equation*} where $γ\in\mathbb{R}$ is some fixed inhomogeneous parameter. It is often interpreted as a statement about visits of $qα\,(\bmod 1)$ to a shrinking target centered around $γ\,(\bmod…

On the edge reconstruction of the second immanantal polynomials of undirected graph and digraph
Tingzeng Wu
https://arxiv.org/abs/2507.14607 https://

On the edge reconstruction of the second immanantal polynomials of undirected graph and digraph
Let $M=(m_{ij})$ be an $n\times n$ matrix. The second immanant of matrix $M$ is defined by \begin{eqnarray*} d_{2}(M)=\sum_{σ\in S_{n}}χ_{2}(σ)\prod_{s=1}^{n}m_{sσ(s)}, \end{eqnarray*} where $χ_{2}$ is the irreducible character of $S_{n}$ corresponding to the partition $(2^{1},1^{n-2})$. The polynomial $d_{2}(xI-M)$ is called the second immanantal polynomial of matrix $M$. Denote by $D(G)$ (resp. $D(\overrightarrow{G})$) and $A(G)$ (resp. $A(\overrightarrow{G})$) the diagonal matrix of ver…

Global solvability for doubly degenerate nutrient taxis system with a wide range of bacterial responses in physical dimension
Bao-Ngoc Tran, Juan Yang
https://arxiv.org/abs/2508.03268

Global solvability for doubly degenerate nutrient taxis system with a wide range of bacterial responses in physical dimension
Motivated by the study of bacteria's response to environmental conditions, we consider the doubly degenerate nutrient taxis system \begin{align*} \begin{cases} u_t=\nabla\cdot(uv\nabla u)-χ\nabla\cdot(u^αv\nabla v)+\ell uv,\\ v_t=Δv-uv, \end{cases} \end{align*} subjected to no-flux boundary conditions and smooth initial data, where $α\in\mathbb{R}$ is the bacterial response parameter. Global solvability of weak solutions to this taxis system is highly challenging due to not only the doubly …

Measurement of the $B^0\rightarrow \rho(770)^{0}\gamma$ branching fraction
LHCb collaboration, et al.
https://arxiv.org/abs/2507.14401 https://

Measurement of the $B^0\rightarrow ρ(770)^{0}γ$ branching fraction
The ratio between the branching fractions of the $B^0\rightarrow ρ(770)^{0}γ$ and $B^{0}\rightarrow K^{*}(892)^{0}γ$ decays is measured with proton-proton collision data collected by the LHCb experiment at centre-of-mass energies of 7, 8, and 13 TeV, corresponding to an integrated luminosity of 9 fb${}^{-1}$. The measured value is \begin{equation*} \frac{{\cal B}(B^0\rightarrow ρ(770)^{0}γ)}{{\cal B}(B^0\rightarrow K^{*}(892)^{0}γ)}=0.0189\pm 0.0007\pm 0.0005, \end{equation*} where the fi…

Existence of normalized solutions to nonlinear Schr\"{o}dinger equations with potential on lattice graphs
Weiqi Guan
https://arxiv.org/abs/2507.04204 …

Existence of normalized solutions to nonlinear Schrödinger equations with potential on lattice graphs
We study the existence of ground state normalized solution of the following Schrödinger equation: \begin{equation*} \begin{cases} -Δu+V(x)u+λu=f(x,u), & x\in\mathbb{Z}^d \\ \Vert u\Vert_2^2=a \end{cases} \end{equation*} where $V(x)$ is trapping potential or well potential, $f(x,u)$ satisfies Berestycki-Lions type condition and other suitable conditions. We show that there always exists a threshold $α\in[0,\infty)$ such that there do not exist ground state normalized solutions for $a\in (0…

Building Intuition for Dynamical Mean-Field Theory: A Simple Model and the Cavity Method
Emmy Blumenthal
https://arxiv.org/abs/2507.16654 https://

Building Intuition for Dynamical Mean-Field Theory: A Simple Model and the Cavity Method
Dynamical Mean-Field Theory (DMFT) is a powerful theoretical framework for analyzing systems with many interacting degrees of freedom. This tutorial provides an accessible introduction to DMFT. We begin with a linear model where the DMFT equations can be derived exactly, allowing readers to develop clear intuition for the underlying principles. We then introduce the cavity method, a versatile approach for deriving DMFT equations for non-linear systems. The tutorial concludes with an application…

Gaussian integrals on symmetric spaces (the complex case and beyond)
Salem Said
https://arxiv.org/abs/2505.22672 https://arxiv.org/pd…

Gaussian integrals on symmetric spaces (the complex case and beyond)
The present work is concerned with Gaussian integrals on simply-connected non-positively-curved Riemannian symmetric spaces. It is motivated by the aim of explicitly finding the high-rank limit of these integrals for each of the eleven families of classical Riemannian symmetric spaces. To begin, it deals with the easier complex case (where the isometry group admits a complex Lie group structure). To go beyond this case, it introduces a variational characterisation of the high-rank limit, as the…

Non-convergence of the principal eigenvalue of elliptic operators for large advection
Xueli Bai, Zhi-An Wang, Xin Xu, Kexin Zhang, Maolin Zhou
https://arxiv.org/abs/2507.04418

Non-convergence of the principal eigenvalue of elliptic operators for large advection
This paper investigates the limit of the principal eigenvalue $λ(s)$ as $s\to+\infty$ for the following elliptic equation \begin{align*} -Δφ(x)-2s\mathbf{v}\cdot\nablaφ(x)+c(x)φ(x)=λ(s)φ(x), \quad x\in Ω \end{align*} in a bounded domain $Ω\subset \mathbb{R}^d (d\geq 1)$ with the Neumann boundary condition. Previous studies have shown that under certain conditions on $\mathbf{v}$, $λ(s)$ converges as $s\to\infty$ (including cases where $\lim\limits_{s \to\infty }λ(s)=\pm\infty$). …

This https://arxiv.org/abs/2403.11479 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

A priori estimates for parabolic Monge-Ampère type equations
We prove the existence and regularity of convex solutions to the first initial-boundary value problem for the parabolic Monge-Ampère equationn $$ \left\{\begin{eqnarray} &&-u_t+\det D^2u= ψ(x,t) \quad\quad\ \text{ in } Q_T,\newline &&u=ϕ\quad\text{ on }\partial_pQ_T, \end{eqnarray}\right. $$ where $ψ,ϕ$ are given functions, $Q_T=Ω\times(0,T]$, $\partial_p Q_T$ is the parabolic boundary of $Q_T$, and $Ω\subset\mathbb{R}^n$ is a uniformly convex domain. Our approach can also be use…

Shape derivative approach to fractional overdetermined problems
Sidy M. Djitte, Ignace A. Minlend
https://arxiv.org/abs/2506.00268 https://

Shape derivative approach to fractional overdetermined problems
We use shape derivative approach to prove that balls are the only convex and $C^{1,1}$ regular domains in which the fractional overdetermined problem \begin{equation*} \left\{\begin{aligned} \Ds u&= λ_{s, p} u^{p-1}\quad\text{in}\quad\Om \\ u &= 0\quad \text{in}\quad\R^N\setminus \Om\\ u/d^s&=C_0\quad\text{on\;\; $\partialØ$} \end{aligned} \right. \end{equation*} admits a nontrivial solution for $p\in [1, 2]$ and where $λ_{s, p}= λ_{s, p}(Ø)$ is the best constant in the family of Subcritic…

Existence and multiplicity of normalized solutions to a large class of elliptic equations on bounded domains with general boundary conditions
Claudianor O. Alves, Zhentao He, Chao Ji
https://arxiv.org/abs/2507.04624

Existence and multiplicity of normalized solutions to a large class of elliptic equations on bounded domains with general boundary conditions
In this paper, by adapting the perturbation method, we study the existence and multiplicity of normalized solutions for the following nonlinear Schrödinger equation $$ \left\{ \begin{array}{ll} -Δu = λu + f(u)\quad & \text{in } Ω, \mathcal{B}_{α,ζ,γ}u = 0 & \text{on } \partial Ω, \int_Ω |u|^2\,dx = μ, \end{array} \right. \leqno{(P)^μ_{α,ζ,γ}} $$ where $Ω\subset \mathbb{R}^N$ ($N \geq 1$) is a smooth bounded domain, $μ>0$ is prescribed, $λ\in \mat…

Upper and Lower Solution Method for Regular Discrete Second-Order Single-Variable BVPs
Shalmali Bandyopadhyay, Kyle Byassee, Curt Lynch
https://arxiv.org/abs/2506.16526

Upper and Lower Solution Method for Regular Discrete Second-Order Single-Variable BVPs
This paper investigates the existence of positive solutions for regular discrete second-order single-variable boundary value problems with mixed boundary conditions, including a nonhomogeneous Dirichlet boundary condition, of the form: \begin{equation*} u^{ΔΔ}(t-1)+h(t,\ u(t),\ u^Δ(t-1))=0 \mbox{ for }t\in[1,\ T+1];~u^Δ(0)=0;~u(T+2)=g(T+2) \end{equation*} where h is continuous on $[1, T + 1] \times \mathbb{R}^2$ and $g: [0, T + 2] \to \mathbb{R}^+$ is continuous. Using the concept of upper …

Nondegeneracy of bubble solutions to the Choquard equation in two dimension
Jinkai Gao, Xinfu Li, Shiwang Ma
https://arxiv.org/abs/2508.02286 https://arxiv…

Nondegeneracy of bubble solutions to the Choquard equation in two dimension
In this paper, we study the following Choquard equation with exponential nonlinearity \begin{equation*} -Δu=\left(\int_{\R^{2}}\frac{e^{u(y)}}{|x-y|^α}dy\right)e^{u(x)},\quad \text{~in~}\R^{2}, \end{equation*} where $α\in (0,2)$. Although the classification of solutions to this equation has been established recently, the nondegeneracy of its solutions remains open. Here, we prove the nondegeneracy by combining the integral representation of solutions with the spherical harmonic d…

Asymptotic behavior of the least energy solutions to the Choquard equation in dimension two
Jinkai Gao, Xinfu Li, Shiwang Ma
https://arxiv.org/abs/2508.02139 https://

Asymptotic behavior of the least energy solutions to the Choquard equation in dimension two
In this paper, we are interested in the following planar Choquard equation \begin{equation*} \begin{cases} -Δu=\displaystyle\left(\int\limits_Ω\frac{u^{p+1}(y)}{|x-y|^α}dy\right)u^{p},\quad u>0,\ \ &\mbox{in}\ Ω, \quad \ \ u=0, \ \ &\mbox{on}\ \partial Ω, \end{cases} \end{equation*} where $Ω$ is a smooth bounded domain in $\mathbb{R}^2$, $α\in (0,2)$ and $p>1$ is a positive parameter. Unlike the higher-dimensional case, we prove that the least energy solutions $u_{p}$ n…

Existence and nonexistence of solutions for singular quadratic quasilinear equations
David Arcoya, Jos\'e Carmona, Tommaso Leonori, Pedro J. Mart\'inez-Aparicio, Luigi Orsina, Francesco Petitta
https://arxiv.org/abs/2508.06375

Existence and nonexistence of solutions for singular quadratic quasilinear equations
We study both existence and nonexistence of nonnegative solutions for nonlinear elliptic problems with singular lower order terms that have natural growth with respect to the gradient, whose model is $$ \begin{cases} -Δu + \frac{|\nabla u|^2}{u^γ} = f & \mbox{in } Ω,\newline \hfill u=0 \hfill & \mbox{on } \partial Ω, \end{cases} $$ where $Ω$ is an open bounded subset of $\mathbb{R}^N $, $γ> 0$ and $f$ is a function which is strictly positive on every compactly contained subs…

This https://arxiv.org/abs/2504.05655 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Multi-bubble solutions for the Dirichlet problem of the $H$-system with higher degree
We consider a Dirichlet problem of the $H$-system \begin{equation*} \begin{cases} Δv = 2v_x\wedge v_y ~& \text{ in }\mathcal{D},\\ v=\varepsilon \tilde g ~& \text{ on }\partial{\mathcal{D}}, \end{cases} \end{equation*} where $\mathcal D\subset \mathbb{R}^2$ is the unit disk, $v:\mathcal D\to \mathbb{R}^3$, and $\tilde g:\partial \mathcal D\to \mathbb{R}^3$ is a given smooth map. As $\varepsilon\to 0^+$, we construct multi-bubble solutions concentrating at distinct points, taking around each po…

This https://arxiv.org/abs/2503.21940 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Normalized vector solutions of nonlinear Schrödinger systems
Given $μ>0$ we look for solutions $ λ\in\mathbb{R}$ and $v_1,\dots,v_k\in H^1(\mathbb{R}^N)$ of the system \[ \begin{cases} \displaystyle -Δv_i+ λv_i+V_i(x)v_i = \sum_{\substack{j=1}}^kβ_{ij} v_iv_j^2 &\text{ in } \mathbb{R}^N, \text{ } i=1,\dots,k,\newline \displaystyle \int_{\mathbb{R}^N} \left(v_1^2+\dots+v_k^2 \right)\mathrm{d} x = μ, \end{cases}\] where $N=1,2,3$, $V_i:\mathbb R^N\to \mathbb R$ and $β_{ij}\in\mathbb{R}$ satisfy $β_{ij}=β_{ji}$ and $β_{ii}>0$. Under …

The fully nonlinear Loewner-Nirenberg problem: Liouville theorems and counterexamples to local boundary estimates
Jonah A. J. Duncan, Luc Nguyen
https://arxiv.org/abs/2507.16383

The fully nonlinear Loewner-Nirenberg problem: Liouville theorems and counterexamples to local boundary estimates
In this paper we give a complete classification of positive viscosity solutions $w$ to conformally invariant equations of the form \begin{align}\label{ab}\tag{$*$} \begin{cases} f(λ(-A_w)) = \frac{1}{2}, \quad λ(-A_w)\inΓ& \text{in }\mathbb{R}_+^n \newline w = 0 & \text{on }\partial\mathbb{R}_+^n, \end{cases} \end{align} where $A_w$ is the Schouten tensor of the metric $g_w = w^{-2}|dx|^2$, $Γ\subset\mathbb{R}^n$ is a symmetric convex cone and $f$ is an associated de…

Spike layered solutions for elliptic systems on Riemannian Manifolds
Anusree R Kannoth, Bhakti Bhusan Manna
https://arxiv.org/abs/2506.20300 https://

Spike layered solutions for elliptic systems on Riemannian Manifolds
In this article, we study the following Hamiltonian system: \begin{equation*} \begin{cases} \begin{aligned} &-\varepsilon^{2}Δ_{g} u +u = |v|^{q-1}v, &-\varepsilon^{2}Δ_{g} v +v = |u|^{p-1}u && \text{ in } \mathcal{M}, & \quad u,v >0 && \text{ in } \mathcal{M}, \end{aligned} \end{cases} \end{equation*} where $\mathcal{M}$ is a smooth, compact and connected Riemannian manifold of dimension $N\geq 3$ without boundary. The exponents $p,q>1$ are assumed to lie be…

Normalized solutions for the nonlinear Schr\"{o}dinger equation with potentials
Matteo Rizzi, Xueqin Peng
https://arxiv.org/abs/2507.20961 https://arx…

Normalized solutions for the nonlinear Schrödinger equation with potentials
In this paper, we find normalized solutions to the following Schrödinger equation \begin{equation}\notag \begin{aligned} &-Δu-\fracμ{|x|^2}h(x)u+λu =f(u)\quad\text{in}\quad\mathbb{R}^{N},\\ & u>0,\quad \int_{\mathbb{R}^{N}}u^2dx=a^2, \end{aligned} \end{equation} where $N\geq3$, $a>0$ is fixed, $f$ satisfies mass-subcritical growth conditions and $h$ is a given bounded function with $||h||_\infty\le 1$. The $L^2(\mathbb{R}^N)$-norm of $u$ is fixed and $λ$ appea…

Boundedness and asymptotic stability of classical solutions to a model for tuberculosis granuloma formation
Masaaki Mizukami, Yuya Tanaka
https://arxiv.org/abs/2506.16752

Boundedness and asymptotic stability of classical solutions to a model for tuberculosis granuloma formation
This paper deals with a problem which describes tuberculosis granuloma formation \begin{align*} \begin{cases} u_t = Δu - \nabla \cdot (u \nabla v) - uv - u + β, &x \in Ω,\ t>0, \\ v_t = Δv + v -uv + μw, &x \in Ω,\ t>0, \\ w_t = Δw + uv - wz - w, &x \in Ω,\ t>0, \\ z_t = Δz - \nabla \cdot (z \nabla w) + f(w)z -z, &x \in Ω,\ t>0 \end{cases} \end{align*} under homogeneous Neumann boundary conditions and initial conditions, where $Ω\subset \mathbb{R}^n$ ($n\ge 2$) is a smooth bounded …

Solutions with large number of peaks for a slightly supercritical nonlinear equation in dimension three
Yixing Pu
https://arxiv.org/abs/2506.01652 https://…

Solutions with large number of peaks for a slightly supercritical nonlinear equation in dimension three
We investigate the existence of solutions to the semilinear equation with a slightly supercritical exponent in dimension three, \begin{align*} -Δu=K(x) u^{5+μ},\quad u>0 ~\text{in}~ \mathbf{B}, \quad u=0 ~\text{on}~ \partial \mathbf{B}, \end{align*} where $μ>0$, $\mathbf{B}$ is the unit ball in $\mathbb{R}^3$, $K(x)$ is a nonnegative radial function under suitable condition on $K$. We prove the existence of positive multi-peak solutions for $μ>0$ small enough. All peaks of our solut…

Existence and multiplicity of normalized solutions for the quasi-linear Schr\"{o}dinger equations with mixed nonlinearities
Qihan He, Hao Wang
https://arxiv.org/abs/2507.00375

Existence and multiplicity of normalized solutions for the quasi-linear Schrödinger equations with mixed nonlinearities
In this paper, we study the existence and multiplicity of the normalized solutions to the following quasi-linear problem \begin{equation*} -Δu-Δ(|u|^2)u+λu=|u|^{p-2}u+τ|u|^{q-2}u, \text{ in }\mathbb{R}^N,~ 1\leq N\leq4, \end{equation*} with prescribed mass $$\int_{\mathbb{R}^N}|u|^2dx=a ,$$ where $λ\in\mathbb{R}$ appears as a Lagrange multiplier and the parameters $a,τ$ are all positive constants. We are concerned about the mass-mixed case $2

Normalized solutions for Choquard equations with critical nonlinearities on bounded domains
Ru Yan
https://arxiv.org/abs/2506.19872 https://

Normalized solutions for Choquard equations with critical nonlinearities on bounded domains
The aim of this work is the study of the existence of normalized solutions to the nonlinear Schrödinger equation with nonlocal nonlinearities: \begin{equation}\nonumber \left\{\begin{aligned} &-Δu =λu+(I_α*|u|^{2_α^*})|u|^{2_α^*-2}u+a(I_α*|u|^p)|u|^{p-2}u,\ x\inΩ, &u>0\ \text {in}\ Ω,\ u=0\ \text {on}\ \partial Ω,\ \int _Ω|u|^2dx=c, \end{aligned} \right. \end{equation} where $c>0,\ α\in (0,N),\ \frac{N+α+2}{N}

Weak comparison principle for widely degenerate elliptic equations
Antonio Giuseppe Grimaldi, Stefania Russo
https://arxiv.org/abs/2506.22128 https://

Weak comparison principle for widely degenerate elliptic equations
We prove a comparison principle for local weak solutions to a class of widely degenerate elliptic equations of the form \begin{equation} -\text{div} \left( \left(|Du|-1 \right)^{p-1}_+\frac{Du}{|Du|} \right) = f(x,u) \qquad \text{ in } Ω,\notag \end{equation} where $p \ge 2$ and $Ω$ is an open subset of $\mathbb{R}^{n}$, $n\geq2$. Moreover, we establish some second order regularity results of the solutions, that yields a weighted Sobolev inequality with widely dege…

On Growth of Sobolev norms for cubic Schr\"odinger equation with harmonic potential in dimensions $d=2,3$
Yilin Song, Ruixiao Zhang, Jiqiang Zheng
https://arxiv.org/abs/2506.17731

On Growth of Sobolev norms for cubic Schrödinger equation with harmonic potential in dimensions $d=2,3$
In this article, we study the growth of higher-order Sobolev norms for solutions to the defocusing cubic nonlinear Schrödinger equation with harmonic potential in dimensions $d=2,3$, \begin{align}\label{PNLS} \begin{cases}\tag{PNLS} i\partial_tu-Hu=|u|^{2}u,&(t,x)\in\mathbb{R}\times\mathbb{R}^d,\\ u(0,x)=u_0(x), \end{cases} \end{align} where $H=-Δ+|x|^2$. Motivated by Planchon-Tzvetkov-Visciglia [Rev. Mat. Iberoam., 39 (2023), 1405-1436], we first establish the bilinear St…

Low regularity well-posedness of nonlocal dispersive perturbations of Burgers' equation
Luc Molinet, Didier Pilod, St\'ephane Vento
https://arxiv.org/abs/2506.17801

Low regularity well-posedness of nonlocal dispersive perturbations of Burgers' equation
We consider the Cauchy problem associated to a class of dispersive perturbations of Burgers' equations, which contains the low dispersion Benjamin-Ono equation, (also known as low dispersion fractional KdV equation), $$ \partial_tu-D_x^α\partial_xu=\partial_x(u^2) \, ,$$ and prove that it is locally well-posed in $H^s(\mathbb K)$, $\mathbb K=\mathbb R$ or $\mathbb T$, for $s>s_α$, where \begin{equation*} s_α=\begin{cases} 1-\frac{3α}4 & \text{for} \quad \frac23 \le α\le 1; \frac 32(1-�…

Existence and asymptotical behavior of normalized solutions to focusing biharmonic HLS upper critical Hartree equation with a local perturbation
Jianlun Liu, Hong-Rui Sun, Ziheng Zhang
https://arxiv.org/abs/2507.14896

Existence and asymptotical behavior of normalized solutions to focusing biharmonic HLS upper critical Hartree equation with a local perturbation
This paper is concerned with the following focusing biharmonic HLS upper critical Hartree equation with a local perturbation $$ \begin{cases} Δ^2u-λu-μ|u|^{p-2}u-(I_α*|u|^{4^*_α})|u|^{4^*_α-2}u=0\ \ \mbox{in}\ \mathbb{R}^N, \\[0.1cm] \int_{\mathbb{R}^N} u^2 dx = c, \end{cases} $$ where $0<α0$, $2+\frac{8}{N}=:\bar{p}\leq p<4^*:=\frac{2N}{N-4}$, $4^*_α:=\frac{N+α}{N-4}$, $λ\in \mathbb{R}$ is a Lagrange multiplier and $I_α$ is th…

Schauder estimates for parabolic $p$-Laplace systems
Verena B\"ogelein, Frank Duzaar, Ugo Gianazza, Naian Liao, Christoph Scheven
https://arxiv.org/abs/2507.15722

Schauder estimates for parabolic $p$-Laplace systems
We establish the local Hölder regularity of the spatial gradient of bounded weak solutions $u\colon E_T\to\R^k$ to the non-linear system of parabolic type \begin{equation*} \partial_tu-\Div\Big( a(x,t)\big(μ^2+|Du|^2\big)^\frac{p-2}2Du\Big)=0 \qquad\mbox{in $E_T$}, \end{equation*} where $p>1$, $μ\in[0,1]$, and the coefficient $a\in L^\infty(E_T)$ is bounded below by a positive constant and is Hölder continuous in the space variable $x$. As an application, we prove Hölder esti…

Normalized solutions for the NLS equation with mixed fractional Laplacians and combined nonlinearities
Shubin Yu, Chen Yang, Chun-Lei Tang
https://arxiv.org/abs/2506.20943

Normalized solutions for the NLS equation with mixed fractional Laplacians and combined nonlinearities
We look for normalized solutions to the nonlinear Schrödinger equation with mixed fractional Laplacians and combined nonlinearities $$ \left\{\begin{array}{ll} (-Δ)^{s_{1}} u+(-Δ)^{s_{2}} u=λu+μ|u|^{q-2}u+|u|^{p-2}u \ \text{in}\;{\mathbb{R}^{N}}, \\[0.1cm] \int_{\mathbb{R}^{N}}|u|^2\mathrm dx=a^2, \end{array} \right. $$ where $N\geq 2,\;00$ and $λ\in\mathbb R$ appears as an unknown Lagrange multiplier. We mainly focus on some special cases, inc…

A dichotomy result for a modified Schr\"odinger equations on unbounded domains
Anna Maria Candela, Giuliana Palmieri, Addolorata Salvatore
https://arxiv.org/abs/2507.18528 …

A dichotomy result for a modified Schrödinger equations on unbounded domains
This article aims to investigate the existence of bounded positive solutions of problem \[ (P)\qquad \left\{ \begin{array}{ll} - {\rm div} (a(x,u,\nabla u)) + A_t(x,u,\nabla u) = g(x,u) &\hbox{in $Ω$,}\\ u\ = \ 0 & \hbox{on $\partialΩ$,} \end{array}\right.\] with $A_t(x,t,ξ) = \frac{\partial A}{\partial t}(x,t,ξ)$, $a(x,t,ξ) = \nabla_ξA(x,t,ξ)$ for a given $A(x,t,ξ)$ which grows as $|ξ|^p + |t|^p$ , $p > 1$, where $Ω\subseteq \mathbb{R}^N$, $N \ge 2$, is an open co…