Papers

We prove an interior $C^{1,\alpha}$ estimate for solutions of the homogeneous linearized Monge–Ampère equation in dimension two under the assumption \[ 0<\lambda\leq \det D^2\varphi\leq\Lambda<+\infty. \] No continuity assumption on the Monge–Ampère density is required. Our result is an affine-invariant analogue of the classical Morrey–Nirenberg $C^{1,\alpha}$ estimate in two dimensions. The core of the proof is the partial Legendre transform. After the transform, the first derivatives of the solution are quotients of adjoint solutions for a uniformly elliptic non-divergence form equation. Bauman's Harnack inequality gives the Hölder control of the quotient, while the Jacobian identity of the partial Legendre transform and a Caccioppoli estimate give its local boundedness. As an application, we prove a Liouville theorem for entire solutions with at most linear growth.

We consider the damped nonlinear Klein-Gordon equation \begin{align*} \partial_t^2u-\Delta u+2\alpha\partial_tu+u-|u|^{p-1}u=0 \end{align*} on $\mathbb{R}^d$, where $\alpha>0$, $2\leq d\leq5$, and $p>2$ is in the energy-subcritical range. We classify global solutions that converge to a superposition of two positive and two negative translates of the ground state, without imposing any symmetry, coplanarity, or a priori geometric condition on their centers. We prove that every such four-soliton configuration is asymptotically either an alternating collinear configuration or an expanding rhombus with alternating signs. We further determine the precise long-time asymptotics of all four centers.

We resolve a question posed by Harvey and Lawson concerning the uniqueness of continuous viscosity solutions to the Dirichlet problem for the Lagrangian mean curvature equation on the unit ball with continuous boundary data. For every dimension $n\geq 2$, we construct a continuous phase $\psi \colon \overline{B_1}\to(-n\frac{\pi}{2},n\frac{\pi}{2})$ and continuous boundary data $g \colon \partial B_1\to \mathbb{R}$ for which the Dirichlet problem \[ \sum_{i=1}^n \arctan\lambda_i(D^2u)=\psi(x)\quad\text{in }B_1, \qquad u=g\quad\text{on }\partial B_1, \] admits a continuum of distinct continuous viscosity solutions.

We study boundary-weighted barycenter spaces, in which an interior support point has cost (or weight) two and a boundary support point has cost one. These spaces are finite-dimensional topological models for the concentration patterns produced by noncompact boundary Euler–Lagrange functionals: an interior bubble carries twice the quantized mass of a boundary bubble, and very negative sublevels are therefore modeled by boundary-weighted rather than ordinary barycenters. The paper gives a systematic algebraic-topological treatment of these spaces. We construct the mixed strata, the closed-stratum poset, and the triangular boundary-weighted colimit filtration; compute the Euler characteristic; and prove a homology decomposition in terms of ordinary barycenter spaces of \(\partial M\) and \(M/\partial M\). We then specialize the formulas to compact connected orientable surfaces with boundary and to hemispheres, obtaining explicit rational Betti polynomials and, for surfaces, the mod-two polynomials required by the mean-field application. Finally, we explain how these polynomials measure the topology at infinity in the resonant Neumann mean-field equation on a compact surface with boundary.

We study quantitative mean-field limits for classical particles with Yukawa (screened Coulomb) interactions in every fixed dimension $d\ge2$, uniformly as the screening parameter $\kappa$ tends to zero. Our main result is a first-order commutator estimate in the natural Yukawa modulated energy, with an additive error of order $N^{-2/d}$ for $d\ge3$ and $(1+\log N)/N$ for $d=2$. It requires only a bounded reference density and a Lipschitz transport field, with no uniform lower bound on interparticle distances and no negative power of $\kappa$. The modified Helmholtz operator $-\Delta+\kappa^2$ creates the main new difficulty. Truncating the potential to a constant inside each truncation ball produces a surface charge and a positive volume charge whose total mass is strictly less than one. We keep the reference density unchanged and control this loss of mass through an exact Green function representation and renormalized energy identities. A stress-energy identity with interface terms and averaging over the truncation radii then give the uniform commutator estimate. Combined with the modulated energy dissipation identity and a normalized quadratic transport cost, this estimate yields weak–strong stability, propagation of chaos, and time-integrated control of the mean-square difference between empirical and mean-field forces. We also prove a quantitative Yukawa-to-Coulomb limit. For smooth product data, $N\to\infty$ and $\kappa\downarrow0$ may be taken simultaneously with no relation between their rates; in dimension three, a direct comparison at the particle level also holds for general symmetric initial laws with finite initial error.

We study uniform $L^2$ non-localization of Dirichlet Laplacian eigenfunctions on planar integrable polygons, with particular emphasis on spectral degeneracy and quantitative dependence on the observation set. For a measurable set $V\subset\Omega$ of positive measure, define \[ C_2(V;\Omega) := \inf_{\lambda\in\sigma(-\Delta_\Omega)} \inf_{0\neq u\in E_\lambda(\Omega)} \frac{\|u\|_{L^2(V)}}{\|u\|_{L^2(\Omega)}}. \] We prove that $C_2(V;\Omega)>0$ for rectangles, isosceles right triangles, equilateral triangles, and hemi-equilateral triangles, uniformly over the complete eigenspaces and hence independently of spectral multiplicity. For rectangles, we obtain a quantitative refinement. If the reflected extension of $V$ has finite perimeter and $\alpha=|V|/|\Omega|$, we derive an explicit sufficient threshold $\lambda_*(\Omega,V)$ such that every eigenfunction with $\lambda\geq\lambda_*(\Omega,V)$ satisfies \[ \frac{\|u\|_{L^2(V)}}{\|u\|_{L^2(\Omega)}} \geq \left[ \frac{\alpha}{2} \left( 1-\frac{\sin(\pi\alpha)}{\pi\alpha} \right) \right]^{1/2}. \] We further establish stability under bounded real-valued potentials on the rectangular branch and under controlled spectral defects, yielding corresponding non-localization results for sufficiently accurate quasimodes and narrow spectral clusters.

We derive an exact and novel expression for an averaged two-point correlation function in the statistically stationary, forced-dissipative two-dimensional Navier-Stokes equations subject to the Coriolis force under the beta-plane approximation. This identity is related to the so-called geostrophic balance: it connects the effect of the Coriolis force to the pressure gradient through a two-point correlation function. Additionally, we provide sufficient conditions under which the asymptotics of the averaged third-order structure function at large spatial scales follow the universal third-order law of two-dimensional turbulence in the absence of the Coriolis force. This complements our previous results on small spatial scales. Together, our results provide a clear picture of the role of the Coriolis force in beta-plane turbulence. On the one hand, the spherically averaged rates of enstrophy and energy transfer are not affected by the Coriolis force. On the other hand, the Coriolis force contributes to anisotropic large-scale organization by altering the spatial distribution of energy and promoting the formation of zonal structures. The proof relies on a new formulation of the Karman-Howarth-Monin relation. For the geostrophic balance, we use a novel antisymmetric projection of the KHM relation under which only the pressure and Coriolis terms survive. For the cascade laws, we show that the Coriolis contribution to the averaged classical KHM relation vanishes identically at any scale.

In this paper, numerical solutions of a class of nonlinear ordinary differential equations are obtained using a collocation method based on the shifted Horadam polynomials. The proposed problem, which is of the fourth-order, satisfies a class of two-point boundary conditions. We first discuss definitions and basic properties of the Horadam polynomials before presenting new and useful differentiation formulae for them. Novel interesting identities are deduced from the differentiation properties. The collocation method under consideration assumes that the solution of the proposed problem can be expressed as a shifted Horadam polynomial series. To determine the expansion coefficients of the series solution, one collocates at the zeros of the shifted Horadam polynomials, and the proposed boundary value problem is subsequently reduced to a set of nonlinear algebraic equations. These algebraic equations are then solved using Newton's iterative method to obtain the numerical values of the expansion coefficients of the shifted Horadam polynomial series solution. Several examples of the proposed nonlinear boundary value problem are considered to demonstrate the accuracy, efficiency, and reliability of the proposed method. Numerical solutions and errors obtained are compared with existing solutions. Comparisons of results, which are shown in tables and graphs, clearly reveal that the shifted Horadam collocation method outperforms the existing methods.

Touching and dividing cell nuclei may appear as connected structures in microscopy images, making it difficult to distinguish neighboring nuclei during segmentation. We introduce and analyze a regularized subjective-surface model designed for this setting. On a smooth bounded domain $\Omega\subset\mathbb{R}^n$ with homogeneous Dirichlet boundary conditions, the model is \[ u_t = \nu\Delta u + \left(\varepsilon^2+|\nabla u|^2\right)^{1/2} \operatorname{div}\!\left( G(x)\frac{\nabla u} {\left(\varepsilon^2+|\nabla u|^2\right)^{1/2}} \right) - \mu\Lambda(x)H_\eta(u-q). \] Here, $\varepsilon>0$ and $\nu>0$ are fixed regularization parameters, $G$ is a strictly positive smooth edge coefficient, and the nonnegative interaction weight $\Lambda$ incorporates fixed information about neighboring nucleus candidates. The principal objective of this work is to establish an existence theory for the proposed model. For compatible $C^{2+\alpha}$ initial data taking values in $[0,1]$, we prove the existence and uniqueness of a global classical solution whose restriction to every finite time interval is Schauder-classical, together with preservation of the physical range, finite-time Schauder estimates, and $L^\infty$-nonexpansive dependence on the initial data. The main analytical step is a global spatial-gradient bound, obtained by combining gradient estimates near $\partial\Omega$ with interior gradient estimates. These results provide a mathematical foundation for applying the model to the analysis of touching and dividing nuclei in 3D and 3D+time microscopy image data.

In this paper, we consider a time-periodic nonlocal dispersal susceptible-infected-susceptible (SIS) epidemic model with saturated incidence and Neumann boundary conditions in a spatiotemporally heterogeneous environment. First, we define the basic reproduction number for the model, which depends on dispersal rates, total population size and saturation parameters, and is quite different from those of models incorporating standard or bilinear incidence. Then, we establish its variational characterization and investigate the impacts of those parameters on it. Next, we explore the existence, uniqueness and global attractivity of the equilibria. We also analyze the asymptotic behaviors of endemic steady states for small saturation parameters, and for both small and large diffusion rates. Our results show that the saturation effect enables the total population size to have a significant impact on the disease dynamics. Furthermore, as the saturation parameter tends to zero, the basic reproduction number and the endemic equilibrium reduce to those of the standard-incidence model, respectively. Finally, we support our findings by numerical simulations.

In this paper, we give a partial answer to Brezis' Open Problem~2.1, which concerns the uniqueness of solutions to the Ginzburg–Landau equation in the unit disc with the degree-one boundary condition. Let $\lambda_1$ be the first Dirichlet eigenvalue of $-\Delta$ in the unit disc and set $\varepsilon_*:=\lambda_1^{-1/2}$. We prove that there exists $\delta>0$ such that the radial solution is the unique weak solution for every $\varepsilon\in(\varepsilon_*-\delta,\infty)$. More generally, we establish the analogous uniqueness result for the Ginzburg–Landau system on bounded connected $C^{1,1}$ domains in $\mathbb R^N$, $2\leq N\leq4$, with nontrivial boundary data. In particular, uniqueness persists slightly below the convexity threshold $\varepsilon_*$, where the strict convexity argument is no longer available. The proof combines strict convexity for $\varepsilon\geq\varepsilon_*$ with a compactness argument, nondegeneracy of the solution at $\varepsilon=\varepsilon_*$ and the implicit function theorem.

We construct surface diffusion flows $f:\mathbb{T}^2\times[0,T)\to\mathbb{R}^3$ that drive smooth closed embedded tori to pinchoff in finite time. The flow remains embedded for $t\in[0,T)$ and develops a curvature singularity only at a distinguished point $p$ as $t\nearrow T$. Away from $(p,T)$ the flow converges smoothly as $t\nearrow T$. We characterise the singularity profile: If $A(t)$ denotes the radius of the waist, and the surface is given locally near the waist by the rotation of a radial graph $(z,t)\mapsto r(z,t)$, then there exists a constant $\mu>0$ and smooth function $U:\mathbb R\to\mathbb R$ such that \[ A(t)=\{4\mu(T-t)\}^{1/4}(1+o(1)), \qquad A(t)^{-1}r(A(t)\zeta,t)\stackrel{C^\infty_{loc}}{\longrightarrow} U(\zeta). \] Here $U$ is a rigorous realisation of the classical fundamental positive even conical similarity profile first computed numerically by Wong, Miksis, Voorhees and Davis and subsequently analysed by Bernoff, Bertozzi and Witelski.

We introduce weighted Laplacian flow (WLF), a deterministic particle-flow framework for sampling from a target distribution known only up to normalization. The proposed method evolves the logarithmic density ratio through a transport equation and determines the particle velocity from a target-weighted Poisson problem, resulting in a nonlocal and kernel-free mechanism for redistributing mass. For bounded smooth domains, we establish global well-posedness of the flow for Lipschitz initial data and prove that the flow contracts the discrepancy between the evolving and target densities at an exact exponential rate in terms of the oscillation of the log-density ratio, as well as in the associated projective metric. The relative entropy also satisfies a precise dissipation relation involving the two directional Kullback–Leibler divergences. In addition, we identify a variational interpretation of WLF. The method can be viewed as a gradient flow of the reverse Kullback–Leibler divergence under a target-anchored metric. Consequently, the method achieves an explicit relaxation scale without requiring log-concavity or spectral-gap assumptions on the target distribution. Numerical experiments on multimodal, heavy-tailed, and high-dimensional targets demonstrate the effectiveness of the proposed approach in capturing long-range mass transport.

We establish the orthonormal Strichartz estimates in the abstract Keel–Tao framework, under precisely the same hypotheses as in their original theorem. As an important consequence, we resolve the endpoint conjecture for the Schrödinger equation in dimensions $d\ge 2$, which was first raised by Frank, Lewin, Lieb, and Seiringer. This consequence also yields an affirmative answer to the conjecture for the free transport equation, which was stated in dual form by Bennett, Bez, Gutiérrez, and Lee. As a further application, we also establish a refinement of the Strichartz estimate for a single function.

For a stochastic partial differential equation posed on a Gelfand triple and satisfying the fully local monotone conditions of Röckner, Shang and Zhang, we compute the logarithmic derivative of the law of the solution at a fixed time along a prescribed direction of the state space. The formula is intrinsic, being expressed through the Hilbert-Schmidt Malliavin derivative $\Phi_r = \mathcal{D}_r X(t)$ and the covariance $\gamma_t = \int_0^t \Phi_r \Phi_r^{*} \,\mathrm{d}r$ alone, so that neither the inversion of the first variation used in finite dimensions nor uniform Malliavin-Sobolev bounds on the Tikhonov family are called upon. It is obtained from an integration-by-parts identity on an abstract Hilbert space, a Moore-Penrose construction of a covering field on Wiener space, and a trace formula for the Tikhonov limit, specialised to the equation through the representation $\Phi_r = Y(t,r)\mathcal{B}(r,X(r))$ of the Malliavin derivative by the first variation; the stochastic $p$-Laplacian and the two-dimensional Navier-Stokes equation are treated in detail. Beyond the variational class, a scalar reduction gives an integration-by-parts identity for the law of a pairing $\langle u(t),\varphi\rangle$ which passes to the renormalised limit for the singular equations of Bruned, Chandra, Chevyrev and Hairer, and which is represented by a logarithmic derivative under second-order Malliavin smoothness and negative-moment hypotheses.

We prove a weighted Bossel-Daners transfer principle for the first Robin eigenvalue of the $p$-Laplacian under $w=m^{1/p'}$, where $p'=p/(p-1)$. The argument combines double-density isoperimetry with spectral admissibility for singular weights and normalized-flux monotonicity. It uses an exact $BV$ zero-extension formula, an $L^{p'}(m\,dx)$ selection lemma, and a nonatomic rank map that remains well defined on positive-measure level sets. For the singular pair \[ m_b(x)=|x|^b, \qquad w_b(x)=|x|^{b/p'}, \] known power-weight isoperimetry provides the geometric input. We establish spectral admissibility up to the critical exponent $p=N$ and, for the positive radial first eigenfunction $z=z(r)$ on the centered ball $B_R$, derive the integrated singular radial equation and center asymptotics and prove directly that \[ \theta_R'(r)>0 \quad(0<r<R), \qquad \theta_R(r)=\frac{|z'(r)|^{p-1}}{r^{b/p'}z(r)^{p-1}}, \] with $\theta_R(0)=0$ and $\theta_R(R)=\beta$. Let $\Omega\subset\mathbb{R}^N$ be a finite union of bounded connected Lipschitz domains whose closures are pairwise disjoint, let $\Omega_b^\sharp$ be the centered ball of equal $|x|^b$-weighted volume, and let $\lambda_{1,\beta}^b$ denote the first eigenvalue for the displayed pair. Consequently, \[ \lambda_{1,\beta}^{b}(\Omega_b^\sharp) \le \lambda_{1,\beta}^{b}(\Omega) \] whenever $N\ge 2$, $1<p\le N$, $-p<b<0$, and $\beta>0$. The range $1<p<N$ is complementary to the previously known $p\ge N$ weighted Talenti theory; at the shared endpoint $p=N$, the present proof also permits the singular contact $0\in\partial\Omega$.

We investigate the existence and non-existence of large-amplitude boundary layer solutions to the outflow problem for the one-dimensional full compressible Navier-Stokes equations on the half-line $\mathbb{R}_+$. Through a delicate global phase-plane analysis, we substantially extend the small-amplitude boundary layer solutions obtained via the center-manifold approach to the large-amplitude regime. Based on the sign of $M_+-1$ (with $M_+$ denoting the Mach number at the right end state) and the Prandtl-number-related parameter $\zeta$, we provide a complete characterization of the existence and non-existence of large-amplitude boundary layer solutions for this half-space outflow problem. In sharp contrast to the corresponding inflow problem, boundary layer solutions to the outflow problem are permitted to be non-monotone, and their existence region in the phase plane can be an open set.

We study positive solutions of the partially confined stationary nonlinear Schrödinger equation $$-\Delta u+|y|^2u+\lambda u=g(u),\quad (y,z)\in\mathbb{R}^d\times\mathbb{R}^{m},\quad 1\leq d<N,\quad m:=N-d.$$ The spectrum of $-\Delta + |y|^2$ is given by $[d,\infty)$. We prove that $\lambda\geq-d$ is necessary for positive solutions in the classes considered here, and the threshold $\lambda=-d$ yields several Liouville type results. In particular, for the pure-power equation $$(-\Delta + |y|^2-d)u=u^p,$$ we prove that any nonnegative $H^1(\mathbb{R}^N)$ weak solution is trivial whenever $1\leq p\leq \max\{(N+2)/(N-2), m/(m-2)\}$ for $m\geq 3$ or $p\geq 1$ for $m=1,2$. The proof combines a half-space oscillator gap, moving planes at the spectral threshold, a Picone inequality, a strict Gaussian second-moment inequality, and anisotropic Pohozaev identities. For $\lambda>-d$, we establish the existence of positive solutions under some standard assumptions. Furthermore, every positive solution decaying at infinity is radially symmetric and strictly decreasing in the confined variables and, up to one common translation, radially symmetric and strictly decreasing in the free variables. \vskip 0.2in Dedicated to our supervisor Prof. Wenming Zou on the occasion of his 60th birthday.

We study the orbital stability of solitary waves to the generalized Benjamin–Bona–Mahony equation $$ u_t+u_x+\kappa\frac{p+1}{2}(u^p)_x-u_{txx}=0, \qquad (t,x)\in\mathbb R^+\times\mathbb R, $$ where $\kappa\in\mathbb R\setminus\{0\}$ and $p\geq2$ is an integer. This equation admits solitary waves of the form $$ u(t,x)=\phi_c(x-ct). $$ At the negative critical endpoint speed $$ c=c_p^-= \frac{p-1}{2(p+1)} \left(1-\sqrt{\frac{p+3}{2}}\right)<0, $$ we prove that the corresponding solitary wave is orbitally unstable whenever either $\kappa<0$, or $\kappa>0$ and $p$ is even. The proof combines a codimension-two coercivity estimate, two-parameter modulation, and a corrected localized virial functional adapted to the degeneracy of the momentum slope at the endpoint.

We show that the inviscid shallow water equations with gravity and surface tension are locally well-posed in dimensions one and two. In contrast to previous works by Benzoni-Gavage, Danchin, and Descombes, we work only in the real-valued phase space and make use of the symmetry of the system. Our alternative proof for local well-posedness sheds some light on the symmetric structure of the shallow water system and can be used for further study, including designing numerical scheme, dispersion estimate, etc.

We study zeros transported by the heat semigroup from stationary solutions of the sparse operators $L_n=c_{n,0}\D^n+c_{0,n-1}x^{n-1}$. The starting point is the exact cubic boundary obtained by Hernández-del-Valle and Guerra-Polania~\cite{HerGuerra}. We show that the hypergeometric function used there is the residue-$1$ component of a naturally graded stationary solution space, and that heat transport preserves this grading after the scaling $x=t^ny$. This identifies the cubic boundary as the transported zero of a specific stationary mode. We then study $L_4=c_{4,0}\D^4+c_{0,3}x^3$. Its $6\times6$ boundary-jet determinant leads to the universal scaling \[ F(t)=\rho\left(\frac t2\right)^4 \Phi\!\left(\rho^2\left(\frac t2\right)^7\right), \qquad \rho=\frac{c_{0,3}}{c_{4,0}}. \] The resulting nonlinear equation for $\Phi$ is singular at the origin. After an explicit desingularization, its initial compatibility condition factors as \[ (5\Phi'(0)-1)(6\Phi'(0)-1)=0. \] Consequently there are exactly two normalized graded analytic boundary germs. We place the desingularized equation in Briot–Bouquet form, prove local existence and uniqueness of both branches, and give a triangular recursion for all their Taylor coefficients. Finally, we prove that the two branches are generated respectively by the residue-$1$ and residue-$3$ stationary solutions. The quadratic and cubic cases provide the established baseline; the fourth-order problem is the first point at which distinct stationary modes produce distinct analytic branches. We do not claim a general branch-count theorem.

Memory effects and anomalous diffusion arising in models of transport in complex media, fractals, and viscoelastic materials can be described by fractional-differential equations, such as the time-fractional diffusion equation $D^\alpha_t u - \Delta_x u = 0$. The paper develops a decomposition of solutions of these equations into a product of spatial and temporal components, corresponding to a freeze-out at long times. For fractional diffusion with the Caputo derivative the spatial factor is the inverse-Laplacian $(-\Delta)^{-1}$ of the initial data, while for the Riemann-Liouville derivative it is the inverse bi-Laplacian $ (-\Delta)^{-2}$. In both cases, the temporal factor of the solution is a scaled negative power of time. This memory artifact explicitly encodes the initial data, which gives a simple and robust way to reconstruct the initial conditions in the backward-in-time inverse problem with solution values measured at long times. We derive characteristic timescales for Mittag-Leffler functions, which correspond to such factorization in anomalous diffusion. This also enables an accurate approximation of the number of real zeros of the Mittag-Leffler function. We apply these results to heat transfer on a comb, a model which manifests subdiffusion arising from the comb's fractal structure.

Electrodiffusion has been extensively studied in the isothermal setting, whereas the mathematical theory of thermally coupled electrodiffusion remains comparatively underdeveloped. We investigate a non-isothermal electrodiffusion model describing the evolution of multiple ionic species with different diffusivities and valences in a two-dimensional incompressible viscous fluid. The coupling to a spatially and temporally varying temperature gives rise to a nonlinear and nonlocal system with logarithmic nonlinearities in the ionic fluxes. We establish local well-posedness for strictly positive initial concentrations and prove global well-posedness when the initial temperature is close to a homogeneous state by developing a new entropy structure tailored to thermodiffusive effects. No smallness assumption is imposed on the initial ionic concentrations or fluid velocity. To overcome the singularity of the logarithmic terms, we develop a novel cutoff-mollification regularization, derive uniform logarithmic estimates, and prove persistence of strict positivity of the ionic concentrations. This positivity removes the singular behavior of the logarithmic nonlinearities and is essential for the uniqueness argument. These results provide a rigorous mathematical framework for the analysis of non-isothermal electrohydrodynamics systems.

It is shown that the solutions to a wide class of semi-linear parabolic equations with analytic (holomorphic) nonlinearity, with spatially periodic boundary conditions, are analytic in time with values in the Gevrey class of analytic functions with respect to the spatial variable.

We describe a Lean formalization of Hamilton's 1982 theorem on closed, connected three-manifolds with positive Ricci curvature. The development contains a short-time existence theorem for Ricci flow and substantial geometric-analysis infrastructure: Riemannian tensor calculus, the Levi–Civita connection, Ricci-flow evolution equations, scalar and tensor maximum principles, three-dimensional curvature algebra, preservation of Ricci pinching, and Hamilton's improved pinching estimate. The formalization follows an alternative blow-up route, rather than Hamilton's original normalized-flow proof. Its time-uniform short-time existence, maximal continuation, no-local-collapsing, and Cheeger–Gromov–Hamilton compactness pipelines have been formalized and are included in the artifact, while we give only a brief account of these companion developments and record the interfaces and consequences used by the Hamilton argument; a detailed exposition of their full constructions is deferred to the second author's forthcoming thesis. We interweave representative Lean declarations with their mathematical meaning and record the status and provenance of every major component. All source-level status claims are tied to the source release identified below.

We prove that stable nonlocal minimal cones in three dimensions are flat, for $s$ sufficiently close to zero. Our approach develops new structural properties of stable $s$-minimal surfaces in every dimension as $s \downarrow 0$, notably a compactness theory and pinching theorems near a half-space.

We construct families of sign-changing solutions for the four-dimensional Brezis–Nirenberg problem \[ -\Delta u=u^3+\varepsilon u\quad\text{in }\Omega,\qquad u=0\quad\text{on }\partial\Omega, \] as $\varepsilon\to0^+$. A Lyapunov–Schmidt reduction shows that the location and relative scales of the bubbles are governed by a signed Green–Robin interaction matrix. We formulate an abstract existence criterion in terms of a simple positive eigenvalue admitting a positive eigenvector and a stable critical set. We then apply it to a positive–negative pair in a general domain and to several symmetric multi-peak configurations, including alternating regular polygons, orthogonal polygons, one central peak surrounded by peaks of the opposite sign, and aligned three-, four-, and five-peak patterns. For the two-peak solution we also prove that it has exactly two nodal domains and, under a natural balance condition and connectedness of the boundary, that the closure of its nodal set meets the boundary.

In this work we derive local differential Harnack estimates for positive solutions of filtration equations on complete Riemannian manifolds with a lower Ricci curvature bound by using Nash–Moser iteration. Inspired by earlier work on parabolic frequency functions, we establish some new frequency monotonicity formulas for these equations. The results cover the porous-medium and fast-diffusion equations, as well as several non-power diffusion laws.

We study inverse problems for the elliptic and parabolic obstacle problems from boundary measurements. For the classical elliptic obstacle problem with strictly superharmonic obstacle function, we show that the Dirichlet-to-Neumann map admits a one-sided linearization at every boundary datum lying strictly above the obstacle. The linearized map is the Dirichlet-to-Neumann map for a rough Dirichlet problem on the a priori unknown non-contact set. We show that these linearized Cauchy data uniquely determine the non-contact set up to set of Sobolev $2$-capacity zero and consequently determine the obstacle. Our result applies to the inverse problem for a parabolic obstacle problem where both the coefficient and the obstacle function are time-independent by reducing to the elliptic inverse problem.

We provide an adjustment of the Hardy–Littlewood–Sobolev inequality for p=1 to the anisotropic setting. Several examples of anisotropic Bourgain–Brezis inequalities are obtained as corollaries of the main theorem.