Papers

We prove a conditional Schoen-type Harnack inequality for positive weak solutions of the critical $p$-Laplace equation $$ -\Delta_p u=g(u),\qquad 1<p<n, $$ under a global critical Sobolev growth assumption and the monotonicity condition that $s^{-(p^*-1)}g(s)$ is nonincreasing. The result is conditional on two inputs, the classification of bounded positive entire blow-up limits as Aubin–Talenti $p$-bubbles and a preliminary singular-rate upper control on the normalized necks. Under these two hypotheses, solutions in $B_{3R}$ satisfy $$ \Big(\sup_{B_R}u\Big)\Big(\inf_{B_{2R}}u\Big)^{p-1} \le C R^{p-n}. $$ The main point is a Pohozaev-neck argument which upgrades the preliminary singular decay rate $|x|^{-(n-p)/p}$ to the sharp $p$-harmonic fundamental rate $|x|^{-(n-p)/(p-1)}$. The argument replaces the Kelvin-transform and moving-sphere methods available in the conformally invariant semilinear case $p=2$, but unavailable for the general $p$-Laplacian.

Uncertainty quantification in neural networks prediction is a main issue for usual applications. Our approach seeks at reducing computation costs by directly evaluating uncertainty using PDE's information on the asymptotic variance, rather than the deep ensemble method which may be seen as a Monte Carlo estimation of the prediction, requiring the training of multiple networks. We thus study the law of the limiting process describing the random fluctuations around the mean-field limit of wide two-layer neural networks trained by stochastic gradient descent in a weak-noise regime. Building on a recent trajectorial central limit theorem, in which this limit is characterized as the weak solution of a linear stochastic evolution equation, we identify its law explicitly. More precisely, we show that it is a centered Gaussian process in the dual of a weighted Sobolev space, and we derive a closed covariance representation for the finite-dimensional distributions obtained by testing it against smooth functions. This covariance is expressed through the solution of a backward transport equation with a nonlocal source term, whose coefficients are driven by the mean-field trajectory. As a consequence, by testing against the activation function at a fixed input, we obtain an expression for the limiting variance of the corresponding network-output fluctuations. We illustrate this result numerically on a one-dimensional regression example.

In this paper we consider an implicit semi-discrete approximation of a degenerate reaction-cross-diffusion system. Due to the symmetry in the parabolic part, this system is known to preserve segregation of densities – initially non-overlapping densities belonging to different species remain segregated for all times, which leads to internal layers between different species. We show that time-discrete approximations exist and converge to a weak solution, as the timestep goes to zero.

This paper establishes the global well-posedness for the 3D anisotropic MHD system with partial dissipation: $\Delta_\mathrm{h}u$ for velocity and $\partial_1^2b$ for magnetic field, near background field $(0,1,0)$. Crucially, only horizontal components $(u^\mathrm{h}_0,b^\mathrm{h}_0)$ need to be small in $H^2(\R^3)$, while $(u^3_0,b^3_0)$ can be arbitrarily large. Our analysis develops novel techniques including component-decoupled energies and iterative control of dangerous nonlinearities using the background field structure. This establishes the global result for anisotropic MHD equations allowing large vertical data, breaking the full-smallness requirement of previous works.

The mass-critical NLS on Euclidean space $\R^d$ exhibits a strong mass rigidity: all positive ground states are generated from a single profile and have the same ground state mass $\widehat{M}(Q)$. By appealing to bifurcation methods, Dancer constructed in his seminar paper \cite{DancerSolution} solutions to the corresponding equation on $\R^d\times\T$ which decay in the noncompact directions and are nontrivially periodic in one direction. Such bifurcation approach, however, does not provide any energetic characterization of the solutions, and in particular does not explain their relation to the Euclidean ground-states. By introducing a new strict monotonicity mechanism for the prescribed-mass energy level, combining the semivirial-vanishing geometry framework developed in author's recent work, we prove that for any mass $c\in(0,2\pi\widehat{M}(Q))$ the semivirial-vanishing variational problem $m_c$ admits a normalized Dancer-type optimizer which also solves the focusing mass-critical NLS on $\R^d\times\T$. This also gives a sharp complement for the existence results deduced in our earlier work \cite{Luo_LegendreFenchel} via the Legendre-Fenchel duality.

We solve interface problems on the line between various constant coefficient linear evolution partial differential equations. Our prototypical examples are the heat, linear Schrödinger, Airy, linearized Korteweg de Vries, and biharmonic Schrödinger equations. In each problem, one of the listed equations is posed on one spatial half line, and another on the other half line, with appropriate interface conditions. These problems are solved by means of novel extensions of Fokas's unified transform method and the explicit solution formulae are evaluated using Filon quadrature.

We study a class of nonlinear nonlocal elliptic equations in $\mathbb{R}^N$ involving superpositions of Hartree-type nonlinearities. Motivated by the Schrödinger-Poisson-Slater system, these equations arise as natural generalizations of problems with a single nonlocal interaction term. More precisely, we consider equations driven by a family of Riesz potentials weighted by a positive Borel measure, which gives rise to a superposed nonlocal operator. To treat this problem variationally, we introduce suitable functional settings, namely the superposed Coulomb space and the associated superposed Coulomb-Sobolev space, and study their main properties. Combining variational methods with a recently developed scaling-based critical point theory, we prove existence and multiplicity results for radial solutions. We also investigate a Brezis-Nirenberg-type problem and obtain multiplicity results near eigenvalues of an associated nonlinear eigenvalue problem. Our results extend previous works on single Hartree-type equations and provide a unified framework for treating superpositions of nonlocal interactions of Hartree type.

We establish robust log-convex interpolation inequalities within the scale of Gagliardo seminorms. We achieve this by deriving some Chebyshev-type integral inequalities for general non-synchronous functions. Our primary motivation for establishing these robust interpolation inequalities stems from the study of the asymptotic nonlocal-to-local stability of weak solutions to the boundary Dirichlet problem associated with the regional fractional $p$-Laplacian. More precisely, if $u_s \in W^{s,p}(\Omega)$ weakly satisfies $(-\Delta)_{p, \Omega}^s u_s = f_s $ in $\Omega$ and $ \gamma^s_0(u_s) = g_s$ on $\partial\Omega,$ with $\frac{1}{p} < s \leq 1$ and $\Omega \subset \mathbb{R}^d$ is bounded Lipschitz, then, under appropriate convergence of the data $f_s$ and $g_s$ as $s \to 1^-$, we establish that $\| u_s - u_1 \|_{W^{\eta,p}(\Omega)} \xrightarrow{s \to 1^-} 0 $ for all $0 \leq \eta < 1$.

We propose a unified framework for dimension reduction of partial differential equations posed on thin domains. Our approach combines three complementary ingredients: a careful boundary-condition analysis, an averaging-based splitting for general thin geometries, and a slow-manifold viewpoint for the resulting fast-slow system. Homogeneous Neumann conditions on the thin faces emerge as the most relevant and physical regime because they preserve the transverse zero mode and therefore lead to a genuine lower-dimensional reduced equation. For general thin domains we derive the averaged fast-slow system and isolate the geometry-induced correction term. We then formulate a splitting-based Lyapunov-Perron construction for an exact slow manifold when a suitable spectral decomposition of the slow variable is available, and we construct approximate slow manifolds and corrected reduced dynamics directly from the invariance equation by asymptotic expansion. Moreover, we propose the Schnakenberg reaction-diffusion system as a canonical test problem for comparing the full thin-domain dynamics, the averaged model, and the manifold-corrected reduced dynamics. Finally, we also extend the framework to thin tubular domains and derive the corresponding rescaled fast-slow system in curved geometry.

We study the dynamics of gradient descent in the Edge of Stability regime, where the learning rate is large enough to induce persistent oscillations in the trajectory, which has been linked to better generalization performance. We introduce the mean–fluctuation dynamics, a tractable continuous-time model coupling the window-averaged trajectory to its fluctuation covariance. Among our contributions, we rigorously derive this model from gradient descent in a sharp-valley framework, characterize its stationary states and their linear stability, and establish precise connections with other effective dynamics. Numerical experiments illustrate these predictions and their finite-time limitations. We also study our model in the overparametrized regime of wide two-layer networks at a fixed learning rate, where we rigorously derive a kinetic equation describing weights and their fluctuations as a Wasserstein-2 gradient flow, for which we prove well-posedness, a mean-field limit, and conditional convergence results.

We study the McKean–Vlasov free energy on the unit sphere associated with the unnormalized self-attention (USA) model for noisy transformer dynamics. We prove a sharp global-minimizer dichotomy in every dimension $d\ge2$. There is a unique $\beta_*^{(d)}>0$ such that \begin{equation*} \frac{I_{d/2+1}(\beta_*^{(d)})}{I_{d/2}(\beta_*^{(d)})}=\frac1d, \end{equation*} where $I_\nu$ is the modified Bessel function of the first kind. For $0<\beta\le \beta_*^{(d)}$, the uniform density remains the unique global minimizer up to the linear-stability threshold \begin{equation*} K_\#^{(d)}(\beta)=\frac{\beta^{d/2}}{2^{d/2}\Gamma(d/2)I_{d/2}(\beta)}, \end{equation*} and the phase transition is continuous. For $\beta>\beta_*^{(d)}$, the uniform density is not globally minimizing at $K_\#^{(d)}(\beta)$, so the critical coupling satisfies $K_c<K_\#^{(d)}(\beta)$ and the transition is discontinuous. This result generalizes the authors' recent $d=2$ work arXiv:2604.16288 to arbitrary dimension. The proof uses the sharp Beckner–Onofri/logarithmic Hardy-Littlewood-Sobolev (HLS) inequality on the sphere, together with a Funk–Hecke/Bessel coefficient computation and a degree-two quartic obstruction.

We study a Lamé-Stokes coupled system arising as the incompressible-inclusion limit of periodic high-contrast elastic composites. The elastic matrix satisfies the Lamé equations, the fluid inclusions satisfy the Stokes equations with a local incompressibility constraint, and the two phases are coupled by continuity of displacement and traction across the interface. The main difficulty is that the incompressibility constraint is imposed only in the inclusions, while the interface structure prevents a direct application of standard elliptic homogenization theory. We develop a quantitative homogenization theory for this model. First, using the Babuška-Brezzi theory, we prove uniform well-posedness of the mixed variational problem, including inf-sup stability and a priori estimates independent of the microscale parameter. Second, by combining formal asymptotic expansions with two-scale convergence, we derive the homogenized effective elasticity equation and prove weak $H^1$ convergence of the microscopic displacement. The effective tensor is characterized by cell problems and is symmetric and strongly elliptic on symmetric matrices. Third, under smoothness assumptions on the interface, we establish piecewise higher-order Sobolev regularity and $L^\infty$ gradient bounds for the cell correctors. These estimates yield $O(\sqrt{\varepsilon})$ convergence rates for the displacement in $H^1$ and for the pressure in $L^2$ over the fluid region.

Convex duality relations are a useful tool for deriving error estimates for challenging nonlinear and non-smooth variational problems. Applied at the continuous level they can deliver nonlinear analogues of the Prager-Synge a posteriori error identity, while at the discrete level they allow the derivation of minimal regularity a priori estimates. By leveraging elementary properties of Bregman divergences, we obtain three results on the error control via convex duality for a general class of problems: first, we prove a local efficiency bound for the duality gap error estimator, secondly, we derive a guaranteed a posteriori bound for non-conforming fields, and finally, we prove a minimal-regularity quasioptimal estimate for a Crouzeix-Raviart discretisation of the $\varphi$-Laplace problem.

The weak Harnack inequality is a fundamental result in the theory of fully nonlinear parabolic partial differential equations. It has several important consequences such as the Hölder continuity of derivatives of solutions to certain fully nonlinear parabolic equations. Classical proofs of the weak Harnack inequality rely on decay of measure estimates, which require delicate localization and covering arguments. We give a new proof based on a detailed study of particular envelopes of solutions to parabolic partial differential equations. This allows us to completely bypass the need of covering arguments. We also prove a $W^{2,\epsilon}$ estimate with an improved dependence of $\epsilon$ on the ellipticity ratio compared to previous results.

We study Haantjes operators, that is, (1,1)-tensor fields with vanishing Haantjes torsion. Our main result is a complete local description of gl-regular Haantjes operators. Additional results include a splitting theorem for general (not necessarily gl-regular) Haantjes operators and, more generally, for operators with vanishing generalised Nijenhuis torsion of an arbitrary level, as well as a complete treatment and understanding of the case when the eigenvalues of a Haantjes operator are complex; the latter case was ignored in many previous papers on this and related topics.

We study the stationary and dynamical Vlasov-Poisson-Boltzmann system in a bounded, convex domain subject to a confining external potential field. For the stationary problem, we construct a unique stationary solution with an inflow boundary condition. A key difficulty is to obtain pointwise regularity for stationary solutions due to the intricate coupling between the self-consistent electric field and the Boltzmann collision operator. To overcome this issue, we establish a $W^{1,p}_{x,v}$–$\alpha C^1_{x,v}$ bootstrap framework and derive an unweighted $C^1_v$ estimate by exploiting the structure of the external potential field. We then investigate the dynamical Vlasov-Poisson-Boltzmann system near the stationary solution. We prove the global existence and uniqueness of solutions for small perturbations and establish exponential convergence toward the stationary state in weighted $L^\infty$ norms. Our results reveal the stabilizing effect of the external potential field and provide a framework for the stationary and dynamical theories of the Vlasov-Poisson-Boltzmann system in bounded domains.

We prove an asymptotic upper bound on the number of nodal domains of eigenfunctions of a class of degenerate elliptic operators. Our proof yields the same constant as in Pleijel's bound for the Dirichlet Laplacian. The operators considered include the Baouendi-Grushin operator and operators with ellipticity degenerating on the boundary.

We study the Cauchy problem for the Tricomi equation perturbed by space-time Gaussian White Noise. To prove existence and uniqueness of the solution, we employ a Fourier transform approach that allows to obtain its representation in terms of certain integrals of the Airy functions. Then, via a careful analysis of the asymptotic behaviour of those integrals, we obtain all the desired properties of the solution, such as square integrability, continuity of its sample paths and stationarity with respect to the space variable. In relation to that stationarity, we also provide the precise description of how the correlation function behaves for small values of the space-lag. We also remark that, in contrast to the findings of the recent paper [11], the properties of the solution to our stochastic Tricomi equation are equivalent to those derived in studying the corresponding problem for the wave operator.

We study the short-time asymptotic behavior of the heat trace associated with the Neumann Laplacian on polygonal domains in the plane. By establishing locality principles for the heat kernel near corners, edges, and the interior, we approximate the heat kernel on the polygon by model heat kernels defined on infinite sectors, half-planes, and the full plane, respectively. Although it is known that the Neumann heat trace of polygons admits a three-term asymptotic expansion followed by an exponentially small remainder, an explicit estimate for the exponent in this remainder term is not known. In this article, we provide such an estimate. We also discuss whether the exponent is sharp, and how it relates to known results. Finally, we discuss issues that arise when trying to extend the results to Robin boundary conditions.

We study the Cauchy problem for a time-fractional Airy-type equation on an infinite metric star graph with edge-dependent Caputo derivatives. By constructing layer potentials generated by fractional Airy fundamental solutions, the problem is reduced to a generalized Abel integral equation. Using Pskhu's solvability results, we establish existence, uniqueness, and an explicit integral representation of the solution. The obtained framework extends previous results where identical fractional orders were assumed on all edges.

We study the existence theory for steady compressible Navier-Stokes-Fourier system in a three dimensional bounded domain for the case of viscosities depending on the temperature in the form $(1+\vartheta)^\alpha$ for $0\leq \alpha\leq 1$ and the hard sphere pressure $p(\varrho,\vartheta) = \vartheta\varrho h(\varrho)$, $h(\varrho)$ is increasing and singular at $\varrho = a > 0$. This paper considers both the heat-flux and Dirichlet boundary conditions for the temperature and Dirichlet boundary conditions for the velocity. The key point to controlling the pressure is based on some estimates of the Bogovskii operator.

We study weak solutions of the one-dimensional pressureless Euler-Poisson-alignment system. When smooth solutions develop singularities, distributional weak solutions are not unique. We introduce an entropy-based selection principle via an associated scalar balance law with time-dependent flux and establish global well-posedness for its entropy solutions. The resulting entropic solution yields a uniquely selected weak solution of the Euler-Poisson-alignment system. In the attractive regime, it is compatible with sticky particle dynamics, while in the repulsive regime atomic states may disperse, revealing a fundamental qualitative difference between the two cases.

We obtain weighted centro-affine Bochner formulas on spherical caps associated with smooth strictly convex hypersurfaces. As a consequence, we prove weighted Poincaré inequalities on caps and on intersections of caps for a class of weights depending on the position vector $X$ of the hypersurface. In the unconditional case, we obtain a centro-affine Poincaré inequality with weight $|X|^2$, which is used to prove a Brunn–Minkowski inequality for the $(n+2)$-nd dual quermassintegral. We also establish an $L_0$-Brunn–Minkowski inequality for the $q$-th dual quermassintegral for $q\in(0,n)$, with equality only for dilates, and an $L_p$-Brunn–Minkowski inequality for $q=n+\alpha$ whenever \[ 0<\alpha\le \frac{2p(1-p)}{2-p}, \] which in particular covers the range $q\in(n,n+6-4\sqrt{2}]$ for suitable $p\in(0,1)$. These Brunn–Minkowski inequalities imply weighted centro-affine Poincaré inequalities and uniqueness results for the $L_{p,q}$-Minkowski problem in the unconditional class. Our main contribution is the introduction of a flat logarithmic centro-affine geometry on the positive orthant $(0,\infty)^n$, adapted to the multiplicative structure of the $L_0$-sum. In this geometry, a Bochner formula yields a sharp Poincaré inequality, as well as a new proof of the centro-affine Poincaré inequality with constant $n$ due to Kolesnikov–Milman, for unconditional bodies and unconditional functions.

We study $n$-dimensional annuli for $n\,\in\,\{2,\dots,N\}$ with $N\,<\,\infty$. We choose a non-dimensional setting such that for any fixed $n $ and given number ${\cal A}>0$ the annuli ${\Omega}_{(n),\cal A}$ are defined as space between two concentrical balls with radii ${\cal A}/2$ and ${\cal A}/2 +1$ in ${ R}^{n}$. For these geometries we provide calculated (precise) Poincaré constants. These depend on ${\cal A}$ and the dimension $n$. Additionally we find a direct match of the Poincaré constants for solenoidal vector fields in ${R}^{n}$ and the Poincaré constants for scalar functions in ${ R}^{n+2}$ (all with vanishing Dirichlet traces). This is based on the relation of the first eigenvalues and one eigenfunction of the (scalar) Laplace and the Stokes operator. In addition we consider the limit ${\cal A}\,\to\,0$. In this context problems in domains ${\Omega}_{(n),\sigma}^{*}$ are investigated. These domains enable us to use the Green's function of the Laplacian with vanishing Dirichlet traces to show that the first eigenvalue here tends to the first eigenvalue of the corresponding problem on the open unit ball in ${ R}^{n}$. On the other hand, we take advantage of the so-called small-gap limit for ${\cal A}\to\infty$.

We consider the isothermal Euler-Poisson system with linear damping on a periodic domain in the large damping regime. For arbitrarily large smooth initial data with density bounded away from vacuum, we prove the global-in-time existence of smooth solutions. The argument is based on a large-damping bootstrap scheme, a modified density estimate revealing hidden parabolic dissipation, top-order weighted cancellations, and a comparison with an auxiliary drift-diffusion–Poisson system. We further prove exponential relaxation to the homogeneous equilibrium and establish a large-data overdamped limit as the damping coefficient tends to infinity. In the slow time scale, the density converges quantitatively to the large smooth solution of the drift-diffusion–Poisson system with the same initial density. After subtracting a fast initial layer from the rescaled flux, the flux converges quantitatively to the corresponding drift-diffusion flux.

This paper investigates the asymptotic behavior of the norm, as a bounded operator in $L^2(\mathbb{R})$, of the $n$-th Calderón commutator $T_{A,n}$ on the graph of a Lipschitz function $A:\mathbb{R}\to\mathbb{R}$. We prove the estimate $\|T_{A,n}\|_{L^2\to L^2} \leq Cn\|A'\|_\infty^n$, thus formalizing a claim by Mateu and Verdera via a symmetrization strategy and the $T1$ theorem. We also show that additional regularity on $A$ yields sublinear growth in $n$. Specifically, for $A$ supported in $[0,1]$, the bound improves to a behavior of the form $\sqrt{n}\|A'\|_\infty^n$ under a Dini condition on $A'$, or if $A'$ belongs to the logarithmic Besov space $B^{1,0}_{1,1}(\mathbb{R})$. This space contains all compactly supported functions in the Sobolev spaces $H^s(\mathbb{R})$ for $0<s<1,$ as well as functions of bounded variation. These refined estimates are established through an alternative framework based on Hörmander-type conditions and interpolation, bypassing the standard $T1$ approach. Counterexamples are provided to demonstrate that the Dini and Sobolev fractional regularity conditions are incomparable.

We establish a small-time large deviation principle for the stochastic Landau–Lifshitz–Baryakhtar equation using the framework of exponential equivalence. This result characterizes the asymptotic behavior of the solution on very short time scales. In particular, it shows that, as the stochastic thermal fluctuations become small, the magnetization remains exponentially concentrated near its initial state, reflecting the short-time stability of the magnetization dynamics. The associated rate function provides a quantitative measure of deviations from the initial state and the resulting short-time stability.

The aim of this paper is to study the well-posedness and finite element approximation of the electrostatic shear Alfvén wave equations, a coupled system of two partial differential equations arising in plasma physics as a simplified sub-model of the drift-reduced Braginskii equations. To this end, anisotropic Sobolev spaces depending on the normalized magnetic field $\b$ are introduced, together with a Poincaré-type inequality along the integral curves of $\b$, which holds under a geometric directedness condition on the magnetic field. Using these tools, existence, uniqueness, and stability of a weak solution are established via the Faedo-Galerkin method. It is also shown that the geometric condition is satisfied in tokamak and stellarator configurations. A numerical scheme is then proposed, combining Lagrange finite elements in space with a Crank-Nicolson discretization in time. The scheme is shown to conserve a discrete energy exactly in the homogeneous case, and a priori error estimates are derived in the natural energy norm. Several numerical experiments are reported in two and three space dimensions, which confirm the theoretical results and indicate that the geometric condition on the magnetic field is necessary for the invertibility of the discrete system matrix.

We consider the Schrödinger equation associated with the harmonic oscillator and show that if the initial data and its Fourier transform are dominated by Gaussian functions of widths $a>0$ and $b>0$, respectively, satisfying $ab<1$, then the evolved solution and its Fourier transform are dominated by a Gaussian of width $\frac{1}{2}\left(\frac{1}{a}+\frac{1}{b}- \sqrt{\left(\frac{1}{a}+\frac{1}{b}\right)^2-4}\right),$ for all times except for a discrete set, and for all times in one dimension. In the one-dimensional case, we prove that these estimates are sharp. Moreover, for a more restrictive class of initial data, we establish sharper time-dependent Gaussian bounds.

We study critical points of the Ginzburg-Landau energy functional $$\mathcal{F}_\varepsilon[u] = \int_\Omega \Big[ \frac{1}{2}|\nabla u|^2 + \frac{1}{2\varepsilon^2} W(1 - |u|^2)\Big]\,dx, \quad u \in H^1(\Omega, \mathbb{R}^N),$$ with $\Omega \subset \mathbb{R}^M$, $\varepsilon>0$, $M,N \geq 2$ and general conditions on the non-negative potential $W$ allowing for super-quadratic behaviour near its zero set. Under a Dirichlet boundary data of unit-length on $\partial \Omega$, we prove the following maximum principle: every critical point $u_\varepsilon$ satisfies the global uniform bound $|u_\varepsilon| \leq 1$ in $\Omega$. Furthermore, if a family of critical points $(u_\varepsilon)$ converges (in energy) to a smooth $\mathbb{S}^{N-1}$-valued harmonic map in the limit $\varepsilon \to 0$, then we prove global uniform bounds for $(\Delta u_\varepsilon)_{\varepsilon>0}$ in $\Omega$ and, in particular, global Hölder convergence of the gradients $(\nabla u_\varepsilon)$ in $\Omega$ as $\varepsilon \to 0$.