Papers

Using Gaussian heat-kernel estimates, we classify planar Dunkl Newton kernels in weighted Euclidean BMO and obtain the corresponding local classification for unit-ball Green kernels. For atomic Newton potentials supported on a regular reflection orbit, orbit BMO detects exactly whether the coefficients are constant along the orbit. We also prove an intrinsic Dunkl–CLMS theorem in the heat-semigroup Hardy space and derive intrinsic and Euclidean-source Newton–Wente estimates, extending the BMO–Hardy-space method of Chanillo and Li to the Dunkl setting. Further consequences include sharp local atomic estimates, same-domain criteria, a localized $L^1$-to-BMO bound, and a Brezis–Merle-type estimate.

We study the homogenization for obstacle problems of convex Hamilton-Jacobi equations in periodic media with obstacle functions depending on both macroscopic and microscopic spatial variables. Based on the methods of analyzing variational representation formulas, we establish the optimal convergence rate $O( \varepsilon )$ for the homogenization limit under the assumption that the obstacle function is Lipschitz continuous. Moreover, we show that this convergence rate may deteriorate due to the presence of the obstacle when the obstacle function is not Lipschitz continuous, by constructing an example.

A classical idea in analysis going back at least to work of Simon (1997) is that Liouville theorems for solutions to elliptic or parabolic PDEs are equivalent to Schauder type regularity estimates. The goal of this course is to describe some recent developments of this idea concerning the regularity of the complex Monge-Ampère equation with respect to singular reference metrics. We will start with a quick look at the classical $C^2$ and $C^3$ estimates of Calabi-Aubin-Yau and then present a new proof of the Evans-Krylov $C^{2,\alpha}$ estimate on a Euclidean ball. Based on this we will consider the case of singular backgrounds such as cylinders and cones, discussing some recent work by Hein, Tosatti, Lee and Klemmensen. Our discussion is far from complete and knowledge of Kähler geometry including the Aubin-Yau theorems is assumed. The course includes five exercises with solutions and a problem list.

We consider aggregation-diffusion models for a density of mass in $\mathbb R^d$, $d\ge 2$, where the diffusion can be either linear or of porous medium type, while the aggregation effect is governed by the attractive Newtonian or Bessel potential. It is well known that the dynamics can be interpreted as Wasserstein gradient flow of the natural free energy of the system, and that, under suitable assumptions on the diffusion exponent and the mass of the initial datum, solutions exist globally in time. In these regimes, we perform the analysis of the discrete variational approach by means of the JKO scheme. We establish a sharp $L^\infty$ smoothing effect, which proves to be of same rate as that of the porous medium equation. Thanks to this estimate, we obtain uniqueness of global gradient flow solutions for initial data of finite energy and, in the diffusion dominated regime, for measure data having finite second moment. We prove the energy dissipation equality and characterize the gradient flow in terms of suitable evolution variational inequalities. In the fair competition regime, we show uniform extinction of solutions for large time under smallness conditions on the mass.

We investigate uniqueness and non-uniqueness of least energy normalized and least energy nodal normalized solutions for nonlinear Schrödinger equations on compact metric graphs. We first prove existence of least energy nodal normalized solutions in the $L^2$-subcritical regime and, at the critical exponent, below a graph-dependent threshold. We then establish a conditional non-uniqueness result for slightly $L^2$-subcritical powers and identify broad classes of graphs for which the required condition either holds or fails. In particular, we prove uniqueness of least energy nodal normalized solutions on the interval for every mass and every $p\in(2,6)$. Finally, using ODE and phase-plane techniques, we show that least energy normalized solutions on the interval are unique for $p$ sufficiently close to $2$. Overall, the results reveal a strong dependence of the uniqueness picture on the nonlinearity power, the topology, and the metric of the graph, and a structural difference between the constant sign and the sign-changing settings.

This reports on an experiment in autoformalization of arxiv:2606.15977, an inverse problems result for piecewise polynomial anisotropic conductivities, using Isabelle/HOL. The proof of the result is relative to a number of external results which were deemed to be well known and generally accepted to be true. The formalization files are made available at a GitHub repository. Translation issues are discussed.

We establish the existence of a stochastic flow on $L^{\infty} (\mathbb{T})$ for the stochastic Allen-Cahn equation with multiplicative noise \[ (\partial_t - \partial_x^2) u = u - u^3 + \sigma (u) \xi \quad \text{on} \quad \mathbb{R}_+ \times \mathbb{T}, \] where $\xi$ is space-time white noise and $\sigma : \mathbb{R} \rightarrow \mathbb{R}$ is sufficiently smooth, bounded, and has bounded derivatives. Our strategy is to obtain pathwise a priori estimates via regularity structures. In fact, we consider a general singular multiplicative equation with superlinear damping, driven by noises of parabolic regularity $\alpha - 2$, for ${\alpha \in (0, 1)}$, which can be lifted to a weakly admissible model. We show that the required estimates hold whenever \[ m > \frac{2 - \alpha}{\alpha} \varepsilon_{\alpha}, \quad \text{where} \quad \varepsilon_{\alpha} = 1 - \alpha \left( 1 - \frac{2}{3 - \alpha} \right) \in (0, 1) . \] Thus the strength of the damping needs to be chosen only as a function of the regularity of the driving noise. Under an additional smoothness assumption on $\sigma$, we show that the stochastic flow is differentiable with respect to its initial condition.

Pantograph arises in electric trains, material modelling, and the modelling of quantum dot lasers. Pantograph integrodifferential equations are integrodifferential equations involving proportional delays; and they appear in fields such as electrodynamics, epidemiology, control theory, astrophysics, economics, and engineering. This research presents an efficient collocation method based on shifted Jacobi polynomials for obtaining numerical solutions of a class of first order pantograph delay integrodifferential equation of Volterra type with an initial condition. The proposed method expresses the solution of the governing equation as a shifted Jacobi polynomial series with expansion coefficients which are to be determined. Collocating at the roots of the shifted Jacobi polynomials, the underlying problem is reduced to a system of algebraic equations in the unknown expansion coefficients of the shifted Jacobi polynomial series solution. Newton's method is subsequently used to solve the resulting system of algebraic equations and numerical values of the expansion coefficients are obtained. The obtained coefficients are substituted into the assumed series solution to obtain the required numerical solutions. The applicability, reliability, efficiency, and accuracy of the shifted Jacobi collocation method are demonstrated through illustrative examples. Results obtained using the proposed method are compared with exact solutions and other published results. Comparisons of errors which are presented in tables reveal that our method approximates the solution better than the methods under comparison.

We study travelling front solutions of a family of degenerate Fisher-KPP equations $u_t=(u^n u_x)_x+u(1-u^n)$, where $n$ is a positive integer. At the minimal wave speed $c_{\rm min}=\tfrac{1}{\sqrt{1+n}}$, these equations admit sharp front solutions, corresponding to an explicit heteroclinic orbit in the travelling wave phase-space. We show how this sharp front is perturbed when the wave speed is increased to $c=\tfrac{1}{\sqrt{1+n}}+\varepsilon$, with $\varepsilon$ sufficiently small. We use a well-motivated geometric desingularisation (also known as blow-up) near the degenerate equilibrium at the leading edge. By analysing the resulting directional and rescaling charts, we construct a singular heteroclinic orbit connecting the relevant asymptotic states, providing a simple geometric proof of the transition from sharp to smooth travelling fronts.

We use Morse Theory to prove that the Burkholder function is quasiconvex on symmetric matrices, which implies several sharp inequalities for the Beurling–Ahlfors transform when acting on real-valued functions. More generally, we prove that the separating functions introduced by Székelyhidi are quasiconvex on symmetric matrices.

I prove that a collapsing charged scalar field with scalar charge $\mathfrak{e}$ that forms an exactly extremal Reissner–Nordström black hole à la Kehle–Unger with radius $r_+$ must satisfy $|\mathfrak{e}|r_+ > \frac{1}{3}$. In contrast, I also show that no such bound exists for subextremal collapse: a scalar field with an arbitrary $\mathfrak{e} \ne 0$ can form a subextremal black hole with an arbitrary (subextremal) charge-to-mass ratio. Complementing the extremal bound, I prove that a Schwarzschild black hole can become extremal if $|\mathfrak{e}|r_+ > \sqrt{\frac{3 + \sqrt{33}}{3}}$. The fact that $|\mathfrak{e}|r_+$ can be taken to be of order unity could be considered evidence that the third law of black hole mechanics in vacuum is false.

We give a rigorous realization of the dilute instanton picture for a semiclassical Schrödinger operator with a symmetric double-well potential on $\mathbb{R}^n$. Using a localized Feynman–Kac representation, we decompose the heat-kernel trace according to the number of passages made by a Brownian bridge between shrinking neighborhoods of the two wells. We identify the weight of one passage with a hopping coefficient $\rho_\lambda$, $\displaystyle \rho_\lambda = \int_{\partial\Omega} \left( \nabla\overline{\varphi_{\lambda,0}^{\Omega}}\, \varphi_{\lambda,0}^{-\Omega} - \overline{\varphi_{\lambda,0}^{\Omega}}\, \nabla\varphi_{\lambda,0}^{-\Omega} \right)\cdot\nu. $ On the exponentially long time scale $\beta=N/\lvert\rho_\lambda\rvert$, the number of passages converges, for every fixed $N>0$, to a Poisson random variable of mean $N$. We identify $-\frac{1}{\lambda}\log\lvert\rho_\lambda\rvert\to S(d,-d)$ and obtain $\displaystyle E_1(\lambda)-E_0(\lambda) = 2\lvert\rho_\lambda\rvert\left(1+o(1)\right). $ Thus the familiar instanton expansion of the double-well eigenvalue splitting emerges directly from a factorization of the heat-kernel trace.

We consider a stationary fluid–structure interaction problem in which the steady Navier–Stokes equations are coupled, through a free elastic interface, with a clamped Euler-Bernoulli beam equation. Using a complexification of the fixed-domain formulation and the holomorphic implicit function theorem, we prove that, in a neighbourhood of the trivial solution, the mapping from the right-hand side to the weak solution is real analytic. As a byproduct we obtain a small-data existence and local uniqueness result for the coupled system. Our motivation comes from data-driven reduced-order modelling for parametric PDEs, where approximation properties are closely related to the regularity of the solution map. Numerically, a manufactured-solution test exhibits approximately second-order convergence in the reported relative $L^2$ errors, while a proper orthogonal decomposition study for a parametric force family shows rapid decay of the empirical reconstruction error until a numerical floor is reached.

Motivated by scattering theory, this paper proves general results about the distribution of zeros in logarithmic neighborhoods of the real axis for a certain class of functions holomorphic in the closed lower half-plane. We define a new indicator function that measures the growth of the holomorphic function along logarithmic curves, and connect this to the distribution of zeros of the function. This is related to classical results on the distribution of zeros in sectors for entire functions of completely regular growth. These results can be applied to the determinant of the scattering matrix of a Schrodinger operator on odd-dimensional Euclidean space, yielding bounds on resonance counting functions for logarithmic neighborhoods of the real axis. As a further application, we study a certain family of potentials in one dimension having jump-like singularities. Using our complex-analytic results we show that the singularities can lead to many – and even infinitely many – strings of resonances along logarithmic curves. We connect the properties of these strings of resonances, including their location and linear density, both to the parameters describing the singularities of the potential and to the asymptotic behavior of the scattering determinant.

We define the notion of regular parabolic (bi-)Lipschitz images as the parabolic (bi-)Lipschitz maps from $n$-dimensional space time which, up to translation, fix the $t$ variable and whose spatial components are each regular parabolic Lipschitz functions. We show that any parabolic Ahlfors-David regular is parabolic uniformly rectifiable if and only if it has big pieces of parabolic Lipschitz images of $n$-dimensional space time if and only if it has big pieces of parabolic bi-Lipschitz images of $n$-dimensional space time. This further extends the David-Semmes theory to the parabolic setting. Our proof combines the ideas of the first authors previous work [BH12,BHH+22] and some ideas of Azzam and Schul [AS12]. The proof easily adapts (and is far less complicated) to the Euclidean case to give an alternative proof of the analogous fact.

We show if $\partial \Omega$ is the graph of a parabolic Lipschitz function, then parabolic surface measure $\sigma$ of $\partial \Omega$ is absolutely continuous with respect to its caloric measure if and only if a (square) Dini-$\beta$ condition is satisfied. More specifically, the (square) Dini-$\beta$ condition is that \[\int_0^1 \hat{\beta}(X,t,r)^2 \frac{dr}{r} < \infty, \quad \text{$\sigma$-a.e. } (X,t) \in \partial \Omega.\] Here $\hat{\beta}$ is a parabolic version of the Jones ($L^2$) $\beta$-numbers. We show that these conditions are satisfied if and only if the graph is covered by a countable collection of regular Lipschitz graphs, that is, graphs with additional in-time regularity in the form of a half order time derivative in the parabolic BMO space. This supports the view that covering by regular Lipschitz graphs is the right notion for qualitative parabolic rectifiability in the context of parabolic PDEs. We also show that if \[\int_0^1 \hat{\beta}(X,t,r)^2 \frac{dr}{r} < \infty\] up to a set of caloric measure zero then the caloric measure is absolutely continuous with respect to surface measure.

We prove a third-order derivative estimate for convex solutions to the real Monge-Ampère equation ${\rm det}\,{\rm Hess}(u) = 1$ on an open set in $\mathbb{R}^{2m}$ under the additional assumption that ${\rm Hess}(u)$ lies in ${\rm Sp}(2m,\mathbb{R})$ at every point. Our method is a geometric interpretation and extension to higher dimensions of Nitsche's classical proof of the Bernstein theorem for the real Monge-Ampère equation on $\mathbb{R}^2$. For $m = 1$ we also improve Nitsche's constant as well as some estimates due to Calabi, and we construct examples of solutions with interesting geometric behavior.

We establish interior Hessian estimates for solutions of the quadratic Hessian equation in arbitrary dimensions.

We study the time-asymptotic stability of solutions to the inflow problem for the one-dimensional Navier–Stokes–Fourier system on the half-line. We consider the most generic wave pattern: the superposition of a degenerate boundary layer, a rarefaction, a viscous contact wave, and a viscous shock. More precisely, if the boundary data belongs to the subsonic region, and the initial perturbation and strengths of the boundary layer, viscous contact wave, and viscous shock are sufficiently small, then the solution to the inflow problem converges to the corresponding superposition, up to a time-dependent shift for a shock. The rarefaction wave, however, is allowed to have arbitrarily large strength. To control the viscous shock, we employ the method of $a$-contraction with shifts. A notable feature of our analysis is that this method can be applied even when the rarefaction wave has large amplitude. In particular, this resolves, in a generic setting, the open problem of the stability of inflow wave patterns containing a viscous shock for Navier–Stokes–Fourier system.

Considering an $n$-dimensional compact Riemannian manifold with a boundary that satisfies a Serrin-type problem, we prove sharp upper and lower bounds for the area of such a boundary. Then, we present the main result of this paper: a divergence formula for a special vector field on a given Riemannian manifold. This divergence formula is closely related to sub-static manifolds and related metrics, e.g., $V$-static, static, and electrostatic manifolds. We show Minkowski-type inequalities for $V$-sub-static manifolds under different Neumann boundary conditions. Moreover, we prove an area-charge inequality for compact (and noncompact) electrostatic manifolds. The main application of the divergence formula presented in this work generalizes the classical result of Boucher–Gibbons–Horowitz: we prove that an asymptotically hyperbolic space of a static spacetime satisfying the null convergence condition must be the hyperbolic space.

We give a necessary and sufficient triangular condition characterizing the toroidal pseudodifferential operators on $\mathbb T^d$ that preserve the positive-frequency cone $\mathbb N_0^d$ and hence act on holomorphic boundary values on the polydisk. For symbols of type $(1,0)$, we prove boundedness on holomorphic Besov and Triebel–Lizorkin spaces throughout the quasi-Banach range. For $S^m_{\rho,\delta}$, $0\leq\delta<\rho\leq1$, we obtain the critical loss $d(1-\rho)|1/p-1/2|$, together with endpoint estimates. A positive-cone oscillatory multiplier proves sharpness of the loss and necessity of the Besov endpoint condition $q\leq t$. As an application, we prove well-posedness for a first-order holomorphic differential operator on these scales.

We derive energy scaling laws for thin elastic sheets with topological defects — disclinations and dislocations — for a fully nonlinear 3D model. For disclinations, the scaling laws are tight in the thickness parameter, and improve upon previous results by applying simultaneously to positive and negative disclinations (e-cones) and by giving an explicit dependence on the defect parameter; this latter dependence is, however, still not tight. For thin bodies with dislocations, these are, to the best of our knowledge, the first rigorous bounds for models of finite thickness, are tight when the Burgers vector is not large with respect to the thickness, and relate to a well-known conjecture from the physics literature about the scaling. A main tool is modeling these bodies in the framework of non-Euclidean elasticity, as bodies with a curl-free pre-strain; the curl-freeness allows us to obtain geometric rigidity estimates for the lower bounds.

We study the higher-order flows of the modified Burgers' equation generated by its standard recursion operator. In particular, we show that the second-order positive flow takes the form of a non-local integro-differential equation. After reducing the equation to a traveling-wave form, we investigate its asymptotic energy behavior. The resulting analysis shows that the non-local energy flux prevents the formation of both topological kink and bell-shaped solitary waves. We show that the non-local dissipation permanently breaks the asymptotic equilibrium between nonlinear convection and linear dispersion, forcing an irreversible energy imbalance. Our analysis shows that the solitary waves are unstable. The non-local term causes continuous energy loss through radiation. As a result, the traveling wave cannot keep a stable and symmetric shape.

We completely classify conformal metrics on the unit ball $(\mathbb{B}^{n+1},|\mathrm{d} x|^2)$, $n\geq4$, with positive constant $Q$-curvature, positive constant $T$-curvature, and minimal boundary. After normalizing the $Q$-curvature, there is a unique conformal metric for each $T$-curvature value in $[0,+\infty)$, up to conformal diffeomorphism. For positive $T$-curvature, these metrics are not Einstein and yield a new family of bubble profiles, distinct from the Aubin–Talenti bubble family except when $T=0$. This new phenomenon has no analogue in either the second-order boundary Yamabe problem or the constant $Q$-curvature problem on closed manifolds. To our knowledge, this is the first classification result for a fourth-order boundary value problem with nonlinear terms both in the interior and on the boundary.

For every $n\geq 3$ we construct a bounded discontinuous map $u\in W^{1,n}(\mathbb{B}^n,\mathbb{R}^{n+2})$ solving an $n$-Laplace system with an antisymmetric potential $\Omega\in L^n$. This shows that a recent regularity result on $n$-Laplace systems with antisymmetric potentials in Lorentz spaces by the authors is sharp in the sense that we cannot move from Lorentz spaces to classical Lebesgue spaces. This gives in particular a negative answer to a question by Rivière.

In this paper, we study a system in which the Cahn–Hilliard system is imposed in the bulk domain and an Allen–Cahn type equation is prescribed on the boundary, connected through a third type transmission condition characterized by a permeability parameter. This setting is closely related to the theory of transmission problems, and the third type transmission condition can be viewed as a remnant of a thin-boundary description, with the permeability parameter controlling the degree of coupling between the bulk and the boundary. The main objective is to perform a rigorous asymptotic analysis with respect to the permeability parameter. We investigate two limits: the parameter tending to zero, corresponding to a completely impermeable boundary, and the parameter tending to infinity, corresponding to a perfectly permeable boundary where the bulk and boundary phases are fully coupled. For each limiting regime, we establish the convergence of solutions to the respective limit problems and characterize the resulting equations.

This paper proposes a framework for treating the inverse obstacle problem for nonlinear elliptic equations with nonlinear materials. The problem is challenging because nonlinear materials exhibit a rich diversity of scenarios to consider, since nonlinearity can take different forms. In this article, after categorizing the nonlinearities into a few fundamental classes, a dedicated imaging method is proposed for each class, derived by combining two powerful concepts: the Monotonicity Principle (MP) and the $p-$Laplace Signature (pLS). The Monotonicity Principle (MP), recently extended to nonlinear materials, provides a monotonic relationship between the material property and the measured quantity (the Average Dirichlet-to-Neumann map) that can be \lq\lq inverted\rq\rq \ to find the shape of anomalies. The $p-$Laplace Signature (pLS) allows for modelling the solution of an elliptic PDE with nonlinear materials, for large or small boundary data, in terms of a proper $p-$Laplace equation that captures the essence (the signature) of the problem. For example, pLS with $p=2$ allows the reduction of a nonlinear elliptic PDE to a linear one, providing a powerful bridge for applying imaging methods and algorithms developed for linear materials. In this contribution, the two pillars of MP and pLS are combined in new imaging methods to enlarge the class of nonlinearity that can be treated within the inverse obstacle problem. Moreover, the theoretical limits of the methods are provided in the ideal case of noise-free measurements: outer-support reconstruction when $p=2$, and convex-hull reconstruction when $p\neq2$.

We study a class of nonlinear elastic energies whose constitutive structure is naturally expressed in terms of a stretch variable but is not directly covered by the standard polyconvex theory formulated in the deformation gradient. The problem is lifted by introducing an independent microrotation $ \overline R \in {\rm SO}(3)$ and the relative stretch $ \overline U = \overline R ^T{\rm D}\varphi$. The curvature variable $ \overline R ^T\operatorname{Curl} \overline R $ controls the full first-order variation of the rotation field and yields strong compactness of minimizing sequences. The identities $\operatorname{Cof}( \overline R^T{\rm D}\varphi) = \overline R^T\operatorname{Cof}{\rm D}\varphi$ and $\det( \overline R^T{\rm D}\varphi)=\det{\rm D}\varphi$ then allow the minors of the lifted stretches to be identified through the weak continuity of the corresponding minors of the deformation gradients. We prove existence in two regimes. The first allows convex dependence on $( \overline U ,\operatorname{Cof} \overline U ,\det \overline U )$ and assumes separate cofactor coercivity. The second depends only on $( \overline U ,\det \overline U )$ and requires no independent cofactor bound. For the constrained model, the weakly closed condition $ \overline R ^T{\rm D}\varphi\in\operatorname{Sym}^{+}(3)$, together with $\det{\rm D}\varphi>0$, implies $ \overline R =R:=\operatorname{polar}({\rm D}\varphi)$ for every admissible pair. The lifted problem is therefore equivalent to a deformation problem containing the curvature of the polar factor. The resulting model is a selectively rotationally regularized nonlinear elastic model of couple-stress type, rather than a purely first-gradient Biot model.

We consider the two-dimensional dissipative generalized surface quasi-geostrophic equations given by \[\partial_t \theta+u\cdot \nabla \theta+\nu (-\Delta)^{\alpha}\theta=0,\quad u=\nabla^{\perp}\Lambda^{\beta-2}\theta\] and establish global existence of weak solutions with initial data in $\dot{H}^{\frac{\beta}{2}-1}(\mathbb{R}^2)$. Our result extends the framework of Marchand's solutions (Comm. Math. Phys. 277 (2008), no. 1, 45-67) to this larger family of active scalar equations.

We prove the existence of small-amplitude traveling shock profiles for the one-dimensional Boltzmann equation of a binary gas mixture with angular cutoff potentials in the full range $-3<\gamma\le 1$. The result extends the classical construction of Caflisch and Nicolaenko from hard potentials to the cutoff soft-potential regime. Indeed, the argument of proofs combines a Lyapunov–Schmidt reduction of the macroscopic component to a Burgers equation with an accelerated backward bi-characteristic method and a weighted $L^2$–$L^\infty$ iteration. Acceleration restores a uniformly positive collision frequency, compensating for the lack of a spectral gap for soft potentials, while the $L^2$–$L^\infty$ framework accommodates the absence of velocity smoothing induced by the cutoff, including a possible singularity along the grazing characteristic $v_1=s$. The shock profile tends to the Rankine–Hugoniot bi-Maxwellians at a mixed exponential rate as $|x|\to \infty$, with a sub-exponential remainder of order $|\varepsilon x|^{2/(3-\gamma)}$.