Papers

We study regularity properties for solutions to the nakedly degenerate elliptic equation $a_{ij}\partial_{ij}u =0$, where the coefficients satisfy $I \ge a_{ij}(x) \ge \lambda(x) I$ and the only assumption is that $\lambda^{-1} \in L^p$. We prove an improvement of oscillation and a Liouville theorem for $p>d-1$, and a Harnack inequality for $p$ sufficiently large depending on dimension. Along the way, we obtain a new $\log-L^\varepsilon$ Weak Harnack inequality for supersolutions. Then, touching subsolutions by double exponential blow-up barriers, we also derive a logarithmic local maximum principle that is new even in the uniformly elliptic case. Both of these results hold for $p>d-1$. Finally, we construct examples showing that there cannot be Harnack or Weak Harnack inequalities in the regime $p<d-1$, nor can there be power-type $L^\varepsilon$ inequalities in the case of any $p<\infty$.

For the Keller-Segel system \[ \left\{\, \begin{aligned} u_t &= \Delta u - \nabla \cdot ( u \nabla v ), \\ v_t &= \Delta v - v + u \end{aligned} \right. \tag{$\star$} \] posed in a planar domain $\Omega$ with Neumann boundary conditions, the existence of classical solutions blowing up at some finite time $T$ has long been established. In fact, it has been shown that for every blow-up point $x$ the quantity $\int_{B_R(x)\cap\Omega} u(\cdot,t )\ln(u(\cdot, t))$ is unbounded as $t\nearrow T$ for all $R > 0$ even though the global mass of $u$ is always conserved. The present manuscript provides some quantitative information on the behavior of such localized $L\log L$ expressions by asserting the existence of $\delta_0=\delta_0(\Omega)>0$ such that any solution to the Neumann problem for ($\star$) blowing up at time $T\in (0,\infty)$ satisfies \[ \limsup_{t\nearrow T} \frac{1}{\ln\frac{T}{T-t}}\int_{B_R(x)\cap\Omega} u(\cdot, t)\ln(u(\cdot, t)) \ge \delta_0 \tag{$\star\star$} \] for all $R > 0$ at each blow-up point $x$. This confirms a certain universality property of the blow-up mechanism seen in the particular examples of radial collapsing solutions constructed in the seminal work [16], especially also beyond the realm of symmetry; apart from that, along with a consequence of ($\star\star$) on the corresponding asymptotics of similarly localized $L^p$ norms of $u$ for $p\in (1,\infty]$, this provides some extension of a known result on non-degeneracy of blow-up points that has concentrated on the choice $p=\infty$ here.

Refining the sharp upper bounds $\mu_{k,d}^* $ obtained by Kröger (1999) for the $k$-th Neumann eigenvalue of a convex domain $\Omega \subset \mathbb{R}^d$, we prove the following inequalities: for any $k\in \mathbb{N}$ there exists a constant $C(k,d) >0$ such that $$D_{\Omega}^2 \mu_k(\Omega) \leq \mu_{k,d}^* - C(k,d) a_2(\Omega)^2/D_{\Omega}^2$$ where $D_{\Omega}$ is the diameter of $\Omega$ and $a_2(\Omega)$ is the second largest semiaxis of the John ellipsoid of $\Omega$. In the planar case, for $k=1$ we also give an explicit value of the constant $C(1,2)$.

We construct examples of complete Calabi-Yau metrics on smoothings of 3-dimensional Calabi-Yau cones that are not products of lower-dimensional Calabi-Yau cones and that have orbifold singularities away from the vertex.

We study a nonlocal diffusion equation of porous medium type featuring a generalised fractional pressure with spatial anisotropy. We construct a finite element method for the numerical solution of the equation on a bounded open Lipschitz polytopal domain $\Omega \subset \mathbb{R}^{d}$, where $d = 2$ or $3$. The pressure in the model is defined as the solution of fractional elliptic problem involving the fractional power of a second order differential operator, in terms of its spectral definition. Under suitable assumptions on the fractional order and the coefficients of the operator, we rigorously prove convergence of the numerical scheme. The analysis is carried out in two stages: first passing to the limit in the spatial discretization, and then in the time step, ultimately showing that a subsequence of the sequence of finite element approximations defined by the proposed numerical method converges to a bounded and nonnegative weak solution of the initial-boundary-value problem under consideration. Finally, we present numerical experiments in two dimensions illustrating the computational aspects of the method and highlighting the interplay between nonlocal effects and spatial anisotropy under different configurations. We also show numerically the failure of the comparison principle and exponential decay of the numerical solution to a steady state.

We study uniqueness and non-uniqueness pairs for the fractional Laplacian in the discrete setting. More precisely, under mild regularity assumptions on a function $f$ satisfying $f=0$ on $\Lambda$ and $(-\Delta)^s f=0$ on $M$, where $\Lambda, M \subset \mathbb{R}^d$ are discrete, we find sufficient conditions on these sets that force $f$ to vanish identically, and we provide examples in which non-uniqueness occurs. The uniqueness proofs combine decay estimates forced by discrete zero sets with analytic continuation, while the non-uniqueness results are obtained from an interpolation theorem.

We consider steady states of the incompressible Euler equation on two-dimensional domains. For non-radial analytic steady states on bounded simply connected domains, it was shown previously that there must be a global functional relationship between the stream function and the vorticity. We show that this does not extend to smooth functions, even under further structural assumptions such as the Morse condition or Arnold's stability criterion. More precisely, we show that a broad class of steady states with multiple critical points can be perturbed to smooth steady states for which the vorticity is not a single-valued function of the stream function. We also establish an analogous flexibility result near the cellular flow on the flat torus, which is a degenerate case. As a consequence of our constructions, there are "branches" of smooth steady states that are isolated from analytic ones. In some cases, the resulting isolated branches can even consist entirely of linearly stable steady states.

This paper investigates homogenization problems for the nonlocal operators with rapidly oscillating coefficients in the cases of periodic and random statistically homogeneous micro-structures. These operators involve the fractional Laplacian and some operators compared to it. Based on the $\Gamma$-convergence method and compactness arguments, we prove the homogenization theorems for these nonlocal operators with product-type and symmetric coefficient-structured kernels respectively. Furthermore, these results are extended to general nonlinear nonlocal equations.

Inverse scattering focuses on recovering unknown scatterers from wave measurements. A fundamental challenge is determining whether an inverse obstacle problem can be resolved from a single far-field measurement, a task particularly demanding for non-convex polytope obstacles under generalized impedance boundary conditions and closely linked to the long-standing Schiffer problem. In this paper, we develop a novel Artificial Test Domain (ATD) framework for single-measurement inverse scattering of impenetrable polytope obstacles. Based on microlocal analysis near exterior-visible flat boundary patches, this approach transcends traditional methods reliant on observable corners. The ATD framework establishes two primary conceptual advancements: a unified generalized impedance hyperplane (GIH) exclusion mechanism, which clarifies the structural role of uniqueness mechanisms, and a unified qualitative–quantitative principle for the generalized impedance setting. Quantitatively, the method yields a far-field–geometry relation where geometric discrepancy is controlled by far-field error, scaled by a leading ATD coefficient. Qualitatively, the non-vanishing of this coefficient reduces to the exclusion of exterior generalized impedance hyperplanes. Once uniqueness is established, this relation produces sharp stability estimates. Within this framework, the classical stability estimates for the sound-soft and sound-hard cases are recovered as special instances of a much more general stability theory. At the same time, we obtain several new sharp stability results that are of significant importance. These results unify currently available single-measurement uniqueness regimes for polytope geometry and provide new insights into the Schiffer problem across multiple generalized impedance settings.

Consider a finite system of diffusing particles coupled through a reactive boundary. Each particle is reflected, but may react with the boundary according to a killing mechanism which depends on the current reactivity of the boundary and the particle's local time along it. With every such reaction, the boundary moves and its reactivity adjusts. We show that this system admits a unique mean-field limit, described by a free boundary problem with nonlinear and nonlocal reactivity. The latter generalises the classical Robin condition for the case of a fixed boundary with constant reactivity. Via Skorokhod's M1 topology and a characterisation of the particles' behaviour near the boundary, we first identify the weak limit points of the empirical measure flows with killing. Then, we combine a probabilistic decoupling technique and energy estimates to prove uniqueness and deduce convergence. Our analysis gives a rigorous mean-field description of a model of epidemic spreading. Moreover, it contributes to the literature on inert drift systems and yields a novel mean-field perspective on the recent encounter-based framework for diffusion-mediated surface reaction from [Phys. Rev. Lett. 125 (2020) 078102].

We investigate the Sobolev regularity required for almost everywhere convergence to the initial datum of solutions to the linear Schrödinger equation along certain tangential curves. In the regime $\alpha<\tfrac12$, we analyze maximal estimates for expressions of the form $e^{it\Delta}f(x+\gamma(t))$ over specific $\alpha$-Hölder curves $\gamma$ and initial data $f\in H^s(\mathbb{R}^n)$. For the model family $\gamma(t)=(t^{\alpha_1},\ldots,t^{\alpha_n})$, where $\alpha=\min_j \alpha_j$, we show that the critical regularity is $s=\max\left\{\frac{1-2\alpha}{2},\frac{n}{2(n+1)}\right\}.$

In this paper we analyze a model for electroporation, a biological process in which a cell membrane exposed to an external voltage becomes permeable due to the formation and growth of nanoscale membrane pores. We prove a local stability result for asymptotic self-similar solutions with a power-law tail. Our method relies on the analysis of an equation for the first moment as well as comparison of solutions of the full problem to solutions of a corresponding transport problem. In particular this shows that the transport term drives the long-time behaviour.

We establish global-in-time decay estimates for the multi-phase Muskat problem in the case where the density takes exactly n+1 distinct constant values. We first linearize the system around a flat stable configuration, followed by the study of associated linearized operator. The asymptotic behavior at low frequencies of eigenvalues yields the decay rate of (1+t)^{-s/2-1/4} for Wiener norm \|f\|_s, which is slower than the classical case, where the decay rate is (1+t)^{-s+\nu}. Afterwards we bound the nonlinear term to close the argument.

In this paper, we study the mass-constrained fractional Choquard equation \( (-\Delta)^s u = \lambda u + \alpha (I_\mu * |u|^{\frac{2N-\mu}{N}})|u|^{\frac{2N-\mu}{N}-2}u + (I_\mu * |u|^p)|u|^{p-2}u \) in \( \mathbb{R}^N \), under the constraint \( \int_{\mathbb{R}^N} |u|^2 \, dx = c^2 > 0 \), where \( N > 2s \), \( s \in (0,1) \), \( \mu \in (0,N) \), \( \alpha > 0 \), and \( 2 + \frac{2s-\mu}{N} \le p < \frac{2N-\mu}{N-2s} \). We first establish a nonexistence result in the \( L^2 \)-critical case \( p = 2 + \frac{2s-\mu}{N} \). Then, in the \( L^2 \)-supercritical range, we prove the existence of normalized ground states in two complementary regimes determined by the quantity \( \mathcal{M}_1(c) \). Our approach is based on constrained variational methods, a min-max construction, and refined estimates for the associated fiber maps.

We establish a Weiss-type almost-monotonicity formula for a broad class of variable-coefficient energy functionals, assuming only minimal regularity of the coefficients. As an application, we classify blow-up limits for the Alt–Phillips problem with variable coefficients under significantly weaker regularity hypotheses than those imposed in Araújo et al. [Calc. Var. Partial Differential Equations, 65, no.~1, Paper No.~24 (2026)]. Moreover, by means of a distinct argument, we extend the corresponding free-boundary regularity result. We conclude with a discussion of further extensions, including two-phase analogues.

We study the limit behavior of Cahn–Hilliard-type functionals in which the derivative is replaced by higher-order fractional derivatives and modulated by an oscillating factor. Depending on the ratio between the oscillation scale and the interface length, we identify three different regimes and prove $\Gamma$-convergence in each regime to a suitable sharp-interface limit functional. In the extreme regimes, we prove a separation-of-scales effect that enables us to highlight the difference relative to the local models.

We study the local stability of the bridge family \[ \Phi(T):=\inf_{u\in\mathcal A_T}\|\nabla u\|_{L^2(\mathbb R^n_+)}, \qquad T>0,\quad n\ge3, \] where \[ \mathcal A_T := \Bigl\{ u\in \dot H^1(\mathbb R^n_+): \|u\|_{L^{\frac{2n}{n-2}}(\mathbb{R}_{+}^n)}=1,\ \|u\|_{L^{\frac{2(n-1)}{n-2}}(\partial\mathbb{R}_{+}^n)}=T \Bigr\}, \] and \(\dot H^1(\mathbb R^n_+)\) is the completion of \(C_c^\infty(\overline{\mathbb R^n_+})\) in the norm \(\|\nabla \varphi\|_{L^2(\mathbb R^n_+)}\). Let \(\mathcal M_T\) denote the set of minimizers of \(\Phi(T)\). We prove that, for every \(T\neq T_E\), there exists \(\alpha_T>0\) such that \[ \|\nabla u\|_{L^2(\mathbb{R}_{+}^n)}^2-\Phi(T)^2 \ge \alpha_T\,d_T(u,\mathcal M_T)^2 +o\!\bigl(d_T(u,\mathcal M_T)^2\bigr) \qquad\text{for all }u\in\mathcal A_T, \] where \(T_E\) is the Escobar threshold and \(d_T\) is the distance in \(\dot H^1(\mathbb R^n_+)\).{for all }u\in\mathcal A_T, \] where \(T_E\) is the Escobar threshold and \(d_T\) is the distance in \(\dot H^1(\mathbb R^n_+)\).

In this paper we study existence and uniqueness of solutions for a very general class of doubly nonlinear diffusion equations on metric graphs, which provide the appropriate mathematical framework to describe complex tubular networks in which axial diffusion is the main focus. Some important particular cases covered in our study are the Porous Medium Equation and the evolution equation for the $p$-Laplacian, but we also consider the case in that diffusion changes from one edge to another, which takes into account the influence of the properties of the tubules forming the network on axial diffusion. Furthermore, the problem is studied under non-homogeneous Neumann-Kirchhoff conditions on the vertices of the graph.

We develop the parabolic weak porosity to characterize the parabolic Muckenhoupt $A_1$ weights with time-lag. Our main result shows that a nonempty closed set is parabolic weakly porous if and only if the parabolic distance function of the set to a negative power is in the parabolic Muckenhoupt $A_1$ class. We apply a novel stopping time argument in combination with the translation and doubling results for the parabolic weakly porous sets.

We study the controllability of a linear KdV-Schrödinger equation on the one-dimensional torus via purely imaginary bilinear controls. Considering controls spanning a suitable finite number of Fourier modes, we prove small-time global approximate controllability in L2(T). The result holds between any pair of states with the same norm and is obtained via the saturation method by following the idea introduced in [Poz24]. We first establish small-time controllability for phase multiplications, and then generate transport operators associated with diffeomorphisms of the torus. Finally, we combine these results to recover global approximate controllability. Note that the controllability property holds independently of the Schrödinger component of the dynamics, which may in particular be taken to vanish.

In this paper, we consider the Cauchy problem for the rod equation in the line. By constructing an explicit smooth initial data, we present a new method to prove that this problem is ill-posed in $H^s(\R)$ with $1< s<3/2$ in the sense of norm inflation, i.e., an initial data is smooth and arbitrarily small in $H^s(\R)$ with $1< s<3/2$, but the solution becomes arbitrarily large in the Sobolev space after an arbitrarily short time.

Let $N \ge 4$, $\Omega$ be a bounded domain in $\mathbb{R}^N$, and let $\Sigma \subset \Omega$ be a smooth closed submanifold of dimension $k$ with $2 \le k \le N-2$. We study the existence of positive solutions $u \in H_0^1(\Omega)$ to the Euler–Lagrange equation \[ -\Delta u + h u = \lambda\, \rho_{\Sigma}^{-s_1}\, u^{2^{*}_{s_1}-1} + \rho_{\Sigma}^{-s_2}\, u^{2^{*}_{s_2}-1} \quad \text{in } \Omega, \] where $h : \Omega \to \mathbb{R}$ is a continuous potential, $\lambda > 0$ is a real parameter, and $0 \le s_2 < s_1 < 2$. For $i=1,2$, the exponents \[ 2^{*}_{s_i} = \frac{2(N - s_i)}{N - 2} \] correspond to Hardy–Sobolev critical growth, and $\rho_{\Sigma} = \mathrm{dist}(\,\cdot\,, \Sigma)$ denotes the distance to the submanifold $\Sigma$. The problem involves two Hardy-type singular nonlinearities with different critical exponents, leading to a lack of compactness. Using variational methods, in particular the mountain pass lemma, together with a suitable construction of test functions, we prove existence results under appropriate assumptions. Our analysis shows that the local geometry of $\Sigma$ and the behavior of the potential $h$ near $\Sigma$ play a crucial role in the existence of positive solutions for this doubly critical problem.

We study the Stokes–Poisson–Boltzmann equations with Dirichlet and Navier boundary conditions. The system consists of the incompressible Stokes equations coupled with a nonlinear Poisson–Boltzmann equation through electrostatic forcing and convective transport effects. To handle the Navier boundary conditions in a unified framework, we employ Nitsche's method for their weak imposition within a conforming finite element setting. We derive a consistent and stable discrete formulation and establish the well-posedness of the resulting problem. By carefully choosing the penalty parameters, the bilinear form is shown to be coercive and continuous. A priori error estimates are proved in the natural energy norms, yielding optimal-order convergence under suitable regularity assumptions. Furthermore, we develop residual-based a posteriori error estimators that incorporate element residuals, inter-element jump residuals, and boundary residuals arising from the Nitsche formulation. The estimators are shown to be reliable and locally efficient. Numerical experiments confirm the theoretical results and demonstrate the robustness and accuracy of the proposed method for the Stokes–Poisson–Boltzmann system.

Dynamic elastography is a widely used, safe, convenient, and cost-effective method to aid in medical diagnosis. It visualizes the wave field propagating through living tissues and quantitatively determines the wave propagation speed from the acquired data, thereby enabling the extraction of the viscoelastic properties of in vivo tissues. Notably, this identification process relies on the mathematical modeling of the viscoelastic characteristics of living tissues. When living tissues are simply modeled as isotropic elastic media, J. McLaughlin and J. Yoon established the uniqueness of the identification in \cite{MY} by reasoning that they called the “shrink and spread argument". Given the realistic viscoelastic nature of biological tissues, generalizing their results by adopting viscoelastic models is of great significance. In this paper, using their reasoning, we prove the uniqueness of identification for two typical viscoelastic media: the isotropic extended Maxwell model and the isotropic extended standard linear solid model. More precisely, we demonstrate that the shear wave speed within a region of interest $\Omega$ can be uniquely determined from a single measurement of the wave field in $\Omega$.

We establish a comparison principle for viscosity subsolutions and supersolutions of a broad class of second-order quasilinear, maximally subelliptic PDEs on general manifolds. In fact, we prove the comparison theorem for a larger class of degenerate subelliptic PDEs. Our result strengthens a recent theorem of Manfredi-Mukherjee, which was established in the setting of Carnot groups. Our main aim is to highlight that maximal subellipticity allows one to obtain a comparison principle for weak solutions in close analogy with the classical elliptic theory.

In this note we derive a space-like quantitative uniqueness result for parabolic operators with Hölder zero-order term that interpolates between the Donnelly-Fefferman and the Bourgain-Kenig estimate. This generalizes a recent result of Teng, Wang and Zhu for the time-independent Schrödinger operator with a Hölder potential.

We consider a bushfire model in a gully. The biological scenario under consideration involves flammable fuel (trees, leaves, etc.) concentrated within the gully, surrounded by rocky hillslopes containing little or no burnable material. The mathematical formulation of the problem is a nonlocal evolution equation of parabolic type. The nonlocality arises from an ignition mechanism that becomes active when the temperature reaches the ignition threshold and is modeled via a kernel interaction with limitrophe areas. The rocky hillsides of the gully impose insulating boundary conditions of Neumann type, while the entrance and exit of the gully are modeled by (not necessarily homogeneous) Dirichlet boundary data, corresponding to prescribed environmental temperatures on the gully's terminals. Given the geometry of the domain, in the asymptotic regime of a narrow gully the model undergoes a dimensional reduction and can be analyzed through a geometric equation posed along the (not necessarily straight) axis of the gully. The reduced equation is supplemented with inner and outer Dirichlet boundary conditions (with no Neumann condition remaining in the limit). The analysis relies on the use of Fermi coordinates to capture the potentially curvilinear geometry of the gully, as well as on parabolic estimates tailored to the specific equation in order to properly account for the ignition interactions. These estimates are delicate, as the domain degenerates and the boundary conditions vary in the limit. To overcome these difficulties, we develop a bespoke reflection technique that provides uniform bounds and enables the passage to the limit.

I study the cubic Fourier-Galerkin truncation of the three-dimensional (3D) incompressible Navier-Stokes equations on the periodic torus after reduction by the full octahedral symmetry group $O_h$. The nonlinear interaction is encoded by a state-dependent orbit-level transfer matrix $M_N(u)$, and the main discrete problem is to estimate orbit-triad incidences in shell slices of translated cubes. Using a face-normalized decomposition, I reduce the local counting problem to the classical two-squares representation function and obtain an incidence bound of order $N^{4+\varepsilon}$ by the shell-counting argument developed in this manuscript. I also derive the exact orbit-level enstrophy identity, the algebraic decomposition $M_N(u)=A_N(u)+V_N(u)$, and deterministic Sobolev row-sum bounds for the raw matrix $M_N(u)$ in the stated range of exponents. These results give an orbit-level description of nonlinear transfer in the truncated system.

We study the threshold scattering problem for the energy-critical nonlinear Schrödinger equation with a repulsive inverse-square potential $\frac{a}{|x|^2} > 0$ in dimensions $d= 4, 5, 6$. On the energy level surface determined by the ground state of the energy-critical NLS without potential, we show that, despite the absence of a ground state in this setting, a strong form of rigidity persists below the kinetic threshold. Specifically, we prove that any solution on this energy surface with kinetic energy strictly below that of the ground state is global and scatters to zero. Our approach combines refined modulation analysis, a center-translated global Virial estimate, and a bootstrap argument to control the modulation parameters.

In the presence of confinement, small-data solutions to nonlinear dispersive equations can exhibit a gradual energy transfer from low to high frequencies, a mechanism driving the emergence of weakly turbulent dynamics. We show that such a forward energy transfer, manifested as arbitrary inflation of higher order Sobolev norms, occurs for small-data solutions of a quasilinear cubic wave equation on the Schwarzschild-AdS black hole exterior with Dirichlet conditions at infinity, for generic values of the mass parameter. This result is motivated by the question of nonlinear stability or instability of Schwarzschild-AdS as a solution to the Einstein vacuum equations, but the strategy of proof applies to a broader class of backgrounds exhibiting stable trapping of null geodesics. As an application, we obtain the analogous norm inflation statement on $\mathbb R \times \mathbb S^3_+$ for generic perturbations of the round metric on the hemisphere $\mathbb S^3_+$ preserving the trapping structure at the boundary.