Papers

This paper is devoted to simultaneously recovering multiple parameters from internal measurements for nonlocal diffusion equations. The uniqueness of the inverse problem is established by employing the asymptotic behavior of solutions, analytic continuation, the Laplace transform, and properties of analytic functions. For numerical reconstruction, we apply the Levenberg-Marquardt method to obtain a stable approximate solution of the inverse problem. Numerical examples are provided to demonstrate the efficiency of the proposed algorithm and to validate our theoretical findings.

In this paper, we propose a perturbation-based conformal prediction framework for uncertainty quantification in operator learning, with a focus on the 2D Navier–Stokes equations. While neural operators provide fast surrogates for expensive PDE solvers, they do not by themselves provide calibrated uncertainty for spatiotemporal field predictions. Our approach wraps a trained Fourier Neural Operator (FNO) with split conformal prediction and constructs the local uncertainty scale by comparing the predictions of two operators trained on nearly identical datasets: one on the original labels and one on labels perturbed by small Gaussian noise. We consider this procedure in the data-scarce regime, where the total label budget is fixed and methods that require a separate uncertainty network must divide training data between multiple models. On the 2D Navier–Stokes benchmark, the perturbation-based method produces substantially narrower conformal bands than existing methods under matched total data budgets while maintaining the target simultaneous coverage. These results suggest that perturbation sensitivity is a practical and sample-efficient uncertainty proxy for conformalized neural operators.

We establish new $L^2$ restriction estimates for toral eigenfunctions. These estimates are sharp in certain cases, and thus prove a conjecture of Huang-Zhang for smooth submanifolds of large codimension. In particular, they provide new progress toward a conjecture of Bourgain-Rudnick. The proof combines a slicing and packing method with the approximation of the discrete spherical multiplier by Magyar-Stein-Wainger and Magyar.

Let $B_1$ be the unit disk in ${\mathbb R}^2$. We consider the harmonic map equation $$ -\Delta u=|\nabla u|^2u,$$ subject to the Dirichlet boundary condition $ u(e^{i\theta})=(R\cos\theta,R\sin\theta,\sqrt{1-R^2}):=g_R$, where $0<R<1$ and $u: B_1\to {\mathbb S}^2$ is understood in the weak harmonic-map sense. In 1983, Brezis and Coron proved the existence of two explicit solutions of this nonlinear Dirichlet problem and showed that they are the unique minimizers in their respective relative homotopy classes. In this paper, we resolve a long-standing open question originally posed in their work, later posed as Open Problem 3.1 in Brezis Favorite Open Problems List. Specifically, we prove that these two explicit maps are the only weak harmonic maps with boundary trace $g_{R}$, thereby providing a definitive affirmative answer to Brezis open problem. The proof is based on a boundary rigidity argument. An auxiliary potential $X$ associated with $u$, the Pohozaev identity for the Hopf differential, and the planar isoperimetric inequality imply $$|u_r|\equiv R, \qquad u_r\cdot u_\theta\equiv0 \qquad\text{on }\partial B_1. $$ Thus the Hopf differential vanishes on the boundary and hence, by holomorphicity, on the whole disk. The problem is then reduced to the conformal case, where a stereographic-coordinate classification gives exactly the two Brezis–Coron maps.

In this article, we study the equation $$ (-\Delta_p)^s u = f(u) $$ in a bounded domain $\Omega\subset \mathbb{R}^n$, where $n\geq 2$, $p>2$, and $f$ is locally Lipschitz. We establish a strong comparison principle in a fairly general setting and use it to derive symmetry results for positive $C^1$ solutions satisfying Dirichlet boundary conditions. We also show that the $C^1$ regularity assumption is indeed satisfied for $p\in \left[2,\frac{2}{1-s}\right)$.

In this paper, we revisit the axisymmetric Nirenberg problem for certain prescribed scalar curvature functions whose critical points include the north and south poles and whose order of flatness at both points is $n-2$. It has been known for some time that the solution set is compact when the parameter points $(K_1,K_2,a_1,a_2)\in\mathbb{R}^4$, determined by the first two leading expansions of the prescribed scalar curvature functions at the poles, stay away from a critical three-dimensional hypersurface $\Sigma^0$, and that loss of compactness occurs near $\Sigma^0$. We construct, in dimensions $n \geq 4$, two-bump axisymmetric solutions for parameter points sufficiently close to $\Sigma^0$ from one side, where the relevant side is determined by the next-order flatness terms. This gives a one-sided refinement of the previously known blow-up phenomenon near the critical hypersurface $\Sigma^0$: two-bump blow-up occurs from this side, whereas the solution set remains compact on the opposite side.

In this article, a rigorous well-posedness result is established for a moist compressible primitive-equation system with phase changes. The model couples the compressible primitive equations to a bulk microphysics model for vapor water, cloud water, and rain water, including evaporation, condensation, auto-conversion, collection, and sedimentation effects. Global strong well-posedness is proved for initial data sufficiently close to constant equilibria.

This paper investigates the nonexistence of solutions to semilinear parabolic and hyperbolic inequalities with positive potentials on metric graphs, including both nonnegative solutions and sign-changing solutions. The Laplacian under consideration is of a nonstandard type, incorporating contributions from both the vertices and edges of the metric graph. We construct a new pseudo-metric and introduce suitable space-time test functions of either coupled or separated type. Under suitable weighted space-time volume growth conditions on the potential, we establish nonexistence results for very weak solutions. More precisely, we show that all such solutions to the inequality must be identically zero.

We characterize the equality case in the Brunn-Minkowski inequality for the Monge-Ampère eigenvalue of general bounded convex domains. This together with the author's previous work completely resolves a question raised by Salani (A Brunn-Minkowski inequality for the Monge-Ampère eigenvalue. Adv. Math. 194 (2005)).

In this paper, we mainly focus on pointwise, arbitrarily high-order interior estimates for the mixed local-nonlocal elliptic equation \begin{equation*} (-\Delta)^{s}u(x)-\Delta u(x)=f(x),\quad x\in B_r(0) \end{equation*} with $0<s<1$. The main challenges in this setting are the absence of an explicit Green function and the ineffectiveness of standard bootstrap arguments. Our approach overcomes these difficulties via the Campanato iteration method, which inductively constructs polynomials approximating the solution. Using the fact that all functions are locally $s$-harmonic up to a small error significantly reduces the computational complexity when studying the regularity of the fractional Laplacian acting on these polynomials. Finally, the regularity estimates are obtained from the analysis of a manageable recursive inequality system.

We study a finite-scale one-component regularity mechanism for suitable weak solutions of the three-dimensional incompressible Navier–Stokes equations. The results are organized in three layers. The first layer is unconditional. Under a fixed scale-invariant local bound Phi(1)=A(1)+E(1)+C(1)+D(1) <= M, smallness of the critical vertical-component quantity C_3(1)=int_{Q_1} |u_3|^3 dx dt yields a positive lower bound, depending only on M, for the local regularity radius at the origin. The proof converts one-component smallness into approximation by the two-and-a-half-dimensional limiting class and then into Caffarelli–Kohn–Nirenberg smallness at a smaller scale. The pressure approximation is measured in a quotient by spatially harmonic functions. This pressure topology reflects a genuine obstruction: time-dependent harmonic pressures may have bounded scale-invariant L^{3/2}-oscillation while their pointwise gradients lie beyond the control provided by the available scale-invariant quantities. The second layer is a conditional logarithmic refinement. A prepared two-shadow comparison package replaces the abstract compactness modulus by a logarithmic modulus and gives a logarithmic finite-scale decay. The third layer is a conditional relaxed-shadowing refinement. The comparison class is enlarged to smooth no-stretching horizontal flows V=(v_h,0), with the comparison pressure allowed to have partial_3 pi not equal to 0. The resulting vertical residual pairs with the small component u_3 in the relative-energy identity. Under the buffered strong-flow and localized relaxed stability inputs stated below, this gives a power-type relaxed harmonic approximation and a power-type finite-scale decay. The unconditional theorem is separated from the logarithmic and power-type assumptions; the latter two layers identify the quantitative stability mechanisms needed to upgrade the compactness modulus.

We discuss the relationship between the chain rule for the divergence operator and the renormalization property for weak solutions of the continuity equation. We construct an example of bounded divergence-free vector field on the plane, which demonstrates that in general the first property is not sufficient for the second one.

In this manuscript we develop the direct and inverse scattering problem for the cubic focusing nonlinear Schrödinger equation and for initial data that are asymptotic to an elliptic travelling wave with distinct phase at $\pm \infty$. We consider the case in which the spectral bands intersect the real axis. We then show that this class of initial data has non zero intersection with the full soliton gas initial data.

We prove the Ashbaugh–Benguria conjecture for bounded domains with Lipschitz boundary in $\mathbb R^m$. More precisely, among all bounded Lipschitz domains of fixed volume, the ball minimizes the sum of the reciprocals of the first $m$ nonzero Neumann eigenvalues. Equality is attained precisely by balls.

It has been a long standing conjecture that the $ \infty $-harmonic functions in the plane have a 1/3-Hölder continuous gradient. It is known that solutions are $ C^1 $ and that the gradient is locally $ \alpha $-Hölder, but $ \alpha $ comes without any positive lower bound. Aronsson's solution $ x^{4/3} - y^{4/3} $ shows that no better general regularity is possible. In the plane there is also a connection between the $ \infty $-Laplace equation and the one-dimensional heat equation, observed already by Aronsson himself. I shall show that this link can be accessed under a certain injectivity condition on the gradient, and that the caloric structure then is enough to prove the 1/3-Hölder continuity. Of course, an injective gradient is by no means a necessary condition, as seen by smooth solutions such as the planes and cones.

We consider a family of pure Neumann $p$-Laplacian problems, including eigenvalue problems, Lane-Emden type equations, and extremal cases such as sign nonlinearities and the $1$-Laplacian. Using variational methods, we develop a unified framework that establishes existence of solutions and characterizes their asymptotic behavior as the parameters vary. This approach reveals a natural asymptotic connection between pure Neumann $p$-Laplacian equations and a relative isoperimetric problem known as the Neumann-Cheeger problem. We describe the shape of minimizers in domains with different geometries and obtain results on regularity, uniqueness, multiplicity, symmetry, and symmetry breaking phenomena.

We establish a Remez inequality for functions in quasianalytic Denjoy-Carleman classes $\mathcal C_M$ on a large family of fat compact sets $K \subseteq \mathbb R^n$ with tame geometry, including all fat compact sets definable in the o-minimal expansion of $\mathbb R$ by restricted $\mathcal C_M$ functions. The inequality generalizes the classical Remez inequality for polynomials and its quasianalytic versions on convex bodies, replacing the polynomial degree by the Bang degree, an integer associated with the weight $M$ and the size of the function. The constants depend explicitly on the Bang degree and the geometry of $K$. We derive a range of quantitative consequences: estimates for the volume of sublevel sets, comparison of $L^p$-norms, effective inequalities of Lojasiewicz, Harnack, and Markov type with explicit constants, and decay estimates for oscillatory integrals.

We analyze fractional phase-transition energies with periodic hetero\-genei\-ties at the critical $H^{1/2}$ scaling. We prove that the $\Gamma$-limit is a sharp-interface functional whose surface energy density combines homogenization and averaging effects. The resulting coefficient is a weighted combination of the minimum and the mean of the oscillatory parameter, reflecting the coexistence of multiple interaction scales. This behavior is specific to the critical regime and differs from the case $s>1/2$.

This paper makes several improvements to existing level set based approaches to computing shape optimizers for the Dirichlet eigenvalues subject to a volume constraint. The most notable changes in formulation include an overhaul of the classical level set construction and root-finding procedures as well the use of a regularized approximation to the standard objective function. Our resulting computational minimizers are either comparable to or improvements on the best known minimizers from the literature. We conclude with a survey of subproblems within the field that may benefit from numerical experiments; these include the existence of cusps on the boundary, the end-behavior of eigenfunction weights in the p-parameterized problem, and the nature of Weyl asymptotics as they relate to the Pólya conjecture.

In this paper, we prove the local boundedness of solutions to systems of partial differential equations in divergence form. More specifically, we consider systems that include the first variations of functionals depending on the spatial variable and exhibiting nonstandard growth with respect to the gradient, such as $$\int_{\Omega} \left( 1+ h(|Du|)\right) ^{\alpha(x)} \, d x,$$ where the convex function $h=h(t)$ does not satisfy the so-called $\Delta_2$ property and does not exhibit the conventional polynomial growth behavior.

We study possible singular endpoints of the three-dimensional incompressible Navier–Stokes equations by parabolic rescaling. Weak limits can lose information through pressure, nonlinear products, concentration, spatial tails, or symmetry, so we keep these quantities as part of one augmented endpoint state and use a fixed whole-space pressure representative throughout the shrinking chain. We prove compact endpoint realization, nonoverlapping assignment of positive defects, two-generation inheritance, rotational alternatives, and a high-critical kinetic normalization centered at the original singular point. We show that positive rotational distance alone does not coerce a productive rotational Reynolds stress, and we give an explicit conditional compactness theorem under which frequency-tight amplitude-fast records pass strongly to a nonzero ancient Euler profile. We then study a possible Euler limit that is invisible under positive heat evolution. Exact heat-orbit nullity forces cancellation separately on each interaction-energy shell and isotropic covariance on every Fourier sphere; independently, any nonzero $L^2$ field normalized along its long heat orbit loses all mass from every fixed spatial ball. Frequency anti-concentration, the accumulating heat-chain passage, and retention of a singular witness remain open; no unconditional regularity theorem is claimed.

We study the axisymmetric Navier–Stokes equations with swirl in the variables $F=u^\theta/r$, $G=\omega^\theta/r$, and $\Gamma=ru^\theta$. Their distinct scaling degrees lead to several critical transition surfaces rather than a single scalar exponent. We prove mesoscopic and temporal critical-face identities, source/contact trace estimates, compact finite-depth preparation obstructions, an affine six-power flattening barrier, a fixed-old-current compression ceiling, and compactness for rescaled strain generators with bounded action and bounded variation. A sharp radial estimate gives exponential control of the time occupied by large swirl-energy records; on parabolic time intervals it yields logarithmic routing from total kinetic records to the meridional component, and an independent cone argument routes non-meridional Type-II records to a scale-critical signed compression action. The source-free circulation also satisfies an exact quadratic dissipation identity, while scalar-weight calculations show that no radial power simultaneously yields a compression-free continuity law and a scale-zero current action. The results exclude several compact and passively prepared recurrence mechanisms under explicit hypotheses; meridional Type-II records, signed critical compression, noncompact high-frequency behavior, and boundary-entry alternatives remain.

The time-harmonic elastic-electromagnetic interior transmission eigenvalue problem (EEITEP) arises when an elastic body becomes invisible to an incident electromagnetic wave. This spectral problem is typically non-elliptic and non-self-adjoint, making its analysis delicate. In this paper, we study the discreteness of transmission eigenvalues and the boundary localization of the associated eigenfunctions. For a general bounded Lipschitz domain, we prove that the set of positive transmission eigenvalues, if non-empty, is discrete with $\infty$ as its only possible accumulation point. For a radially symmetric domain, we demonstrate the existence of a sequence of transmission eigenvalues and derive their asymptotic behavior. We rigorously show that the corresponding transmission eigenfunctions exhibit boundary localization in their electromagnetic components, whereas the elastic displacement field remains globally distributed throughout the domain. Finally, we derive lower bounds for the $L^{\infty}$-norms of the electromagnetic gradients normalized by their $L^2$-norms, quantifying their blow-up behavior near the boundary along this sequence. These findings reveal a potential spectral mechanism for developing super-resolution imaging methods in elastic-electromagnetic scattering.

We show that vectorial absolute minimisers of general $k$-th order supremal functionals in $W^{k,\infty}(\Omega,\mathbb R^N)$ satisfy a maximum principle of the form $$ \max_{\overline U} \mathrm{H} \big(\cdot, u, \mathrm D u, ..., \mathrm D^{k}u\big)=\max_{\partial U}\mathrm{H} \big(\cdot, u, \mathrm D u, ..., \mathrm D^{k}u\big), \qquad\forall\ U\subseteq\Omega \mbox{ open}, $$ suitably interpreted. This is only necessary for absolute minimisers, whilst it characterises a relevant weaker notion of absolute minimality involving compactly supported variations. Further, we obtain an existence result to the Dirichlet problem for such weaker absolute minimisers, as an application of the Baire Category method. Finally, via different methods, we supplement our results by establish a gradient maximum principle for $p$-harmonic maps for $p<\infty$.

The ensemble Kalman filter (EnKF), originally developed in the geophysical sciences, is now widely used for state and parameter estimation problems in various domains of application. It may be viewed as a robust, cheap-to-implement, alternative to the optimal, Bayesian, filter. However, despite its empirical successes, theoretical understanding of its properties, in relation to the optimal filter, is in its infancy. In this paper we contribute novel theoretical understanding, studying the behavior of the mean-field limit of the ensemble Kalman filter (MFEnKF) in the linear setting. In this setting the MFEnKF coincides with the Kalman filter itself, for Gaussian initial data. We study the MFEnKF with general, non-Gaussian, initial data. Under the assumptions of nondegeneracy of the signal noise and detectability of the signal-observation pair we make three specific contributions. First, we derive the MFEnKF via a variational approximation of the Bayes update using a covariance-weighted optimal transport metric. Secondly, we use this metric to show that the MFEnKF is a strict contraction towards the subspace of Gaussian measures, giving explicit geometric rates in terms of the covariances, in both discrete and continuous time. Finally, in the continuous-time setting, we deduce a stable form of almost sure accuracy of the MFEnKF: given any pair of (nondegenerate) initial data for the MFEnKF and the optimal filter, any uniformly continuous moments of the two filtering distributions coincide in large time, for almost every observation path.

In this paper we construct numerically a new class of unstable self-similar solutions of the incompressible Navier-Stokes equations in $\mathbb{R}^3$. Our solutions are axially symmetric and homogeneous of degree $-1$ at $\infty$, and are unstable in the sense that the linearization around these solutions contains unstable modes. Solutions of this type have been discovered numerically by Guillod and Šverák and Hou, Wang, and Yang, and have applications to proving non-uniqueness results. The main novelty in this paper is that we discover the existence of such solutions in the space of axially symmetric swirl-free (ASSF) vector fields. These approximate solutions are defined on all of $\mathbb R^3$ and achieve global pointwise residuals of order $10^{-10}$. We discuss the numerical construction of these solutions in detail, as well as their relevance to the problem of non-uniqueness of solutions of the incompressible Navier-Stokes equations in 3D, in the space of ASSF solutions.

We construct locally minimizing $(1,2)$-clusters whose exterior interfaces are asymptotic to various prescribed singular area-minimizing cones. For $n+1 \leq 7$, Bronsard & Novack characterized all minimizing $(1,2)$-clusters as standard lenses, whose exterior interface is planar. For $n+1 \in [8,2700]$, the authors together with Bronsard showed the existence of a locally minimizing $(1,2)$-cluster whose exterior interface blows down to some (unknown, possibly non-unique) singular area-minimizing hypercone. For $n+1=8$, this was shown independently by Novaga, Paolini & Tortorelli. Here we develop a refined construction using the Hardt-Simon foliation that realizes prescribed cones. For a singular area-minimizing hypercone $C$ that has an isolated singularity or is cylindrical, we show that if $C$ satisfies an explicit energy bound, then there is a locally minimizing $(1,2)$-cluster whose exterior interface is asymptotic to $C$ with quantitative rates. In fact, if $C$ is an area minimizing Lawson cone satisfying this energy bound, we produce a countably infinite family of distinct locally minimizing clusters asymptotic to $C$, distinguished by their prescribed asymptotic decay to leading order. We verify this energy bound for the generalized Simons cones $C_{k,k}$ in every even ambient dimension $n+1 = 2k+2\geq 8$, and for the cylindrical cone $C_{3,3}\times\mathbb{R}$ in $\mathbb{R}^9$, where $C_{3,3}$ is the Simons cone, therefore answering the cone realization problem in these cases. This in particular removes the upper bound of 2700 on the ambient dimension when $n+1$ is even in our preceding work.

We investigate the long-time behaviour of a one-dimensional compressible viscous fluid with general capillarity and density dependent viscosity, driven by a stochastic additive noise. In particular, we prove the existence of invariant measures by applying the Krylov-Bogoliubov method in a setting where the dynamics is supported on a non-complete phase space. This analysis is further enhanced by the derivation of a refined stability result determining the continuous dependence with respect to the initial data. The present paper exhibits some properties and results for Korteweg fluids which are not known in absence of the capillarity tensor. In particular, we prove that the Markov semigroup associated with strong solutions is Feller and we can consider ranges of the adiabatic and viscosity exponents $\gamma$ and $\alpha$ larger than those available in the current ergodic literature for compressible fluids. Also the interplay between the choice of the physical domain and the use of a damping term is discussed.

We study local well-posedness for a class of stochastic reaction-diffusion equations driven by multiplicative, possibly colored, noise. The interaction between rough stochastic forcing and polynomial nonlinearities naturally leads to solutions with low spatial regularity, making the treatment of the nonlinear terms delicate. Our main contribution is a general local existence and uniqueness theory for SPDEs with rough noise and highly irregular initial data. The framework also yields new results in standard noise regimes, including trace-class noise and space-time white noise. We identify the critical initial-data spaces for a wide range of nonlinearities, and we establish instantaneous parabolic regularization, general blow-up criteria, and sufficient conditions for positivity preservation. We apply the abstract theory to several prototypical models, including the stochastic Allen-Cahn, Burgers, Fisher-KPP, and coupled Gray-Scott equations. Finally, in the one-dimensional space-time white-noise setting, we combine our local theory with existing global a priori results in a highly singular regime.

This paper is concerned with a study of a natural generalization of a classical Frank-Kamenetskii model of thermal explosion in the presence of a vortical flow in a two dimensional setting. This model describes possible stationary temperature distributions in a combustion vessel which boundary is maintained at a constant temperature. The model constitutes a Dirichlet boundary value problem for a certain semi-linear elliptic equation that depends on a parameter $\lambda,$ called Frank-Kamenetskii parameter. A remarkable property of this problem is that it admits a classical minimal solution when the Frank-Kamenetskii parameter does not exceed some critical value $\lambda^*$ and no classical solutions for $\lambda>\lambda^*$. The absence of a classical solution, in the framework of Frank-Kamenetskii theory, is associated with the thermal explosion event. Consequently, in the context of combustion, $\lambda^*,$ commonly called an explosion threshold, is a maximal value of the Frank-Kamenetskii parameter which allows to attain a thermal equilibrium within a combustion vessel and thus provides a sharp characterization of the thermal explosion. A critical temperature distribution corresponding to $\lambda^*$ is called an extremal solution. In this paper, we show that, under an assumption of sufficiently fast growth of the reaction term, there exists a regular vortical flow that allows to adjust an explosion threshold by reversing its direction, provided a combustion vessel is not a disk. We also give rather detailed description of extremal solutions. In particular, we show that extremal solutions are always classical.