Papers

We study the nonparametric statistical inverse problem of recovering the space-dependent permittivity in a coupled Fokker-Planck-Darcy system from discrete, noisy observations of the Fokker-Planck solution. We consider the coupled parabolic-elliptic system on a bounded domain with the physically natural no-flux boundary condition for the Fokker-Planck equation and homogeneous Dirichlet boundary conditions for the Darcy equation. The inverse problem is indirect, since the unknown coefficient enters only through the elliptic equation and is observed only through its effect on the density in the parabolic equation. We first develop the analytical theory of the forward problem required for the statistical analysis, establishing well-posedness and a priori bounds uniform over the admissible set of permittivities. We then derive two stability estimates for the inverse problem, namely a Lipschitz-type forward estimate and a generalized backward estimate. Placing a rescaled Gaussian process prior on the log-shifted permittivity, we show that the posterior contracts around the truth at an explicit polynomial rate in the number of observations, with the posterior mean converging at the same rate. Numerical experiments in one and two dimensions, using preconditioned Crank-Nicolson and ensemble Kalman filter algorithms, complement our theoretical results.

In this paper, we verify a bifurcation phenomenon regarding the multiplicity of weak solutions, subject to the Dirichlet boundary condition, of a regularized two-phase free boundary problem associated with the $p$-Laplacian. In fact, we prove the existence of a mountain pass solution when the boundary data is small. In this case, three weak solutions of the regularized problem exist: the $p$-harmonic function, a minimizer of the corresponding functional, and the mountain pass solution. Finally, we consider the special case of radially symmetric solutions. By solving the associated nonlinear ordinary differential equation in the unregularized case, we show explicitly how the Bernoulli condition at the free boundary gives rise to the bifurcation of solutions.

We prove a sharp quantitative stability result for the affine \(L^p\)-Sobolev inequality, for \(1<p<2\), introduced by Lutwak–Yang–Zhang (J. Differential Geom., 62 (2002),17–38). Moreover, the stability exponent is shown to be optimal, and equal to \(2\).

We construct forward discretely self-similar (DSS) solutions of two-dimensional Navier-Stokes equations with arbitrarily large data in $L^2_{loc}(\mathbb{R}^2\setminus\{0\})$. For such rough initial data, we first establish a sharp linear estimate under the discrete scaling, which identifies the \(L^2\)-norm of the initial datum on a fundamental annulus with the Dirichlet energy of its caloric lift over one period in logarithmic time. This, together with a structural cancellation between the linear profile and the nonlinear remainder over one period in logarithmic time, yields a uniform \(\dot{H}^1\)-bound for the remainder, where the natural $L^2_{loc}$-regularity of the initial data seems to be sharp. Since the two-dimensional Leray operator has no \(L^2\)-coercivity, we recover uniform energy control for the remainder by exploiting \(L^p\)-estimates ($1<p<2)$, together with a key bootstrap argument and periodicity. We then first construct smooth DSS solutions using further weighted estimates and a fixed-point argument as long as the initial data belong to $C^2_{loc}(\mathbb{R}^2\setminus\{0\})$. These stronger weighted bounds are not uniform under approximation of rough data. Instead, the passage to \(L^2_{\rm loc}\) data relies only on the unweighted energy estimates. Finally, we use strong time continuity of the limiting periodic profile to recover the uniform spatial tightness needed to verify the initial data.

In this paper, we establish local and global Schauder estimates for classical solutions of a class of fully nonlinear elliptic partial differential equations, which are not necessarily convex or concave, under suitable Hölder continuity assumptions on the data. Our approach is based on a robust blow-up argument, combined with geometric tangential analysis and compactness techniques. This strategy is strongly influenced by methodologies developed in the contemporary theory of nonlinear elliptic PDEs.

We prove a long time well-posedness result for a quadratic non-linear Schrödinger equation with random well prepared initial conditions in dimension d $\ge$ 1. In some regimes, we derive the Wave Kinetic equation associated with the equation in dimension d $\ge$ 5, almost up to the kinetic time to the power d/(d + 1). Our analysis relies on the ideas introduced by Deng and Hani.

Motivated by a recent article of R. Vanlaere (2026), we investigate the spectral asymptotics of a singular harmonic oscillator arising from the radial reduction of the magnetic Schrödinger operator on the unit disk in the presence of a constant magnetic field and an Aharonov-Bohm flux. The corresponding eigenvalue problem is equivalent to the study of the zeros of the Kummer confluent hypergeometric function with respect to its first parameter. Combining now standard semiclassical methods with the uniform asymptotic theory of Whittaker and confluent hypergeometric functions developed by Dunster (1989), Gabutti-Gatteschi (2001) and improved quite recently by Dunster (2026), we obtain a comprehensive description of the different spectral regimes. When the boundary lies in the classically forbidden region, we derive the exponentially small asymptotic correction to the eigenvalue caused by the tunneling effect. For energy levels above the tunneling regime, we derive a shifted Bohr-Sommerfeld asymptotics for highly excited states at a fixed magnetic field. Finally, for the transition regime where the energy level touches the boundary, we derive a three-term asymptotic expansion of the eigenvalue. In an appendix, we translate these spectral results into explicit asymptotic formulas for the zeros of the Kummer function.

We give a counterexample showing that an asymptotic mean value property for $p$-harmonic functions in $\mathbb R^d$ does not necessarily hold in the pointwise sense for $d\geq 3$ and $p>2$ near $2$. The proof uses a perturbation argument from $p=2$ and the Implicit Function Theorem to establish the counterexample. We also use a computer assisted proof to give an explicit range of values of $p$ for which the counterexample holds. Finally, we also establish a counterexample for the pointwise asymptotic variational mean value property.

We prove that the Dirichlet-to-Neumann map associated to the Schrödinger operator $-\Delta+q$ on a planar bounded Lipschitz domain uniquely determines an arbitrary complex-valued potential $q\in L^p$, $p>1$, whenever zero is not a Dirichlet eigenvalue. Our argument uses Nachman's d-bar equation only at sufficiently large complex frequencies, where exceptional points are absent. A new annular frequency estimate converts the first asymptotic coefficient of the complex geometric optics solutions into a pointwise bound for a primitive of the potential difference, reducing the inverse problem to a unique-continuation argument.

Let $\Omega\subset\mathbb R^2$ be a bounded smooth domain. We prove that the weak Dirichlet-to-Neumann map for $-\Delta+V$ uniquely determines every complex-valued potential $V\in L^p(\Omega)$, $p>1$, provided that zero is not a Dirichlet eigenvalue. This extends the previously known range $p>4/3$ to all $p>1$. The proof uses Bukhgeim's quadratic-phase solutions and an average of Alessandrini's identity over the phase center. After separating the Neumann-series tails, we show that each remaining Born term tends to zero. The fixed-order estimates combine bounds for the absolute kernels with oscillatory cancellation and a duality argument for the product of the two Cauchy transforms.

We show functional inequalities with logarithmic weights as a critical case of Caffarelli-Kohn-Nirenberg inequalities. Original CKN inequalities have polynomial weights and are shown via scaling argument. In this paper, we develop this argument into it for nonlinear scaling. Our inequalities are generalizations of critical Hardy type inequalities.

In this paper, we investigate energy conservation for weak solutions to the two-component Camassa-Holm system. We derive a local energy balance equation in the sense of distributions with an explicit defect term. As an application, we determine that the Onsager exponent is 1, and as a consequence, we also obtain the uniqueness of weak solutions.

We study two-dimensional steady gravity water waves with general vorticity in a homogeneous, inviscid fluid of finite depth, subject to a surface pressure that tends to distinct constants upstream and downstream. Starting from a fixed supercritical shear flow, we construct two global branches of monotone fronts connecting distinct laminar states, with increasing and decreasing profiles corresponding to the two signs of the pressure transition. Small-amplitude fronts are first obtained via the implicit function theorem and then extended by analytic global continuation.

We prove local well-posedness for the initial-boundary-value problem for the Korteweg–de Vries equation on the right half-line at the critical regularity $s=-3/4$, with initial and boundary data in $H^{-3/4}(\mathbb R_+)$ and $H^{1/12}(\mathbb R_+)$, respectively. This reaches the endpoint left open in the previous half-line results of Holmer (2006) and Bona, Sun, and Zhang (2006), who established local well-posedness for $s>-3/4$. The proof is based on a contraction argument in the Besov-type Bourgain space introduced by Kishimoto. The main difficulty is to construct a boundary forcing operator compatible with the endpoint $b=1/2$ and the required $\ell^1$-summability in the modulation variable. We construct such an operator using a Laplace-transform representation of the linear half-line problem and a carefully designed extension to $x<0$. The extension combines the required regularity across the boundary with prescribed vanishing spatial moments, yielding favorable low-frequency behavior of the associated kernel. A pointwise kernel estimate then provides the crucial dyadic summability needed for the $X^{-3/4,1/2,1}$ estimate. Together with a time-trace estimate for the Duhamel term at $b=1/2$, this yields the critical local well-posedness result. The ideas used here have prospects to be used in other nonlinear dispersive equations on the half-lines.

Let $G$ be the group $\mathbb{R}_+\ltimes \mathbb{R}^n$ endowed with Riemannian symmetric space metric $d$ and the right Haar measure $\mathrm{d} \rho$ which is of $ax+b$ type, and $L$ be the positive definite distinguished left-invariant Laplacian on $G$. Let $u=u(t,\cdot)$ be the solution to $u_{tt}+Lu=0$ with initial conditions $u|_{t=0}=f$ and $u_t|_{t=0}=g$. In this article we show that for a fixed $t \in\mathbb{R}\setminus\{0\}$, \begin{align*} \|u(t,\cdot)\|_{L^{1,\infty}(G,\mathrm{d}\rho)}\leq C\ (1+|t|)\big(\|(\operatorname{Id}+L)^{\alpha_0/2}f\|_{L^1(G,\mathrm{d}\rho)} +\|(\operatorname{Id}+L)^{\alpha_1/2}g\|_{L^1(G,\mathrm{d}\rho)}\big) \end{align*} if and only if $\alpha_0\geq {n/2}$ and $\alpha_1\geq n/2-1$. Moreover, the polynomial-in-time growth $1+|t|$ in the above estimate is best possible.

We consider the global dynamics of finite-energy solutions to the $2$-equivariant harmonic map heat flow from $\bbR^2$ to $\bbS^2$ and $\dot H^1$-bounded radial solutions to the energy-critical nonlinear heat equation in dimension $6$. Building upon soliton resolution, we obtain a classification of their multi-bubble dynamics. First, we prove that all such solutions are global. For the harmonic map heat flow, the number and sign of the bubbles are determined by the energy class of the initial data, while for the nonlinear heat equation the bubble signs necessarily alternate. In both models, the largest scale is determined, up to a positive multiplicative constant, by the spatial tail of the initial datum. The remaining scales concentrate according to exponential and iterated-exponential laws determined by the largest scale. Finally, global bubble trees with an arbitrary number of bubbles exist for both models. For the harmonic map heat flow, this follows directly from the classification; for the nonlinear heat equation, we construct alternating bubble trees with any prescribed radial $\dot H^1$ tail, thereby realizing the scale laws arising in the classification.

We prove an upper bound on the homogenization length scale which is a fixed power of the coarse ellipticity ratio for coefficient fields that satisfy a finite range of dependence assumption.

We establish the extension of Milnor's inequality, relating total curvature with the bridge index, to the $C^1$-closure of a knot class, without any restriction on the self-intersections of the limit curve. Together with recent work of Reiter–von der Mosel, this resolves the circular elastic knot conjecture, first predicted by Gallotti–Pierre-Louis in 2007 and then formulated as a mathematical conjecture by Gerlach–Reiter–von der Mosel in 2017. More precisely, if the bridge and braid indices of a tame knot class coincide, then the multiply covered circle is the unique elastic knot.

We study the parabolic vectorial Allen–Cahn equation in two space dimensions for a fixed smooth potential with finitely many non-degenerate wells. Our result provides a parabolic extension of Bethuel's elliptic compactness theorem: at almost every positive time, the diffuse energy, potential energy, and gradient tensor converge to measures concentrated on a countably \(1\)-rectifiable interface and satisfy all the Bethuel-type relations. In the scalar case, the tangential defect vanishes and the usual mean-curvature-flow structure is recovered. For general vectorial potentials, the limiting stress and dissipation yield a two-mobility system: the normal energy flux balances weighted curvature, while a tangential internal mobility transports residual tangential energy. Thus the result extends the Allen–Cahn-to-Brakke framework of Ilmanen from scalar to vector-valued systems, with an additional non-negative dissipation defect in the localized energy inequality.

We prove that every minimizing $p$-harmonic map from $B^3$ into $\mathbb S^3$ is locally $C^{1,\alpha}$ for some $\alpha\in(0,1)$, for every $p>p_0$, where $p_0=\frac{7-\sqrt{17}}{2}\approx 1.44$. This closes the gap between the previously established regularity ranges $[2,2.642]\cup[2.961,3]$ and, in particular, yields full interior regularity for all $p\ge2$. The result also extends regularity to the subquadratic range $p_0<p<2$.

Following the work [27], we further study spacetime observability for the wave equation on a one-dimensional interval. The main task is to characterize the observable symmetry condition (OSC) under both Dirichlet and Neumann boundary conditions, which is more complex than the torus setting due to boundary reflection, and to discuss the relation between OSC under different settings. We prove that OSC, together with the geometric control condition (GCC), is necessary and sufficient for observability. We also provide a necessary and sufficient condition for the related unique continuation property.

In this paper, we consider a nonlocal elliptic equation involving a nonnegative parameter $a$ that quantifies the strength of the nonlocal term, posed in a smooth bounded domain with homogeneous Dirichlet boundary conditions and a given nonnegative source term $f$. This class of problems arises in several applications, including the modeling of thermistor behaviour and population dynamics with crowding effects. We focus on the breakdown of the maximum principle as the parameter $a$ increases. We characterize the two types of degeneracy that may occur, possibly simultaneously, at the critical parameter $a^*$: the solution may vanish at an interior point of the domain, possibly forming a dead-core region, or its normal derivative may vanish on the boundary. In particular, we obtain sufficient conditions for the emergence of dead-core solutions, thereby answering an open question raised in the recent literature. Several one- and two-dimensional examples are constructed to illustrate each possible scenario. Additionally, we study an optimization problem motivated by biological considerations, where the goal is to maximize the critical threshold $a^*$ over an admissible class of resource distributions. For sources with prescribed total mass, a uniform upper bound, and an interior-contact condition, we show that the critical threshold is unbounded and that the corresponding total population can be arbitrarily small. We also establish the existence of a maximizer in a class with a prescribed positive lower bound on the total population, a uniform bound on the sources, and the same interior-contact condition.

We investigate the optimal damping design and exponential stabilization for a class of higher-dimensional weakly degenerate damped wave equations. In weighted Sobolev spaces, we establish well-posedness and uniform exponential energy decay, and verify the existence of long-time optimal damping potentials. To overcome analytical difficulties arising from system non-self-adjointness, we use finite-time approximation via semigroup operator norms. Through linearization and adjoint-state analysis, we derive first-order necessary optimality conditions and show finite-time optimal dampings possess a bang-bang structure.

We establish a quantitative sound-soft scattering limit for a finite bubbly crystal near the upper edge of the first Bloch band of the corresponding infinite crystal. The inclusions form a dense periodic array of period $\varepsilon$ in a bounded Lipschitz domain, and their density contrast is $\delta=\varepsilon^2$, with fixed positive wave speeds. Under coordinate-reflection symmetry, we prove that the normalized capacitance symbol has a unique nondegenerate maximum on the Brillouin torus. Its Hessian defines a positive Dirichlet elliptic operator governing the limiting spectral detunings. A mean-constrained variational formulation allows us to compare the actual finite-array capacitance matrix with the truncated infinite-lattice operator and to prove exponential localization of their difference near the sample boundary. We obtain an $O(\varepsilon)$ norm-resolvent approximation and connect it to the full acoustic scattering problem, retaining the second-order Bloch correction required by the frequency scaling. For rescaled detunings outside the effective Dirichlet spectrum, the exterior $L^2$ and far-field discrepancies are $O(\varepsilon)$, while the interior $L^2$ field is $O(\varepsilon^{1/2})$. Numerical experiments illustrate the far-field convergence and the effective spectral modes. Explicit one-dimensional calculations describe finer Fabry–Pérot transmission windows and show that a fixed frequency strictly inside the first band need not have a unique scattering limit.

The exponential decay of weak solutions towards the constant steady state of general cross-diffusion systems in bounded domains with no-flux boundary conditions is investigated. The proof of quantitative decay rates is based on the relative entropy method and involves two parameters: the exponent determining the entropy density and the exponent in the entropy production integral. The key ingredient are novel convex Sobolev inequalities, which extend existing results in the literature to a broad range of values of the two parameters. These inequalities are established using convexity arguments and the Gagliardo-Nirenberg inequality.

In our previous results, we have proposed a new free boundary problem representing the bread baking process in a hot oven. The problem consists of conservation laws for the internal energy and water mass, the Stefan condition, the boundary and initial conditions. We note that the boundary condition for the water content contains the temperature at the boundary. Accordingly, we do not expect a strong solution for $W^{1,2}$-initial data. To address this difficulty, we established strong solvability by assuming stricter conditions for the initial functions in [3]. In contrast, this paper aims to prove the existence of a weak solution, when these assumptions for initial functions are relaxed.

This paper is concerned with the inverse source problem for the stochastic fractional Helmholtz equation driven by white noise. For every fractional order $0<\alpha<1$, we prove the existence and uniqueness of the outgoing distributional solution to the direct problem with resolvent estimates at high frequencies, and establish its stochastic representation. For the inverse problem, we demonstrate that the variance of the source can be uniquely determined by the correlated random exterior data at a single frequency. We further establish an increasing stability estimate for the inverse problem by using the multifrequency correlated exterior data. Our stability result shows that as the upper bound of the bandwidth of the utilized frequency increases, the stability will also improve. The analysis employs the construction of geometric optics solutions, which connects the correlated data to the X-ray transform of the variance.

We study trust-region minimization of a smooth, possibly nonconvex functional plus a convex functional composed with a bounded linear operator. A relative primal–dual gap condition controls both the error in an approximate proximal-gradient step and its linear-model decrease. Together with a computable absolute stationarity test, it yields a finite Cauchy search, convergence of the proximal stationarity measure to zero, and an $O(\varepsilon^{-2})$ bound on outer trials. The outer analysis allows the linear operator to take values in a Banach space and does not require dual attainment. When the operator takes values in a Hilbert space and the regularizer is finite and Lipschitz, the dual proximal-gradient method produces finite gaps tending to zero, provided the required proximal maps and functional values can be evaluated. We prove $O(j^{-1})$ gap bounds for both recovered and averaged primal candidates and give a sharper bound on the primal error for exactly recovered points. A semilinear elliptic control problem with unsmoothed total-variation regularization and an $L^2$ control cost illustrates the method in the full $H^1$ metric. Across five meshes, outer and state Newton counts remain constant, while interior-point iteration counts vary mildly.

We study the uniqueness and boundary behavior of nonzero convex Aleksandrov solutions to $\det D^2 u=M|u|^p\nu$ with zero boundary values on bounded convex domains in $\mathbb{R}^n (n \geq 2)$. For $0<p<n$, we prove the uniqueness of nonzero convex solutions in the finite-energy class when $\nu$ is a locally finite Borel measure with positive mass and $\int_\Omega \text{dist }(\cdot,\partial\Omega)\,d\nu<\infty$. For $p>n$, we construct an explicit two-shell measure on the unit ball for which the problem has at least three radial solutions that are globally Lipschitz and have finite energy. In the case of $\nu=\text{dist }(\cdot,\partial\Omega)^{-\alpha}\,dL^n$, $0\leq\alpha<2$, we prove global Lipschitz continuity when $p-\alpha>n-2$ and obtain sharp upper and lower estimates on domains with a flat boundary part when $n(\alpha-1)-2<p-\alpha\leq n-2$. When $\alpha=0$ and $p=n-2$, our log-Lipschitz lower estimate has the same exponent as the known upper estimate. This answers the question raised by Le (Global Lipschitz and Sobolev estimates for the Monge-Ampère eigenfunctions of general bounded convex domains. Ann. Fac. Sci. Toulouse Math. (6) 35 (2026)). We also give a sufficient condition for finite Monge-Ampère energy on every bounded convex domain, prove its necessity when the boundary contains a flat part, and apply it to prove the uniqueness of the Monge-Ampère eigenvalue among all nonzero convex solutions.

Short-horizon forecasts of atmospheric temperature are needed to support climate-aware digital-twin systems, but such forecasts must be produced where thermal observations are incomplete. This study evaluates a physics-informed neural network for potential-temperature forecasting, constrained by a pressure-coordinate thermodynamic advection-source equation and a diabatic-source closure fit from the preceding 12-hour period and frozen before future-time training. Using hourly ERA5 reanalysis at three pressure levels, the model is evaluated as a conditional hindcast at lead times of one, two and three hours against persistence, local-trend, and two matched neural-network baselines, one of which receives the same future meteorological forcing as the PINN, helping distinguish the physical constraint from access to future forcing. In an Oklahoma development case, mean RMSE improvement over the strongest baseline grew from 8.1% at one hour to 23.8% at three hours; under an observation-density sweep down to 5% of candidate locations, this 3-hour advantage remained 14.6–16.9%, with no evidence that lower density improves performance. Under a fixed protocol transferred to an Alabama heat event with three virtual-observation layouts, three-hour improvement ranged 19.7-24.4% with consistent origin-level wins. A parallel Montana stress test, in which fixed pressure levels intersected complex terrain, produced a three-hour degradation of roughly 17.5%, identifying a terrain-related applicability limit of the formulation. Together, these results indicate that the physics constraint's benefit grows with forecast horizon, persists under severe observation sparsity, and transfers across regions, but is bounded by the validity of a fixed vertical-coordinate representation over complex terrain, evidence relevant to physics-constrained components of climate-aware forecasting and digital-twin systems.