Papers

We consider the nonlinear Schrödinger equation on a two-dimensional strip with an attractive $\delta$ interaction and power nonlinearity. We investigate the transverse stability and bifurcation of line solitons as the width of the strip varies. We first establish local well-posedness in $H^1$, conservation of mass and energy, and global existence in the $H^1$-subcritical regime. We then identify a critical width $L_*$ at which the line soliton undergoes a transverse instability. More precisely, we prove orbital stability for $L<L_*$ and orbital instability for $L>L_*$. At the critical width, a simple eigenvalue of the linearized operator crosses zero, and we construct, via the Lyapunov-Schmidt reduction, a branch of positive nontrivial stationary solutions bifurcating from the line soliton. We determine the direction of this bifurcation by computing the second-order variation of the width along the branch. Finally, we investigate the orbital stability of the bifurcating solitons and obtain a stability criterion which can be evaluated in the regime of sufficiently small interaction strength.

We study the boundary regularity of weak solutions to nonlocal nonlinear elliptic equations with bounded measurable coefficients. Our main result establishes that a capacity density condition is equivalent to the validity of a uniform boundary Hölder estimate for solutions with Hölder continuous exterior data. More generally, we derive a fine capacitary estimate on the modulus of continuity that captures how regularity is inherited from the exterior datum to the solution.

In this paper we provide Monte Carlo and deep neural network approximations for stochastic representations of solutions to linear elliptic partial differential equations with constant diffusion, drift and killing. Building on the modified Walk-on-Spheres algorithm of Beznea et al. (arXiv:2209.01432), we introduce Monte Carlo estimators that explicitly incorporate sampled random times arising in the analyzed stochastic representations. We establish uniform error bounds for these estimators and show that, under suitable assumptions, a prescribed approximation accuracy is achieved with sample complexities growing at most polynomially in both the inverse accuracy and the problem dimension. Furthermore, we prove a deep neural network approximation result for the stochastic representations. Assuming suitable neural network representations of the boundary data and the distance function to the boundary, we use the constructed Monte Carlo to design deep neural networks that approximate the representation uniformly with a number of parameters growing at most polynomially in the inverse accuracy and the problem dimension. These results extend previous complexity analyses to a broader class of elliptic equations involving drift and killing.

In this paper, we consider the initial-boundary value problem of a three-component reaction-diffusion system with dual-dependent motility, which depends on both the chemical concentration and the nutrient level. We systematically establish the global existence, boundedness, and asymptotic behavior of the classical solutions to the system with no-flux boundary conditions through classified discussion and by exploiting specific structural properties of the motility function. In turn, by comparing the results obtained in the distinct cases, we demonstrate that the solution converges to different constant steady states, thereby indicating the crucial role of the dual-dependent motility function in the long-term behavior of the system.

In this article, we propose a novel hyperbolic relaxation system for a general class of fourth-order nonlinear partial differential equations arising in the modelling of thin film flows or the phase separation of binary mixtures. The approximations are constructed to dissipate energies which recover the energy (Lyapunov) functional of the limit equation when the relaxation parameters vanish. Using the relative energy framework, we prove the convergence of weak entropy solutions of the relaxation system to smooth solutions of the limit equation. We validate our analysis with a series of numerical examples for thin film equations and Cahn-Hilliard equations.

We show low-regularity well-posedness of the Korteweg-de Vries equation with spatially quasi-periodic initial data. To this end, we employ frequency-dependent time localization and a bilinear version of the Córdoba–Fefferman square function estimate to show a bilinear Strichartz estimate for quasi-periodic functions. The solutions are proved to preserve the Sobolev regularity of the initial data, which was not the case in earlier works. The argument extends to other dispersion relations and higher order nonlinearities.

We consider the magnetic Neumann Laplacian on bounded domains in $\mathbb{R}^2$ with outward peaks. The operator is associated with a large magnetic field depending on a parameter $\lambda$, and we investigate the behaviour of the eigenvalues as $\lambda$ tends to $+\infty$. We show that their asymptotic expansion is influenced by the sharpness $q$ and geometry of the peak, and that its main term is of order $\lambda^{\frac{2}{q+1}}$. This is an extension of previous works on magnetic Neumann Laplacians in smooth domains and domains with corners.

We establish a uniform pole-subtracted limiting absorption principle for a fixed cluster of \(N\) disjoint three-dimensional high-contrast elastic resonators when the contrast tends to infinity and \(\omega=\delta^{1/2}\tau\) approaches the zero threshold. The exterior Dirichlet-to-Neumann map and a variational Grushin–Feshbach reduction give an exact decomposition of the cutoff resolvent into a uniformly bounded regular part and a finite-rank term governed by \[ \cM_\delta^\pm(\omega) = \delta K-\omega^2I_m \mp\ii\delta\omega\Gamma_0 +\mathcal O(\delta^2+\delta\omega^2). \] The cutoff-resolvent norm is uniformly equivalent to \(1+\|(\cM_\delta^\pm(\omega))^{-1}\|\); hence the finite-dimensional channel carries every loss of uniformity. An elastic optical identity factors the leading radiation matrix through one total-force map into \(\C^3\). Thus \(\rank\Gamma_0=3\) and \(\dim\Ker\Gamma_0=6N-3\). Compression to a static eigenspace of dimension \(r\) leaves at most three leading radiative channels. Simple bright poles have width \(O(\delta)\), whereas force-dark poles with a positive second radiation form have width \(O(\delta^2)\); the corresponding real-axis peaks have orders \(\delta^{-3/2}\) and \(\delta^{-5/2}\). We also treat multiple static eigenvalues, compute the spherical coefficients, and derive a conditional two-parameter crossover for a symmetry-broken dimer.

We define Sobolev spaces associated with the Grushin operator with or without drift. We study some of their norm equivalences and establish embedding and algebra properties. We also prove the boundedness of the Riesz transforms on these spaces.

We study regularity, Harnack inequalities, Liouville rigidity, and principal eigenvalues for viscosity solutions of singular or degenerate fully nonlinear elliptic equations $\Phi(x,|\nabla u|)F(D^2u)-H(x,\nabla u)+c(x)|u|^{i(\Phi)}u=h(x)$ in a bounded domain $\Omega$, where $\Phi$ describes the singular or degenerate dependence on the gradient and $H$ is a Hamiltonian. We first prove global $C^{1,\gamma}$ regularity for the Dirichlet problem without lower-order terms. The argument combines a global $L^\infty$ estimate from the Alexandroff–Bakelman–Pucci inequality, boundary barriers yielding a global Lipschitz bound, and a compactness-based iterative approximation scheme. We next establish an additive Harnack inequality for nonnegative viscosity solutions by sliding from below a cusp function of the form $-|x|^{1/2}$. Under an additional homogeneity assumption, this yields the classical Harnack inequality and Liouville-type theorems in $\mathbb{R}^n$. Finally, under suitable homogeneity and comparison assumptions, we develop a generalized Dirichlet principal-eigenvalue theory for the full operator. We prove the existence of principal eigenfunctions and characterize the associated eigenvalues through maximum and minimum principles. These results provide a unified framework for global regularity, Harnack estimates, Liouville rigidity, and principal eigenvalues for a broad class of singular and degenerate fully nonlinear equations with Hamiltonian terms.

We study Bochner–Riesz summability for a one-dimensional model of noncompact manifolds with ends. Each end has an effective Euclidean dimension, and these dimensions may differ from one end to another. The central point is the contrast between the symmetric and asymmetric cases. When the end dimensions agree, the model follows Herz's radial theory for the Euclidean Laplacian. When they are unequal, a Fefferman-type obstruction appears, analogous to the ball multiplier obstruction in higher-dimensional Fourier analysis, even though the model itself is one-dimensional. We give a complete characterisation of the \(L^p\)-boundedness of the corresponding spectral projections and Bochner–Riesz means. In the asymmetric case, the boundedness range contains an additional restriction depending on the difference between the end dimensions. This restriction is absent from Herz's radial model and shows that Bochner–Riesz summability on spaces with unequal ends is governed not only by the maximal end dimension, but also by the interaction between the ends.

We study the discrete diffusive exchange-driven growth (EDG) equations on a bounded smooth domain of arbitrary dimension subject to homogeneous Neumann boundary conditions. The system belongs to the class of infinite systems of semilinear partial differential equations with nonlinear source terms of quadratic type. Global-in-time existence of solutions is established for separable exchange kernels of the form \(K_{i,j}=b_i a_j\), where the donor rates exhibit at most linear growth while the receiver rates are sublinear. The analysis is based on a uniform Fisher information estimate obtained from an entropy-entropy dissipation identity. This estimate yields renormalized solutions to a truncated system with finitely many species. A compactness argument then enables passage to the limit in the exchange operator, leading to the existence of global-in-time solutions for the full system.

In this article we study the spectral problem related to the fractional Laplace-Beltrami equation on $(\mathbb{R}^d,g)$ and establish its equivalence with the classical anisotropic Helmholtz equation. The proof is based on Seeley's construction of complex powers of elliptic operators and the pseudodifferential symbolic calculus. As an application, we describe the related fixed-frequency inverse scattering problem of recovering the metric from the scattering amplitude.

We study the recovery of a time-dependent potential in a parabolic equation on a conformally transversally anisotropic manifold. The initial state and a lateral Dirichlet datum supported near the front face are used as inputs, while the terminal state and the Neumann trace near the back face are observed. Assuming injectivity of the attenuated geodesic ray transform on the transversal manifold for all sufficiently small constant attenuations, we prove that these partial input-output data uniquely determine the potential. No simplicity assumption is imposed on the transversal manifold. The proof combines a conformal reduction, transversal Gaussian beam quasimodes, boundary Carleman estimates, and geometric optics solutions with prescribed lateral supports.

We investigate positive weak solutions of the critical $p$-Laplace equation $$ -\Delta_p u = u^{p^*-1} h(u), \qquad 1 < p < n, $$ where $h$ is a positive, bounded, continuous, and nonincreasing function. Our first main result is a complete classification of normalized, bounded, positive entire solutions for every equation in a compact family determined by $h$: Any such solution must coincide with an Aubin–Talenti profile. Moreover, the existence of an Aubin–Talenti profile as a solution implies that $h$ is constant on the entire interval of values attained by that profile. Subsequently, applying this classification result, we establish the following scale-invariant Schoen-type estimate $$ \left(\sup_{B_R} u\right)\left(\inf_{B_{2R}} u\right)^{p-1} \le C R^{p-n} $$ for nonnegative weak solutions defined in $B_{3R}$. As a direct corollary, we obtain a fully unrestricted Liouville theorem: Every positive entire solution must coincide with an Aubin–Talenti profile. For the purely critical equation, we also show that the corresponding Liouville classification is equivalent to a Schoen-type Harnack inequality. The arguments in this work give quasilinear versions of the Kelvin transform and the method of moving spheres, which was previously available only in the semilinear setting, and also yield alternative proofs of the classical results. The technique developed here can likely be extended to a wider class of Liouville-type problems for critical equations.

In this paper, we investigate the following critical Trudinger–Moser inequality on $\mathbb R^2$ under sharp $L^p$-perturbations: $$ S(\lambda,p) := \sup_{\substack{u\in H^{1}(\mathbb R^{2})\\ \int_{\mathbb R^2}(|\nabla u|^2+|u|^2)\,dx\le 1}} \int_{\mathbb R^2} \left(e^{4\pi u^2}-1-\lambda |u|^p\right)\,dx . $$ For $2<p\le 4$, we prove the existence of a critical value $\lambda^{\ast}\in(0,+\infty)$ such that $S(\lambda,p)$ is attained when $\lambda<\lambda^{\ast}$ and is not attained when $\lambda>\lambda^{\ast}$. Moreover, we show that the nonattainment in this range is caused by a vanishing phenomenon. For $p=2$, combining our analysis with the nonexistence results for $L^2$-perturbed Trudinger–Moser inequalities obtained in \cite{Chenluzhu}, we establish the existence of two finite thresholds $\lambda_{\ast}>-\infty$ and $\lambda^{\ast}<+\infty$ such that $S(\lambda,2)$ is attained when $\lambda_{\ast}<\lambda<\lambda^{\ast}$, and is not attained when $\lambda<\lambda_{\ast}$ or $\lambda>\lambda^{\ast}$. In contrast, for $p>4$, we prove that $S(\lambda,p)$ is attained for all admissible values of $\lambda$. Our results indicate that, in the whole-space setting, the $L^p$-perturbation term affects the existence and nonexistence of extremals through either concentration or vanishing phenomena, which is fundamentally different from the bounded-domain case, where existence or nonexistence is governed solely by concentration phenomena. These results provide a complete characterization of how sharp $L^p$ perturbations determine the existence and nonexistence of extremals for critical Trudinger–Moser inequalities on the entire $\mathbb R^2$. The resulting existence and nonexistence theory exhibits a threshold structure with respect to the $L^p$ pertubation reminiscent of the classical Brezis–Nirenberg phenomenon in the whole space $\mathbb R^2$.

We establish a Schauder estimate for a nonlocal Cauchy problem with drift. The leading operator is the generator of a non-degenerate $\alpha$-stable process with $\alpha\in(0,1)$, and the drift is $\beta$-Hölder continuous with $\beta \in (1-\alpha,1)$. The proof relies on a refined conic Littlewood–Paley decomposition and a one-dimensional one-sided dyadic maximum principle.

We investigate the optimal Bianchi-Egnell-type quantitative stability constant for the critical nonlocal Sobolev inequality on the Heisenberg group $\mathbb{H}^{n}$, \begin{equation}\label{nS} S_{HL}(Q,\mu)\left(\int_{\mathbb{H}^{n}}\int_{\mathbb{H}^{n}} \frac{|u(\xi)|^{Q^{\ast}_{\mu}}|u(\eta)|^{Q^{\ast}_{\mu}}} {|\eta^{-1}\xi|^{\mu}}\,d\xi d\eta\right)^{\frac{1}{Q^{\ast}_{\mu}}} \leq \int_{\mathbb{H}^{n}}|\nabla_{\mathbb{H}}u|^{2}d\xi, \qquad u\in S^{1,2}(\mathbb{H}^{n}), \end{equation} where $Q=2n+2$, $n\geq1$, $0<\mu<Q$, and $Q^{\ast}_{\mu}=(2Q-\mu)/(Q-2)$. Let $\mathfrak{M}$ denote the manifold of Jerison-Lee bubbles, and let $H_{NS}$ be the infimum of the quotient between the deficit in \eqref{nS} and $\mathrm{dist}(u,\mathfrak{M})^{2}$. The central issue is that the Euclidean nonlocal problem and the local Folland-Stein-Sobolev problem each possess their own spectral and compactness structures, whereas the present problem couples the HLS interaction with the noncommutative conformal geometry of $\mathbb{H}^{n}$. Under the stated conditions on $(n,\mu)$, these strict thresholds, together with a Heisenberg-group profile decomposition and a nonlocal Br{e}zis-Lieb splitting, imply that $H_{NS}$ is attained. The minimizer then yields the strict comparison $H_{NS}>H_{BE}$ with the optimal stability constant for the local Folland-Stein-Sobolev inequality. We further prove that the sharp universal upper constant for the deficit-to-distance comparison is $1$ and characterize equality. Finally, for the associated Euler-Lagrange equation, we formulate the corresponding residual quotient and derive a strict single-bubble upper bound; this critical-point statement requires a separate expansion and does not follow from attainment of $H_{NS}$.

For Gamow's liquid drop model, we improve the nonexistence bound of Frank-Killip-Nam from 8 to 7.5. The key additional input is a geometric perimeter inequality arising from the capillarity problem. This yields a quantitative gain in the averaged slicing argument and shows that the variational problem admits no minimizer for $A\geq 7.5$.

Inspired by graph theory, Neumann Boundary Value Problems on $p$-adic analytic manifold are formulated. Existence and uniqueness of weak solutions is proven in the ultracontractive case. For this task, a $p$-adic Gauss-Green identity and an extended eigenvalue formula for wavelets with small supports on the manifold are proven. As an application, the closeness to a zero of an analytic function on the manifold is shown to be detectable via a Neumann Boundary Value Problem.

Each quasi-static load step of phase-field fracture is a bound-constrained minimization of a nonconvex, coupled displacement-damage energy under an irreversibility bound on the damage. Monolithic Newton stalls once the nonlinearity localizes at the advancing crack front, and staggered (alternate-minimization) schemes converge slowly there. We present an on-demand nonlinear-elimination preconditioned trust-region Newton method: an energy Steihaug-Toint trust region, a primal-dual active set for irreversibility, and a bound-constrained field-split sweep that eliminates an algebraically-identified "hard set" spanning both fields before each step. The elimination is applied on demand – triggered by the coupled Newton's own stalling and otherwise skipped – so the method reduces to monolithic Newton at no surcharge where the step is already healthy. We find the robustness to come from the energy trust region: with that globalization fixed, monolithic Newton already completes every loading history without cutbacks, where residual-merit Newton death-spirals, alternate minimization stalls, and the full-field sweep loses robustness. Against that well-globalized baseline, the on-demand elimination cuts outer nonlinear iterations by 19-25% (brittle) and 17% (ductile), with always-on elimination reaching 26-28% and about $39\%$ at the ductile nucleation step. Measured machine-independently, as a full-mesh-equivalent assembly-work proxy rather than wall-clock, it is competitive with – not faster than – monolithic Newton (within about 10%), whereas an always-on sweep adds up to 30%. Nonlinear elimination is thus an iteration-reduction mechanism whose overhead the on-demand gate bounds, with no demonstrated total-work advantage over well-globalized monolithic Newton.

Recent work of Colbrook and Stepaniants produced the first counterexamples to the planar Pompeiu and Schiffer conjectures and introduced the conformal fixed-disc, disk-polynomial, and validated-tail machinery used here. By adapting this framework to the complementary Dirichlet endpoint, we disprove the unrestricted planar Berenstein conjecture. Specifically, we construct a bounded simply connected domain $\Omega$ with real-analytic Jordan boundary, which is not a disc and for which there exist $k\in(27.4381178838,27.4381198839)$ and a nonzero real-valued function $u\in C^\omega(\overline\Omega)$ satisfying $(\Delta+k^2)u=0$ in $\Omega$, with $u=0, \partial_\nu u=\text{constant}\ne0$ on $\partial\Omega$. Thus the overdetermined Dirichlet–Neumann data do not characterize the disc without an additional sign assumption on $u$. The domain has dihedral symmetry of order $26$, but is neither a disc nor centrally symmetric, and the corresponding eigenfunction changes sign. Equivalently, its boundary arclength measure satisfies $\widehat{\sigma_{\partial\Omega}}(k\omega)=0$ for $\omega\in\mathbb S^1$. After conformally transferring to the unit disc, exact support identities and quantitative disk-polynomial estimates yield rigorous control of the infinite-dimensional tail. A Newton–Kantorovich argument then reduces existence to finitely many explicit inequalities, which are certified using interval arithmetic. The extension from the Pompeiu–Schiffer problem is not formal. The earlier construction absorbs both boundary conditions into a single inverse-Laplacian equation. At the Dirichlet endpoint considered here, the nonzero Neumann datum forces the harmonic source modes to remain, producing a coupled interior–boundary system involving the full zero-Dirichlet inverse and its Neumann trace, together with a separate sign-recovery problem.

We study the long-time asymptotics of heat kernels on Abelian covers of compact manifolds and on Abelian covers of principal bundles. For Abelian covers, we establish three complementary asymptotic expansions: a distributional expansion describing correlations of the heat semigroup, a local pointwise expansion of the heat kernel on compact subsets, and a global expansion valid on the natural diffusive scale, revealing the Gaussian profile governing heat propagation. We then extend these results to horizontal heat kernels on principal bundles under natural holonomy and curvature assumptions. In this setting, we show that the leading asymptotics are entirely determined by the geometry of the underlying Abelian cover, while the nontrivial fiber modes decay exponentially fast due to a uniform spectral gap. The proof combines Floquet theory on Abelian covers with the Borel–Weil decomposition on principal bundles.

We study graphical mean curvature flow in arbitrary codimension over the half-space and over smooth bounded domains, subject to homogeneous Dirichlet boundary conditions. At the scaling-critical Lipschitz regularity, we prove local well-posedness for initial graphs that can be approximated in $W^{1,\infty}$ by smooth profiles compatible with the boundary condition. Within this class, if the initial Lipschitz seminorm is sufficiently small, the corresponding solution is global; on a bounded domain, it also converges exponentially to the flat graph. Positive-time regularization is quantified by time-weighted Hölder estimates whose weighted quantities remain bounded as $t\downarrow0$. The main analytic ingredient is a boundary Schauder theory for variable-coefficient parabolic systems. On the half-space, it combines coefficient freezing with parity extensions and boundary identities intrinsic to the graphical system. On curved domains, localization and boundary flattening lead to anisotropic estimates, from which normal derivatives are recovered recursively.

We study quantitative unique continuation for bi-Laplace equations \[\Delta^{2}u+V(x)u=0 \] by introducing some new weighted frequency functions. We establish quantitative three-ball inequalities and vanishing-order bounds for bounded and Hölder continuous potentials. Three-ball inequalities are built on rescaling invariant weighted frequency functions. The vanishing order results are shown by related, but different weighted frequency functions.

Late-time wave tails decay at different rates along future null infinity and along timelike worldlines at finite radius. A compactified numerical evolution must represent both the slower decay at null infinity and the faster interior decay, producing an increasingly sharp transition between the two regimes. We address this difficulty for semilinear wave equations in Minkowski spacetime using homothetic hyperboloidal coordinates adapted to the scaling structure of the tail. In these coordinates, the tail approaches a smooth radial profile with the same decay rate at every compactified radius. The formulation therefore avoids the steepening of the radial profile seen in stationary hyperboloidal evolutions, and it reaches late times in a number of steps that grows only logarithmically with retarded time. We demonstrate this approach using pseudospectral simulations in 3+1 dimensions and reproduce the generic decay rates conjectured by Rinne. We also provide numerical evidence consistent with a nongeneric codimension-one cancellation of the leading tail coefficient at null infinity, resulting in a faster decay rate.

We consider the evolution of a viscous incompressible fluid in three-dimensional distorted pipes, of finite length, modeled through the Navier-Stokes equations with mixed boundary conditions. Specifically, the inflow is given by an arbitrary datum, the outflow is subject to a dynamic condition including a directional do-nothing boundary condition; standard no-slip assumptions are imposed on the remaining walls of the domain. We introduce a new functional framework that accommodates the dynamic boundary condition and the incompressibility constraint. Within this framework, we establish the existence of weak solutions. Moreover, under a smallness assumption on the data, we prove the existence and uniqueness of global strong solutions. The newly introduced functional setting enables us to deal, at first, with the associated Stokes problem, and then with the full Navier-Stokes system, via the Galerkin method implemented with a basis of eigenfunctions of a suitably modified Stokes operator.

We establish a local Lewy-type theorem for \(p\)-harmonic function with non-zero gradient in dimension three space $R^3$. Let \(1<p<\infty\), and let \(u\in W^{1,p}_{\mathrm{loc}}(\Omega)\) be a weak \(p\)-harmonic function in a domain \(\Omega\subset\mathbb R^3\), assume it satisfies \(|Du|>0\), we prove that a locally homeomorphic gradient map \(Du\) must have non-vanishing Hessian determinant. Hence \(Du\) is a local diffeomorphism.

Our first main purpose is to establish a framework for initial value problems for time-fractional evolution equation of order $\alpha \in (0,1)$ in Banach space $X$: $$ \pppa (u(t)-a) = Au(t) + F(t), \quad 0<t<T. \eqno{(*)} $$ Here $u: (0,T) \rrrr X$ is an $X$-valued function defined in $(0,T)$, and $a \in X$ is an initial value. The operator $A$ satisfies a decay condition of resolvent which is the same as a generator of analytic semigroup. Based on $X$-valued Laplace transforms, we establish a solution formula yielding the well-posedness for (*). In particular, we can directly treat a case $X=L^p(\OOO)$ over a bounded domain $\OOO$ and a uniform elliptic operator $A$. Our theory is feasibly applicable to other topics such as regularity of solutions, inverse problems and control problems.

We prove a sharp isoperimetric inequality for the harmonic mean of the first $n$ nonzero Neumann eigenvalues of the Witten-Laplacian on origin-symmetric Lipschitz domains in space forms, endowed with radial log-concave measures. The main novelty is that we establish the sharp harmonic mean inequality under general radial log-concave measures, without assuming the weight function to be non-increasing. This extends previous results that were restricted to specific or more restrictive weighted settings. The proof relies on a refined analysis of the first eigenfunction on geodesic balls, a monotonicity property derived from a convexity condition on the radial weight, and a matrix trace inequality.