Papers

While infinite-dimensional inverse problems are traditionally analyzed assuming continuous data, practical applications rely on finitely many discrete measurements. Recent deterministic approaches establish that unknowns belonging to a $d$-dimensional model class can be stably recovered from finitely many measurements. However, for severely ill-posed problems, such as the Calderón problem and inverse scattering, the known constructions may require a number of measurements that is exponential in $d$. We show that random sampling reduces this count dramatically if one asks only for exact identifiability. By exploiting the analytic geometry of the forward maps, we prove that, whenever the infinite-data problem is injective on the model class, $2d+1$ random scalar measurements determine the unknown uniquely, almost surely. Applications are given to the Calderón problem, with both infinite- and finite-dimensional boundary sampling, and to inverse medium scattering from randomly sampled far-field values.

In this paper, we study the Cauchy problems for a semilinear damped wave equation with a time-dependent coefficient for the damping term belonging to the class of non-effective damping terms with critical decay rates involving iterated logarithmic factors. As nonlinearities we consider both $|u|^p$ and $|u_t|^p$. Assuming nonnegative and compactly supported initial data, we establish the blow-up for weak solutions in the sub-Strauss range $1 < p \leq p_{\mathrm{Str}}(n)$ for the power of the nonlinear term $|u|^p$. The proof in the sub-critical case relies on an iteration frame for the space average of the solution, obtained by employing a time-dependent multiplier related to the coefficient of the damping term. On the other hand, in the limit case $p=p_{\mathrm{Str}}(n)$, we work with solutions of the homogeneous equation with separated variables, and we investigate the properties of a fundamental system of solutions for the corresponding time-dependent ODE, in order to derive an iteration frame for a suitable weighted space average of the solution. Finally, for the derivative type nonlinearity $|u_t|^p$ we prove the blow-up of weak solutions in the sub-Glassey range $1 < p \leq \frac{n+1}{n-1}$ by using a comparison argument for a suitable time-dependent function associated with the corresponding local in time solution.

For a locally symmetric space $\Gamma\backslash G /K$ we study the spectrum of the invariant differential operators of $G/K$ on $L^2(\Gamma\backslash G/K)$. This gives a purely analytical proof of the recent theorem of arXiv:2402.02530 that the quasiregular representation $L^2(\Gamma \backslash G)$ is tempered if the limit cone $\mathcal{L}_\Gamma$ of $\Gamma$ is contained in the interior of the Weyl chamber except if $G$ has real rank one or is locally isomorphic to $\mathfrak{sl}_3(\mathbb K)$, $\mathbb K=\mathbb{R},\mathbb{C},\mathbb H$, or $\mathfrak{e}_6^{-26}$.

For reaction-diffusion with blow-up nonlinearities, we consider the question whether the sup norm of any positive blow-up solution must be eventually monotone nondecreasing in time. While some sufficient conditions are known, especially for radial solutions, this natural and basic question for the blow-up theory does not seem to have been addressed so far in full generality. We construct surprising (nonradial) counter-examples of blow-up solutions with oscillatory $L^\infty$ norm, for any Sobolev supercritical power nonlinearity, which show that this property may fail. In addition, this provides examples of type II blow-up for any supercritical power, which considerably increases the known range of powers for which type II blow-up may occur. Moreover, whereas all the type II blow-up rates known so far were at most polynomial, the blow-up in our counter-examples can be arbitrarily singular. As a related question, we clarify the gradient estimates obtained and used in previous works. In particular we show that these estimates hold only at times when the $L^\infty$ norm is maximal with respect to the past.

In this paper, we study the loss of regularity for solutions to one-dimensional degenerate wave equations. We first consider the linear equation \( u_{tt}=\bigl(c(t,x)^2u_x\bigr)_x \) with \(c(t,x)\sim x^p\), and then the quasilinear equation \( u_{tt}=(u^{2a}u_x)_x. \) The initial data are assumed to satisfy \( u_0(x)\sim x^\alpha \) and \( u_1(x)\sim x^\beta \) near the degenerate point \(x=0\). In the linear problem, \(p\) describes the strength of the degeneracy, while \(\alpha\) and \(\beta\) describe the regularity of the initial data. In the quasilinear problem, \(\alpha\) also determines the initial degeneracy through \(u_0(x)^a\sim x^{a\alpha}\). Our main concern is the actual occurrence of loss of regularity, namely, the phenomenon in which the regularity of \(u(t,\cdot)\) near \(x=0\) becomes lower than that of \(u_0\) for \(t>0\). Previously, such a loss had mainly been established for special linear equations with coefficients depending only on time. In our previous work on the quasilinear equation, we proved local well-posedness without loss of regularity when \(\beta\geq\alpha\). In the present paper, we show that this condition is also necessary. More precisely, if \(\beta<\alpha\), then \( C_1tx^\beta\leq u(t,x)\leq C_2x^\beta \) near \(x=0\) for sufficiently small positive time. Thus the order of the solution changes from that of \(u_0\) to that of \(u_1\), and an actual loss of regularity occurs for both linear equations with time-space dependent coefficients and degenerate quasilinear equations.

We study a two–dimensional discrete nonlinear Schrödinger equation with saturable nonlinearity on the lattice $\mathbb Z^2$. Using a variational approach based on the Nehari manifold, we establish the existence of nontrivial periodic ground states on finite lattices and establish the existence of exponentially localized ground states in $\ell^2(\mathbb Z^2)$. A principal result is the rigorous passage from periodic to localized states: we show that, up to lattice translations, periodic ground states converge strongly in $\ell^2(\mathbb Z^2)$ to a localized ground state as the lattice periods tend to infinity. The analysis combines variational methods, spectral properties of the discrete Laplacian, and concentration–compactness techniques adapted to the two–dimensional discrete setting. We further derive qualitative properties of the resulting solutions, including positivity and exponential localization, and establish a conditional orbital stability result within the Grillakis–Shatah–Strauss framework. Numerical computations illustrate the theoretical results and confirm the predicted convergence and localization behavior.

We establish the optimal stability of Dirichlet boundary value problem for the regional (fractional) $p$-Laplacian $(-\Delta)^s_{p,\Omega}$ with $0<s\leq 1$, $\frac{1}{s}<p<\infty$ and $\Omega\subset \mathbb{R}^d$ bounded Lipschitz. More precisely, if $u_s \in W^{s,p}(\Omega)$ satisfies $(-\Delta)^s_{p,\Omega} u_s = f_s$ in $\Omega$ and $u_s = g_s$ on $\partial \Omega$, then under appropriate condition on the date $f_s$ and $g_s$ we show that $\|u_s - u_1\|_{W^{s,p}(\Omega)} \to 0 \quad \text{as } s \to 1^-.$ We also obtain an analogous optimal stability of the normalized Dirichlet eigenpairs $(\lambda_s,\varphi_s)$ associated with $(-\Delta)^s_{p,\Omega}$.

We develop a systematic adiabatic perturbation theory for the integrable $F=1$ spinor nonlinear Schrödinger equation under nonvanishing boundary conditions, formulated entirely within the framework of the associated Riemann–Hilbert problem. In this setting, localized nonlinear excitations are characterized by discrete spectral data consisting of a complex eigenvalue and an associated polarization vector. For a general class of small perturbations preserving the background, we derive the perturbation-induced evolution of the scattering data directly at the level of the Riemann–Hilbert problem. In the one-soliton sector, this yields a closed finite-dimensional dynamical system governing the slow evolution of the effective soliton parameters, including the spectral variables, the soliton center and phase, the residue amplitude, and the internal polarization state. The latter evolves according to a constrained dynamical equation with no scalar analogue. For localized perturbations, the modulation equations are expressed in explicit integral form in terms of the one-soliton eigenfunctions, providing a fully computable description of the dynamics. In the limit of vanishing boundary conditions, the resulting system reduces to the perturbation theory obtained by E. V. Doktorov, et al, Phys. Rev. A 77 (2008), no. 4, 043617.

We investigate the dynamics and spectral stability of traveling kink and antikink solutions in a dissipative sine–Gordon equation with two distinct fourth–order regularization mechanisms: a mixed space–time (inertial) term and a purely spatial (elliptic) term. The model includes damping, bias forcing, and higher–order dissipative effects, and is motivated by refined descriptions of fluxon dynamics in long Josephson junctions. Using a collective–coordinate reduction, we derive a Melnikov–type condition for speed selection, yielding explicit predictions for asymptotic propagation speeds, which are validated by direct numerical simulations of the full partial differential equation. Spectral stability is analyzed using Evans function techniques adapted to the singular slow–fast structure induced by the regularization. By formulating the linearized problem on a consistent–splitting domain, we show that no additional point spectrum bifurcates from the origin. Near the edges of the essential spectrum, a square–root transformation is used to resolve branch singularities and establish analyticity in a lifted spectral variable. Numerical Evans function computations near $\lambda=0$ and near the essential spectrum edges confirm the analytical results, indicating absence of unstable eigenvalues for both kink and antikink solutions.

In this paper, we study the nonlinear fractional Choquard equation \begin{equation}\label{eq:0.1a} (-\Delta)^su+Vu=(|x|^{-\gamma}*|u|^2)u \quad in \quad \mathbb{R}^N, \end{equation} where $0<\gamma<4$, $0<s<1$, $N\geq 4$ and $V\in C^1(\mathbb{R}^N)$ is a positive potential. Set $s_0=\frac {\gamma}{4}$. Under suitable assumptions on $V$, we prove that the equation admits a nonnegative ground state solution for $s\in(s_0,1)$, whereas no ground state solution exists for $0<s\le s_0$. Furthermore, we show that any ground state solution $u_s$ blows up and concentrates at a minimum point of $V$ as $s\downarrow s_0$. Finally, up to a subsequence, the ground state solution $u_s$ converges to a ground state solution of the classical Choquard equation as $s\uparrow 1$.

We analyze the existence and stability of bright and dark pulses in a damped nonlinear Schrödinger (NLS) equation with periodic forcing. This Lugiato-Lefever equation is a canonical model in nonlinear optics describing pattern formation in a dissipative Kerr cavity driven by a bichromatic laser pump. Bifurcating from the bright and black NLS solitons, we rigorously construct bright and dark single-pulse solutions to the Lugiato-Lefever equation. Using a recently developed toolbox for concatenating and periodically extending pulse solutions in spatially periodic systems, we then obtain periodic multipulse solutions corresponding to experimentally observed broad-bandwidth frequency combs. While the existence and stability of the bright pulses follow from standard Lyapunov-Schmidt reduction and Krein index arguments, the construction of the dark pulses is substantially more delicate because the nonzero asymptotic states of the black NLS soliton yield neutral essential spectrum of the linearization. By carefully tracking the associated small spatial Floquet exponents along the bifurcation and employing exponential dichotomies, we establish the existence of dark pulse solutions. These consist of two domain walls connected by a long plateau with critical absolute spectrum. To our knowledge, this constitutes the first rigorous bifurcation result in the Lugiato-Lefever equation from the black NLS soliton.

The existence of global weak solutions to a broad class of Navier-Stokes-Korteweg equations is established for large data in the three-dimensional torus, including the diffuse-interface and quantum Navier-Stokes systems as special cases. The model consists of the compressible Navier-Stokes equations with degenerate density-dependent viscosity and a general nonlinear third-order Korteweg term. The existence proof combines a priori estimates provided by the energy and Bresch-Desjardins (BD) entropy inequalities with a carefully designed approximation scheme. A crucial ingredient of the analysis is a new dissipation inequality associated with the Korteweg term, obtained via the systematic integration-by-parts method. To construct the solutions, artificial drag terms and a quantum Korteweg regularization are introduced, which are subsequently removed in the limit by establishing a renormalized formulation.

We study nondegenerate $\mathbb Z_2$ harmonic functions on Euclidean spaces with branching loci given by ellipsoids and planar conics. We first prove that the parameter map associated with Yan's ellipsoidal family is injective. Together with Yan's surjectivity result, this identifies the ellipsoidal models, up to Euclidean motions and overall sign, with nondegenerate harmonic quadratic polynomials of index \(n-1\) and positive critical value. In \(\mathbb R^3\), we use modified ellipsoidal coordinates to construct a nondegenerate $\mathbb Z_2$ harmonic function in a neighborhood of any planar hyperbola. We then show that no global critical $\mathbb Z_2$ harmonic function with a planar hyperbola as branch locus can have finite Almgren frequency at infinity. Using modified paraboloidal coordinates, we construct a global nondegenerate $\mathbb Z_2$ harmonic function with any prescribed planar parabola as branch locus and obtain a precise asymptotic expansion and scale-down limit. Finally, we identify the parabolic model as the infinitesimal two-valued graph potential of a family of special Lagrangian three-folds constructed by Joyce.

We establish the limiting absorption principle for time-harmonic elastic scattering of plane waves by a periodic rigid diffraction grating. By perturbing the frequency with a small positive imaginary part, we regularize the ill-posed problem at propagative wavenumbers (that is, when uniqueness fails under the classical Rayleigh expansion condition) and characterize the limiting solution via a singular perturbation result from functional analysis. The limiting solution satisfies the original scattering problem together with an additional constraint that ensures uniqueness. Both incident pressure and shear waves are considered, and the same constraint condition is obtained in both cases. The results provide a rigorous selection mechanism for physically admissible solutions at resonance frequencies. Our framework extends naturally to the Neumann (cavity) boundary condition as well as other transmission conditions, in particular when guided waves exist in periodic structures.

In this paper, we investigate the orbital stability of vortex patches in the two-dimensional incompressible Euler equations, extending the penalized energy variational framework pioneered by Abe and Choi \cite{abe2022stability} for Lamb dipoles. The recent work by Abe, Choi and Jeong \cite{Abe2025StabilityOL} (which removes $L^1$ constraint) and Dong and Luo \cite{Dong2026StabilityOV} (which treats domains lacking scaling or translation invariance) left open the challenge of a unified $L^p$ stability theory without any a priori $L^1$ or $L^p$ bounds on two-dimensional domains. We establish a unified $L^p$ stability theory on three typical two-dimensional domains. These domains are: the half-plane, strips of any width, and domains satisfying a weak finite volume condition. For each domain, we prove that the penalized energy functional admits a minimizer for suitable $p$, and that every such minimizer satisfies the elliptic equation $\omega^{p-1} = \lambda(\psi - W x_2)_+$. Furthermore, we demonstrate that the set of minimizers is orbitally stable under the Eulerian dynamics. The absence of spatial scaling and horizontal translation invariance necessitates novel strategies: on the strip, we refine a concentration-compactness argument to prove strict subadditivity; on weak finite volume domains, we bypass the need for subadditivity by exploiting the inherent decay rate $q$ of the domain to enforce compactness. This work synthesizes the approaches of \cite{abe2022stability}, \cite{Abe2025StabilityOL}, \cite{abe2025existence}, and \cite{Dong2026StabilityOV} into a comprehensive framework, significantly expanding the scope of provably stable vortex structures.

For the fractional powers of operators, the classical formulation by V. Balakrishnan inherently requires positivity or specific sectorial conditions of the generator (i.e., the generation of analytic semigroups). To overcome these limitations, this paper presents a novel approach to constructing fractional powers of operators $D^k$ through the $\delta$-regularized logarithmic representation of infinitesimal generators $D$, based on the framework of $C^0$-semigroup theory for abstract evolution equations in Banach spaces. The proposed method bypasses the conventional geometric constraints by employing an algebraic $\delta$-regularization for bounded operators. Specifically, by applying a complex-analytic logarithmic representation to a family of bounded operators derived from the resolvent of the semigroup, we mathematically prove that, under the minimal algebraic assumption of invertibility, the fractional powers for generators of more general $C^0$-semigroups are uniquely and rigorously well-defined, independent of the choice of the regularization parameter $\delta$. The theoretical framework established in this study offers extensive potential for applications, including regularity estimates for non-analytic semigroups and non-autonomous systems where the infinitesimal generators depend explicitly on the time variable.

We consider the nonlinear Schrödinger equation in one dimension with nonlinearities of type $|u|^\alpha u$ for any power $\alpha>0$ and review two different methods for obtaining solutions, namely, local well-posedness, with initial data either in $L^2$ or $H^1$, or in the weighted subspace of $H^1$. One approach is based on the Strichartz estimates, and thus, $H^1$ well-posedness typically holds for nonlinearities with power $\alpha \geq 1$. The other one is a direct application of weighted estimates commuting with derivatives and a certain infimum condition on the initial data, and thus, can treat nonlinearities for the whole range $0 < \alpha < \infty$; furthermore, it can handle a sum of different nonlinearities. We then conclude with an application of the second approach to the NLS with finitely many combined nonlinearities, important for physical applications (e.g., in laser optics), as it is more challenging, if at all possible, to obtain local well-posedness with the first method due to the lack of scaling invariance.

In this article, we prove the sharp form of the fractional isoperimetric inequality in the plane, originally posed by Maz'ya [Problem 1, Integral Equations Operator Theory, 2018]. More precisely, we show that, for any given $s\in(0,1)$ and any bounded domain $\Omega \subset \mathbb{R}^2$ with $C^1$ boundary, $$ P_s(\Omega) \le \frac{\pi^{s-\frac{1}{2}}\Gamma\left(\frac{3-s}{2}\right)} {s(1-s)\Gamma\left(\frac{4-s}{2}\right)}\left[\mathcal{H}^1(\partial\Omega)\right]^{2-s}, $$ where $P_s$ denotes the fractional $s$-perimeter, $\Gamma$ denotes the Gamma function, and $\mathcal{H}^{1}$ denotes the $1$-dimensional Hausdorff measure on $\mathbb{R}^2$. The constant is sharp and equality is attained by the disk. The proof proceeds in three steps: reducing the fractional perimeter to a chord functional, reducing connected domains to convex bodies, and proving the sharp convex chord inequality via a fractional Willmore-type inequality, a variational formula along outer parallel bodies, and asymptotic analysis.

Let $N\in\mathbb N$ and $\gamma\in[-1,0)$. In [Anal. PDE 17 (2024)], Brezis, Seeger, Van~Schaftingen, and Yung asked how, in the exceptional range $\gamma\in[-1,0)$, $\dot{\mathrm{BV}}(\gamma)$ and $\dot W^{1,1}(\gamma)$ on ${\mathbb R}^N$ are related to other function spaces, especially to Hardy–Sobolev spaces, and whether these spaces are normable. In this article, we prove that the homogeneous Hardy–Sobolev space is strictly embedded, respectively, into $\dot{\mathrm{BV}}(\gamma)$ and $\dot W^{1,1}(\gamma)$, and neither $\dot{W}^{1,1}(\gamma)/\mathbb R$ nor $\dot{BV}(\gamma)/\mathbb R$ is normable, which answers the above two questions.

Brezis, Merle, and Riviére [Arch. Rational Mech. Anal. 1994] proved that, if a smooth solution $u:\mathbb{R}^2\to\mathbb{C}$ of the planar entire Ginzburg–Landau equation $$ -\Delta u = u(1-|u|^2)\quad \text{in}\quad \mathbb{R}^2 $$ satisfies the finite potential energy estimate that $$ \int_{\mathbb{R}^2}\left[1-|u(x)|^2\right]^2\,dx < \infty, $$ then it has the asymptotic property that $$ |u(x)|\to 1\quad \text{as}\quad |x|\to\infty. $$ The converse problem whether the asymptotic property implies the finite potential energy estimate was originally posed in their work and later formulated by Brezis as Open Problem 2.5 in [Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 2023]. In this article, we give an affirmative answer to this question. Our proof relies on the Kelvin inversion, a Morrey-type energy decay for the translation Jacobi system, the $L^4$-integrability of the phase form, and a linearised amplitude equation; all these tools together imply the $L^2$-integrability of the function $1-|u|$ on an exterior domain, without any a priori integrability assumption.

In this paper we consider nonlinear biharmonic equations with $p$-Laplacian ($p\ge2$) of the form $$ \left\{ \begin{array}{l} \Delta^2 u - \Delta_p u + V (x) u = f (x, u) \text{,} u \in H^2 (\mathbb{R}^N) \text{,} \end{array} \right. $$ where the potential $V(x)$ may be indefinite. Using local linking and Morse theory, nontrivial solutions are obtained. In case the nonlinearity $f(x,\cdot)$ is odd, we obtain a sequence of large energy solutions. In the second part of the paper, for bounded positive potential, we get multiple solutions for the case that $$f(x,u)=\lambda g (x) | u |^{q - 2} u + | u |^{m - 2} u$$ with exponent $m$ critical or subcritical.

We study a kinetic–fluid system in which a Vlasov or Vlasov–Fokker–Planck equation is coupled to the compressible Navier–Stokes equations with density-dependent viscosities through a locally averaged Brinkman force. The averaging is chosen in a conservative form so that the coupled system preserves the total momentum and satisfies a natural energy-dissipation balance, while avoiding the pointwise evaluation of a possibly rough fluid velocity. The first main result is a conditional exponential relaxation estimate for sufficiently regular solutions. The estimate is proved under positive and negative density moment bounds and a Muckenhoupt $\calA_2$ condition on a power of the density, which replace uniform pointwise upper and lower bounds. The proof combines a modulated energy and hypocoercivity analysis with a compensating functional for the density fluctuation. A key ingredient is a weighted Calderón–Zygmund estimate for the associated elliptic corrector, which allows us to control the viscous contribution under the $\calA_2$ condition. The second main result concerns global finite-energy Bresch–Desjardins entropy weak solutions in admissible low-dimensional regimes. Using the additional entropy structure associated with the density-dependent viscosities, we verify the density assumptions required by the conditional relaxation theorem and obtain exponential relaxation for the weak solutions. In the diffusionless case, the particle distribution aligns toward a mono-kinetic state, while in the diffusive case the relaxation is towards a Maxwellian equilibrium.

We establish a sharp rigidity/flexibility threshold for stationary solutions of the Krieger–Strain equation, an isotropic model of the Landau–Coulomb equation. For every $1< p < \frac65$, we use Nash iteration to construct nontrivial, nonnegative solutions in $L^1(\mathbb{R}^3) \cap L^p(\mathbb{R}^3)$ with arbitrarily strong exponential localization. Conversely, every stationary weak solution in $L^{\frac65}(\mathbb{R}^3)$ is trivial, identifying $L^{\frac65}(\mathbb{R}^3)$ as a new sharp integrability threshold. To our knowledge, this is the first use of Nash iteration for a nonlinear equation from collisional kinetic theory. The construction is based on a high–high–low cancellation within the Krieger–Strain operator and suggests that such mechanisms may occur more broadly in kinetic theory. The construction must accommodate kinetic features unusual for the method including a strongly nonlocal collision operator; an equation fundamentally posed on the whole space—not the periodic box; and a positive scalar unknown. At this low level of regularity, the usual formulations of the collision operator are not a priori well-defined, so a central part of the problem is specifying in what sense the constructed objects solve the equation. We isolate the notion of mollifier confluence, a simple and canonical way to interpret a nonlinearity below naive thresholds related to multiplying distributions. We complement this definition with a systematic treatment of weak solution notions and several explicit formal computations and clarifying examples that may be of independent interest.

We construct regularized distance functions for star-shaped domains in space forms $\mathbb N^n(K)$ and derive explicit formulas for their Hessians in terms of the principal curvatures of the boundary. As an application, we use these regularized distance functions to construct barriers and prove the existence of an admissible $C^{1,1}$ solution to the degenerate $\sigma_k$ equation on ring domains in $\mathbb R^n$ and $\mathbb H^n$, under suitable conditions on the boundary components.

In this article, we study formally determined inverse problems for wave equations in the presence of a variable sound speed. We prove that by measuring the boundary data of finitely many plane waves and their complementary solutions, one can uniquely recover the unknown coefficients of the highest order terms of a second order hyperbolic operator with time independent coefficients. This improves earlier rigidity results in [10], [11] which compared the wave operator generated by a Riemannian metric with the wave operator generated by the Euclidean metric. We compare two general second order hyperbolic operators with time independent coefficients, with the same lower order terms. However, we require the geometry associated with one of the operators to satisfy a pseudoconvexity condition, a no-caustics condition, and a spanning condition. In particular one of the operators could be a wave operator with the sound speed close to a constant and the other operator could be arbitrary. To prove the results, we introduce the notion of a complementary solution for a generalized plane wave solution generated by an incoming plane wave. The complementary solution extends smoothly, across an interface, the generalized plane wave. The unknown coefficients appear in a transport equation at the interface. We show that the unknown coefficients in the interior can be extracted from this transport equation, from the boundary data, via a sequence of Carleman estimates for the wave operator.

We prove the existence of a genuinely time-dependent Brakke flow starting from $\Gamma_0 \subset \mathbb{R}^{n+1}$ whose associated multiplicity-one varifold is stationary, provided that, at some singular point, the scale-invariant $L^2$ distance of $\Gamma_0$ from an $n$-dimensional plane has sufficiently small limsup as the scale tends to zero. This yields the dynamical instability of $\Gamma_0$, a notion recently introduced by Stuvard and Tonegawa, and hence the non-uniqueness of Brakke flows starting from $\Gamma_0$. A notable feature of our result is that it holds without assuming the uniqueness of tangent cones at the singular point.

We prove that for every integer $N\geq 3$ and $\alpha\geq \frac{1}{2}$, Beckner's inequality \[ \frac{\alpha}{2}\int_{\mathbb{S}^N}u(P_{N}u) dw+(N-1)!\int_{\mathbb{S}^N}u dw-\frac{(N-1)!}{N}\log\int_{\mathbb{S}^N}e^{Nu} dw\geq 0 \] holds for every $u\in H^{\frac{N}{2}}(\mathbb{S}^N)$ whose center of mass is at the origin. The proof is mainly based on an integral representation formula and a rigidity theorem for stable critical points. Hence, we answer the generalized Chang-Yang conjecture positively for every integer $N\geq 3$.

We study the fundamental gap of the Dirichlet $p$-Laplacian on bounded convex domains. We first prove the sharp one-dimensional estimate $\Gamma_p(I_D,V)\ge (p-1)(2^p-1)(\pi_p/D)^p$, $I_D=(-D/2,D/2)$, for all $1<p<\infty$ and convex potentials $V$, with equality exactly for constant $V$. An analysis of collapsing convex domains shows that $p=2$ is exceptional: for $p\ne2$, the higher-dimensional gap is not governed by the sharp one-dimensional constant. For $V\equiv0$, define $G_{p,N}:=\inf_{\Omega\subset\mathbb R^N}\operatorname{diam}(\Omega)^p\Gamma_p(\Omega,0)$ over bounded convex domains. We prove $G_{2,N}=3\pi^2$ and $G_{p,N}=0$ for $1<p<2$, $N\ge2$, while for every $p>2$, $G_{p,N}>0$ and the infimum is attained. We obtain explicit dimension-dependent bounds and $G_{p,N}\asymp_p N^{p-2}$ as $N\to\infty$, while for fixed $N$, $\lim_{p\downarrow2}G_{p,N}=3\pi^2$ and $\lim_{p\to\infty}G_{p,N}^{1/p}=4$. We also characterize the collapse as $p\downarrow2$: minimizing domains degenerate, and every limiting normalized ground-state measure is supported on a unit segment and, up to rigid motions, equals $2\cos^2(\pi t)\,dt$ on $(-1/2,1/2)$. The proof establishes log-concavity of the first $p$-eigenfunction with convex potential, a degenerate weighted Poincaré inequality for the ground-state measure, and a lower gap bound in terms of the first eigenvalue and the associated weighted Poincaré constant. These estimates also give the compactness and equality analysis for the collapsing minimizers.

We perform a linearized stability analysis of the Vlasov–Fokker–Planck equation obtained as the mean-field description of a stochastic spontaneous aggregation model with memory. Memory is represented by a chain of $K$ internal variables. We characterize the spatially homogeneous equilibria and derive a scalar dispersion relation for spatially inhomogeneous perturbations. When all relaxation rates are equal, we show that every Fourier mode that is unstable in the memoryless model possesses a unique critical relaxation parameter: sufficiently long memory stabilizes the mode, whereas it remains unstable for short memory. Although memory may destabilize individual modes associated with negative Fourier coefficients of the sensing kernel, we prove that, for radially symmetric distance-decreasing kernels, it cannot destabilize a homogeneous equilibrium that is stable in the memoryless model. Thus, memory cannot create an overall instability of an otherwise stable homogeneous state, although it may change the set of the unstable modes. We present a numerical example for the normalized top-hat kernel, demonstrating that increasing the effective memory length successively stabilizes the Fourier modes, with higher frequencies being stabilized before lower ones. This is consistent with the coarsening effect observed in recent particle simulations. Finally, under suitable nondegeneracy and regularity assumptions, we use the Crandall–Rabinowitz theorem to obtain branches of spatially inhomogeneous stationary solutions bifurcating from the homogeneous state.

We study the stochastic resistive Hall–magnetohydrodynamic (Hall–MHD) system on bounded convex polyhedral domains, subject to nonlinear perfectly conducting boundary conditions. The system is driven by multiplicative Gaussian forcing in the momentum equation together with structured curl-type noise in the induction equation arising from stochastic perturbations of the resistive and Hall parts in the generalised Ohm law. We develop a fully discrete, linearly implicit, structure-preserving mixed finite element method based on a compatible discrete de Rham complex, which preserves the magnetic Gauss law exactly. We prove subsequential convergence of the discrete solutions to a finite-energy weak martingale solution, thereby obtaining a constructive existence result for the stochastic Hall–MHD system with curl-type noise. Numerical experiments illustrate the stochastic dynamics, exact divergence preservation, and convergence of the method.