Papers

Revisiting the rationale provided by Picard in the seminal paper [13], we provide an independent, self-contained proof of the Maxwell compactness property for weak Lipschitz domains in the Euclidean setting, detouring technical complications like the differential forms setting, Gaffney's inequality for smooth domains, Calderon's extension theorem or regularity theory for PDEs, or systems thereof, on smooth domains that are used in this context. As a by-product we provide an independent proof of the classical Gaffney estimate for cubes using only $L^2$-completeness of the Fourier bases.

In this paper we prove the sharp weak estimates for the Riesz potentials $$\|I_{s}(f)\|_{L^{\frac{n}{n-s},\infty}}\leq \gamma_{n,s} v_n^{\frac{n-s}{n}}\frac{\Gamma(s/2)\Gamma((n+2-s)/2)}{\Gamma(n/2)}\|f\|_{L^1},~~ \text{when}~~0<s<\min\{n,2\}$$ and $$ \|I_sf\|_{L^{\frac{n}{n-s},\infty}} \leq \gamma_{n,s}v_n^{\frac{n-s}{n}}\|f\|_{L^1}, ~~\text{when}~~2\leq s<n,$$ where $\gamma_{n,s}=2^{-s}\pi^{-\frac{n}{2}}\frac{\Gamma(\frac{n-s}{2})}{\Gamma(\frac{s}{2})}$ and $v_n$ is the volume of the unit ball. The isoperimetric inequality for the Riesz capacity and the Newtonian isocapacitary inequality play a crucial role in our approach.

We characterise the spectrum of the Neumann-Poincaré operator acting on a mean-zero subspace of the Sobolev-Slobodeckij space $W^{s,p}(\Gamma)$, where $\Gamma$ is a non-Lipschitz domain, consisting of an infinite self-similar chain of disjoint smooth domains accumulating at a limit point. We exploit the discrete geometric scaling of the chain to obtain an exact block-Toeplitz representation of the Neumann-Poincaré operator, as well as the single- and double-layer operators. The corresponding operator-valued symbol depends entirely on the single scaling parameter $\alpha=(d-1)/p-s$ and belongs to the Wiener algebra whenever $0<\alpha<d$. This allows us to exploit block-Toeplitz operator theory to characterise the essential spectrum and Fredholm regions within this regime. In the energy space $H^{-1/2}_0(\Gamma)$, corresponding to $\alpha = d/2$, we prove the spectrum is real and characterise any isolated eigenvalues outside the essential spectrum through an operator-valued Wiener-Hopf factorisation. Furthermore, we apply an operator-valued Szegő limit theorem to derive the asymptotic spectral distribution for large finite truncations and establish an eigenvalue counting formula based on the operator symbol. We illustrate these results through explicit analytical computations for a chain of concentric annuli and numerical approximations for a chain of disks.

In this article, we present a new pointwise estimate for a weighted rough singular integral operator in the setting of stratified Lie groups. This operator, $T_{\Omega, \varpi}$, is based on a kernel $\Omega$ and a weight $\varpi$, where the kernel satisfies a natural size condition and a cancellation property with respect to the weight $\varpi$. Moreover, we do not assume any kind of regularity on these objects. This weighted rough singular integral operator, applied to a function $f$, is estimated through a combination of information involving a weighted maximal function of the gradient of $f$ and a weighted Morrey space. We also deduce from this pointwise estimate some new weighted functional inequalities and, as an application, we obtain a uniqueness result for a rough version of the stationary Navier-Stokes equation over the Heisenberg group.

In this paper, we consider the anisotropic Allen-Cahn equation $-\operatorname{div} a(Du)+W'(u)=0$ in $\mathbb{R}^N$, where $a(p):=DH(p)$ with $H(p)=\frac{1}{2}F(p)^2$ and $F$ a uniformly elliptic integrand, and $W(u)=\frac{1}{4}(1-u^2)^2$. Based on the Mooney-Yang anisotropic minimal graph, we prove that the anisotropic Allen-Cahn equation admits a stable solution for $N\geq4$ in the weak sense, whose level sets are not hyperplanes. As a byproduct, we also construct a smooth solution of the above anisotropic Allen-Cahn equation for $N\geq5$ that is monotone in one direction but is not one-dimensional.

The sufficient conditions of Ladyzhenskaya & Ural'tseva \cite{lu70} for interior gradient estimates in nonuniformly elliptic problems are found to be also necessary. Specifically, nonparametric minimal surfaces exhibit the maximal nonuniformity rate compatible with the local Lipschitz regularity of solutions, and the classical theory of Finn \cite{fin54} and Bombieri & De Giorgi & Miranda \cite{bdm69} doesn't extend beyond Bernstein genre $2$.

This work provides a traffic flow interpretation of the model recently introduced in \cite{ABS2025}. We adapt the conservation law with discontinuous gradient-dependent flux to a macroscopic traffic setting, assuming concave flux functions $f(u)$ and $g(u)$ defined on the density domain $[0,u_{\max}]$. To prevent violations of the maximal density constraint, we introduce modified Riemann Solvers tailored to this setting and generalize the existence theory for weak solutions developed in \cite{ABS2025}. Finally, we construct Riemann Solvers for a broader two-phase model incorporating a free-flow regime, in which $f(u)=g(u)$ on a subset of $[0,u_{\max}]$, and establish existence of weak solutions in this more general framework.

For a nonconstant planar $p$-harmonic function $u$, with $1 < p < \infty$, and according to previous results, most notably Aronsson's fundamental analysis of the hodograph representation in a neighborhood of an isolated critical point, we introduce the flux-normalized frequency $$ N_*(x_0,r) = \frac{r\displaystyle\int_{B_r(x_0)} |D u|^p\,dx} {\displaystyle\int_{\partial B_r(x_0)} |D u|^{p-2}(u-u(x_0))^2\,dS}. $$ This ratio constitutes the natural counterpart of Almgren's frequency function in the nonlinear setting, as it recovers precisely the degree of homogeneity for every homogeneous $p$-harmonic profile. We further establish the corresponding gauge-corrected version of Almgren-type monotonicity and reinterpret, in terms of the frequency function, the classical planar unique continuation property. In addition, we derive a quantitative pinching estimate for the flux-normalized frequency and identify the specific obstruction that prevents a straightforward extension of the method to higher dimensions.

We consider a Hele–Shaw-type tumor growth model obtained from the incompressible limit of a porous-medium equation. The limiting model preserves the patch structure of the tumor density and therefore admits a single free-boundary formulation. We investigate how a pressure-feedback parameter associated with the homeostatic pressure affects boundary stability. Using asymptotic analysis, we study boundary perturbations of finite-radius tumors and planar traveling fronts and derive the corresponding stability criteria. For planar fronts, the stability thresholds are further shown to be bifurcation points from which nonsymmetric traveling waves emerge. Our results indicate that pressure feedback can induce boundary instability in parameter regimes where the model without pressure feedback remains stable.

In a recent breakthrough, Bamler-Lai proved the mean-convex neighborhood conjecture in all dimensions by showing that any nontrivial ancient asymptotically cylindrical mean curvature flow is – up to splitting Euclidean factors – either a translating bowl, or an ancient oval, or a translating oval-bowl. In this paper, we provide a short alternative argument for two of the three scenarios. Specifically, using more elementary/traditional methods, we show that if the convergence to the round cylinder is fast then the solution is a bowl times a Euclidean factor, and if the convergence is slow then the solution is an ancient oval. Moreover, the present paper also yields a new proof of the mean-convex neighborhood conjecture for neck-singularities that substantially simplifies and streamlines the approach from our prior work joint with Hershkovits (Acta '22) and Hershkovits-White (Inventiones '22).

The purpose of this work is to provide a broader framework for analyzing the inhomogeneous nonlinear Schrödinger equation (INLS) \[iu_t + u_{xx} \pm |x|^{-b}|u|^{\alpha-1}u=0, \quad 1<\alpha< 5-2b\; \text{and}\; 0< b\leq 1/4.\] Specifically, we establish low-regularity well-posedness in the Fourier-Lebesgue $\widehat{L}^{p}$ spaces for $4/3<p<8$. The analysis is carried out differently for the cases $p<2$ and $p>2$. Primarily, in both cases, we prove global well-posedness for arbitrarily large initial data via the data decomposition method adapted for the Fourier-Lebesgue spaces. Furthermore, we obtain analogous results for INLS in modulation spaces $M^{p,p'}$ for $4/3<p<2$.

We study heat semigroups on bounded Lipschitz domains $\Omega \subset \mathbb{R}^{d}$ with dimension $d>2$ under non-local Robin boundary conditions. The boundary operator $B \in \mathcal{L}( \mathrm{L}^{2}(\partial\Omega))$ is allowed to destroy the positivity of the semigroup. We assume that there exists a positive operator $C \in \mathcal{L}(\mathrm{L}^{2}(\partial\Omega))$ such that $$ |Bu| \leq C|u| \quad\text{for every }u\in \mathrm{L}^{2}(\partial\Omega), \qquad C{\bf 1}\in \mathrm{L}^{\infty}(\partial\Omega). $$ Under this assumption, we prove that the semigroup generated by the corresponding uniformly elliptic operator is ultracontractive. More precisely, its norm from $\mathrm{L}^{2}(\Omega)$ to $\mathrm{L}^{\infty}(\Omega)$ has short-time order $t^{-d/4}$. The main step is the construction of a positive eigenfunction $\phi$ of a dominating elliptic operator such that $$ 0 \ < \ \delta \ \leq \ \phi \ \leq \ M $$ almost everywhere in $\Omega$. The upper bound is obtained by power truncations and an iteration of Sobolev exponents. The lower bound follows by comparison with the Neumann semigroup. The two-sided estimate gives an $\mathrm{L}^{\infty}$-bound for the comparison semigroup. Nash's inequality and duality then yield ultracontractivity.

We prove that a point atom in an endpoint kinetic-energy measure of a three-dimensional incompressible Navier-Stokes flow on the flat torus forces quantitative concentration of the actual pressure. At each restart time tau, a same-state constraint-response comparison evolves u(tau) by the pressure-free componentwise advection-diffusion equation driven by u, and subtracts the resulting passive field z_tau. Writing r_tau = u - z_tau and g_tau = Q z_tau, we obtain an exact relative pressure-work identity. If the atom has mass m, Nash smoothing makes z_tau terminally non-atomic while r_tau retains terminal atomic mass at least m; hence, after every fixed restart, the terminal limsup of the accumulated pressure work is at least m/2. Mesoscopic localization yields scale-explicit lower bounds for gauge-invariant L2 pressure oscillation and for the local L2 mass of the pressure gradient in shrinking terminal cylinders. Consequently, on every terminal neighborhood of the atomic point, the pressure modulo functions of time and its gradient fail to be square integrable. The relative pressure-work principle extends to constrained solenoidal Oseen evolutions: as tau approaches T*, the zero-initial responses vanish in weak-tail topologies, although every fixed restart retains the same lower bound. This identifies a necessary pressure-work mechanism, generated by the incompressibility constraint, that links endpoint atoms to local pressure concentration, without requiring a full regularity hypothesis.

We study higher regularity and its relation to nonresonant behavior for time-periodic solutions of boundary value problems for one-dimensional linear and nonlinear nonautonomous first-order integro-differential strictly hyperbolic systems. The boundary conditions include integral operators and various types of boundary reflections. We prove that continuous and classical solutions have $C^k$-regularity, provided the coefficients are sufficiently smooth and a suitable number of nonresonance conditions is satisfied. In the linear case, these conditions involve the principal coefficients, the diagonal lower-order coefficients, and the boundary reflection coefficients. In the nonlinear case, they also depend on the nonlinearities and on the solution itself. For nonautonomous hyperbolic systems, higher regularity generally requires additional nonresonance conditions, whose number depends on the desired order of differentiability. These conditions are not only sufficient but, in general, also necessary, revealing a distinctive feature of nonautonomous hyperbolic PDEs. By contrast, in the autonomous case, a single nonresonance condition (if one is needed at all) suffices to obtain arbitrarily high regularity. We also identify a class of nonautonomous hyperbolic problems for which no nonresonance conditions are required. In this case, the higher regularity of solutions is determined solely by the regularity of the data. The main technical tool underlying the proofs is an abstract regularity principle formulated in the setting of vector spaces.

We consider the massive Dirac operator (with positive mass) in the plane with an attractive Lorentz-scalar $\delta$-shell interaction of strength $\tau\in(-\infty,0)\setminus\{-2\}$ supported on a $C^\infty$-smooth curve $\Sigma\subset\mathbb{R}^2$ being a local deformation of the broken line. This singular interaction is defined by imposing a suitable transmission condition on the curve $\Sigma$ in the operator domain. Such a Dirac operator is self-adjoint and has a gap in the essential spectrum, whose size is explicit and depends on the mass and the interaction strength. We show that the number of discrete eigenvalues in the gap is finite. Under the assumption that one of the domains bounded by $\Sigma$ is convex, we prove that the corresponding Dirac operator has a non-empty discrete spectrum, provided that $\tau$ is either sufficiently small or sufficiently large in absolute value. The result holds for any perturbation of a broken line of any opening angle as described above, and this discrete spectrum is induced by the geometry, since for the same type of a singular interaction supported on the straight line the discrete spectrum is empty.

In a remarkable work published in Analysis and PDE 8 (2015) 1541-1563, M. Fazly, J. Wei, and X. Xu establish the following pointwise estimate \[-\Delta u \geq \frac 2{n-4} \frac{|\nabla u|^2}u + \sqrt{\frac 2{p+1- \frac 8{n(n-4)}}} |x|^{\frac \sigma 2} u^\frac{p+1}2 \quad \text{in } \mathbf R^n\] for any bounded, positive, $C^4$-solution $u$ to the the fourth-order Hénon equation \[\Delta^2 u = |x|^\sigma u^p \quad \text{in } \mathbf R^n\] with $n \geq 5$, $\sigma \geq 0 $ but implicitly near $0$, and $p>(n+4+2\sigma)/(n-4)$. Their argument relies on a sophisticated iteration argument in the fashion of the standard Moser proof. In this paper, we provide an alternative approach which relies on the Bernstein technique via the maximum principle. The heart of our argument is a suitable choice of an auxiliary function allowing us not only achieving the same pointwise estimate, which still holds even for $p = (n+4+2\sigma)/(n-4)$, but also relaxing the boundedness of solutions. In addition, the pointwise inequality actually holds for any $\sigma \geq 0$. This estimate leads to an interesting consequence of the existence of conformal metric of $\mathbf R^n$ having positive scalar and $Q$ curvatures. We also show that the coefficient $2/(n-4)$ is sharp, hence providing the first answer to the question in the op. cit. paper. Our proof appears to be simpler and constructive.

We establish sharp endpoint mapping properties for the wave operators $W_\pm(H,-\Delta)$ of two-dimensional Schrödinger operators $H=-\Delta+V$ with real-valued decaying potentials $V$. Together with the known non-endpoint $L^p$ theory, our results give a complete classification of the $L^p$ mapping properties of the two-dimensional wave operators, and reveal an unexpected reversal of the usual threshold paradigm at the endpoints $p=1$ and $p=\infty$. When zero is a regular point of $H$, the wave operators fail to be bounded on $L^1(\mathbb{R}^2)$ and on $L^\infty(\mathbb{R}^2)$, but they satisfy the atural substitute estimates of Calderón–Zygmund type: $$ L^1(\mathbb{R}^2)\longrightarrow L^{1,\infty}(\mathbb{R}^2),\ \ \ \mathcal{H}^1(\mathbb{R}^2)\longrightarrow L^1(\mathbb{R}^2),\ \ \ L^\infty(\mathbb{R}^2)\longrightarrow \mathrm{BMO}(\mathbb{R}^2). $$ When zero is instead a threshold singularity of the first kind—an s-wave resonance with no other threshold obstruction, the wave operators are bounded on both endpoint spaces $L^1(\mathbb{R}^2)$ and $L^\infty(\mathbb{R}^2)$. Thus, in dimension two, an s-wave resonance improves the endpoint behavior of the wave operators, in sharp contrast with dimensions $n\ge3$, where the only regular case is the favorable one. We also determine the endpoint behavior in the remaining zero-energy spectral configurations of $H$. A p-wave resonance obstructs both the $L^1$- and the $L^\infty$-boundedness of the wave operators, while in the zero-eigenvalue case we obtain necessary and sufficient conditions for endpoint boundedness, expressed in terms of the presence of s- and p-wave resonances and of explicit second-order harmonic moment cancellations satisfied by the zero-energy eigenfunctions.

For any $\gamma \in \left[ {0,\frac{1}{3}} \right)$, we construct $\gamma$-Hölder-continuous martingale solutions to the two-dimensional stochastic incompressible Euler equations. These martingale solutions exhibit dissipative behavior and intermittency, thereby establishing the flexible part of the intermittent Onsager theorem. The proof rests on a stochastic and intermittent variant of the Newton–Nash iteration combined with the Littlewood–Paley decomposition scheme. This composite iteration scheme incorporates new stochastic pressure, Reynolds stress and intermittent perturbations to formalize the stochastic and intermittent fluctuations.

We study non-negative solutions of the indefinite sublinear Dirichlet problem $$ \left \{\ \begin{array}{ll} -\Delta u=\lambda u+m(x)|u|^{\alpha -1}u&\quad\hbox{ in $\Omega$},\\ u=0&\quad \hbox{ on $\partial \Omega$}, \end{array} \right . $$ where $\Omega \subset \mathbb{R}^{N}$ is a smooth bounded domain, $0<\alpha <1$, $m\in L^{\infty }(\Omega )$ is a sign-changing or non-positive weight, and $\lambda \in \mathbb{R}$.

Starting from localized energy estimates, we prove finite speed of propagation for kinetic solutions to stochastic porous media equations with nonlinear conservative noise, the existence and uniqueness of which has recently been established. In particular, we propose a novel iteration technique which allows us to obtain a Stampacchia-type inequality involving one single integral quantity, despite the possibly different scaling behaviors of the porous media and the noise term. This allows us to apply stochastic filtering arguments developed for the case of linear source-type noise. Using related ideas, we identify flatness conditions on initial data which guarantee locally the occurrence of a waiting time phenomenon, i.e., the onset of forward propagation of the solution's support is locally delayed. The condition for the latter matches the one for the deterministic porous media equation up to a logarithmic correction in the case of critical nonlinearity in the noise, but it requires more and more flatness of the initial data as the nonlinearity tends towards linear conservative noise. This is in line with the expected behavior: In the case of linear conservative noise, instantaneous forward motion is possible due to the effects of stochastic transport, no matter how flat the initial profile is.

This paper concerns the grow-up rate of solutions to a semilinear heat equation in the unit ball with the exponential nonlinearity and an inhomogeneous term $f$. When $f=0$, it is known that the large-time behavior of a solution changes at the critical dimension $N=10$, and the grow-up phenomenon occurs for the case $N\ge 10$. For the inhomogeneous case with $N\ge 10$, the present author and a coauthor showed in [12] that the grow-up phenomenon disappears once $f$ exceeds a threshold. In this paper, we provide a quantitative characterization of the disappearance of the grow-up phenomenon by obtaining the grow-up rate in the critical dimension. Our result shows that a qualitatively different type of grow-up behavior occurs in the critical dimension compared with the case $N\ge 11$ studied in [12]. The difference is caused by a change in the outer behavior of the solution. Even in the case $f=0$, our result is new in that it provides a rigorous justification of the formal computation by Galaktionov and King [10].

Starting from a three-dimensional model of linearized magnetoelasticity, we investigate the asymptotic behavior of a thin shell as its thickness approaches zero. Our analysis focuses on a family of linearly elastic elliptic membrane shells. Using $\Gamma$-convergence, we rigorously derive a two-dimensional linear magnetoelastic shell model and establish its relationship to the original three-dimensional model in terms of the convergence of minimizers.

We consider the barotropic Euler equations in dimension $d > 1$ with decaying density at spatial infinity. The phase portrait of the nonlinear ODE governing the equation for spherically symmetric self-similar solutions has been introduced in the pioneering work of Guderley. The existence of smooth decaying solutions through the sonic line has been obtained in the work (F. Merle et al., 2022) in a suitable range of parameters. We work in this paper in a different range of parameters and show the existence of holosphere like solutions which are smooth through the sonic line, but display vanishing density at a sphere.

Let $\lambda(u,x)$ be the local Arnold multiplicity of a quasi-plurisubharmonic function $u$. Di Nezza–Guedj–Lu asked whether every maximal weak solution $\varphi_t$ of the twisted Kähler–Ricci flow satisfies $\lambda(\varphi_t,x)=\max\{\lambda(\varphi_0,x)-t,0\}$. We give counterexamples on the Hirzebruch surface $\mathbb F_e=\mathbb P_{\mathbb P^1} (\mathcal O_{\mathbb P^1}\oplus\mathcal O_{\mathbb P^1}(-e))$, $e\ge2$. Let $S$ be its negative section and $F_1,\ldots,F_k$ be distinct fibres. If $a,b_i>0$, $\sum_i b_i>ea$, and the initial current is $a[S]+\sum_i b_i[F_i]$, then $\lambda(\varphi_t,x)=a-\min\{k/e,1\}t$ for $x\in S\setminus\bigcup_iF_i$ and $0<t<\min\{a,b_1,\ldots,b_k\}$. Thus the formula fails for $k<e$; the slower decay is forced by the Zariski negative part of the residual class. Under explicit SNC and positivity hypotheses, we also prove an exact surface formula for pure divisorial data. Consequences include nonlocality with fixed background data and explicit multiplier ideals. Products with smooth factors give counterexamples in every complex dimension $n\ge2$.

We prove the strong unique continuation property for the fractional Schrödinger equation \[ (-\Delta)^s u=Vu \] with scaling-critical potentials $ V\in L^{\frac{n}{2s}}_{\mathrm{loc}}$ for all $0<s<1$. Our result constitutes the nonlocal counterpart to the classical Jerison–Kenig result for Schrödinger operators.

Berestycki and Graham proved large-dilation uniqueness for bounded positive solutions of \[ -\Delta u=f(u)\quad\hbox{in }\kappa\Omega, \qquad u+\alpha\partial_\nu u=0\quad\hbox{on }\partial(\kappa\Omega),\] when $\alpha$ is fixed, and remarked that the dilation threshold should not depend on $\alpha$. We show that it does not, including at the Dirichlet and Neumann endpoints, for possibly unbounded uniformly $C^{2,\gamma}$ domains. The half-space linearizations have a common positive spectral gap over the compactified boundary parameter. The Dirichlet end requires a separate compactness argument because the Robin coefficient diverges there. After rescaling by its inverse, the equation has a harmonic half-space limit. A Liouville lemma rules out a nonzero limiting trace, and the resulting endpoint compactness, together with the half-space gap and localization, yields the uniform uniqueness statement. When $\partial\Omega\neq\varnothing$, we further obtain convergence of the spectral bottom to its half-space value with error $O(\kappa^{-1/2})$. For bounded $C^{4,\gamma}$ domains the boundary layer also has a first mean-curvature correction, with remainder $O(\kappa^{-1-\gamma}+\kappa^{-2})$ on each fixed boundary strip.

We establish local-in-time dispersive and sharp Strichartz estimates for the linear wave equation with Neumann boundary conditions inside a cylindrical convex domain $\Omega \subset \mathbb{R}^3$ featuring a non-empty smooth boundary $\partial\Omega$. The cylindrical geometry dictates that the nonnegative radius of curvature vanishes along the axial direction. By utilizing an explicit spectral analysis based on the zeros of the Airy function derivative, alongside a tailored Airy–Poisson summation formula, we capture the microlocal behavior of multi-reflected waves and caustics. Finally, we apply these sharp Strichartz estimates without loss of derivatives to prove local well-posedness and small-data global well-posedness for the energy-critical quintic nonlinear wave equation (NLW) subject to Neumann boundary conditions.

For periodic Schrödinger operators $H_V=-\Delta+V$ with bounded real-valued potentials on $\mathbb R^d$ with $d\ge2$, we show that for generic potentials, each endpoint of every spectral gap is attained by a single Bloch band at only finitely many quasimomenta, and has a nondegenerate Hessian at every attaining point. This proves the Spectral Edge Conjecture for periodic Schrödinger operators.

We prove that for every integer $n\geq 2$ and $8\pi(n-1)<\rho<8\pi n$, the singular Liouville equation $\Delta u+\e^u=\rho\delta_0$ on a rectangular torus $E_{\mathrm{i}b}=\mathbb{C}/(\mathbb Z+\mathrm{i} b\mathbb Z)$ has exactly $n$ solutions, which are all axisymmetric. Together with previous results by Chen-Lin and Lin-Wang, this yields that \begin{itemize} \item $E_{\mathrm{i} b}$ admits no spherical metrics with a conical singularity of angle $2\pi\vartheta$ as long as $\vartheta$ is a positive odd integer. \item For every integer $n\geq 1$, $E_{\mathrm{i} b}$ admits exactly $n$ spherical metrics with a conical singularity of angle $2\pi\vartheta$ for each $\vartheta\in (2n-1, 2n+1)$. \end{itemize} The basic idea is to prove that the linearized equation has only trivial solutions in the space of axisymmetric functions. The previous method of analysing nodal domains via Bol's isoperimetric inequality only works for $\rho\leq 8\pi$. We develop a unified approach for all $\rho\in (0,+\infty)\setminus 8\pi\mathbb{N}_{\geq 1}$ by exploring the deep connection with the monodromy of the classical Lamé equation.

We establish Łojasiewicz–Simon type gradient inequalities for the Yamabe functional on bounded smooth domains near configurations consisting of finitely many concentrating bubbles, with or without a regular component. A weighted decomposition separates the finite-dimensional spectral and bubble parameters from an infinite-dimensional coercive remainder. We obtain quantitative estimates for the parameters, the remainder, and the corresponding energy gaps, in both the regular-plus-bubbling and the pure-bubbling regimes. These inequalities quantify the deviation from such surfaces and are essential for studying the dynamics of related parabolic flows.