Papers

The analytic minimal model program proposed in \cite{SongTianSingularities, JST} seeks to describe the birational transitions and fibre collapsing of the Kähler–Ricci flow through the geometry of its singularities. A fundamental conjecture of \cite{JST} predicts Type I bounds for the scalar curvature and diameter of every fibre contratced by the limiting cohomological class at the finite singular time. This paper is a continuation of \cite{Splitting} that establishes the Type I conjecture for collapsing solutions on Fano bundles with arbitrary smooth Fano fibres. If the fibre admits a smooth Kähler-Einstein metric, the Type I bound holds for the full curvature tensor.

We study the master equation $ (\partial_t -\Delta)^{s} u(x,t) = R(|x|)f(u(x,t))\quad\mbox{in}\ \mathbb{R}^n\times\mathbb{R}, $ where $s\in(0,1)$, $R$ is positive and strictly decreasing in the radial variable, and $f$ is positive, locally Lipschitz, and satisfies $f'(0)<0$. Existing symmetry results for this operator require $f(0)=0$ with $f'(0)\ge 0$ (or, more generally, that $f$ be non-decreasing near the origin), because the sign of the moving-plane comparison inequality is then controlled by the nonlocal diffusion term. When $f'(0)<0$, the linear term in the comparison inequality appears at the same order as the nonlocal diffusion but with the opposite sign, destroying the comparison principle underlying the standard cut-off perturbation argument. We prove that, despite this obstruction, every positive bounded classical solution with uniform spatial decay is radially symmetric and strictly radially decreasing in $x$ for each $t$. The proof reconciles the direct method of moving planes with this adverse sign by exploiting the interplay between the spatial decay of $u$ and the strict radial monotonicity of $R$. This appears to be the first symmetry result for the master operator that accommodates a local damping mechanism at the origin.

We prove an $\varepsilon$-regularity theorem for $m$-varifolds with mean curvature in $L^p$, $p>m$, with respect to an anisotropic integrand satisfying a Michael-Simon inequality and the quadratic exposed condition: near sufficiently flat density-one points, such varifolds are representable as $C^{1,\alpha}$ graphs. Combined with the recent proof of the anisotropic Michael-Simon inequality, this establishes an anisotropic Allard regularity theorem in arbitrary codimension for a large class of anisotropic integrands, including those close to the area functional. We also exhibit the first examples of anisotropies satisfying both the uniform scalar atomic condition and the Michael-Simon inequality, that are not close to any ellipsoidal norm. These include the $\ell^q$ norms in every dimension and codimension, for explicit ranges of $q$, and a new class of axisymmetric anisotropies.

We prove that for each $3\le k\le n-1$ a convex $C^2$ entire solution of $\sigma_k$ equation in $\mathbb{R}^n$ with positive constant right hand side is a quadratic polynomial, without a growth assumption on the solution itself.

We establish sharp convergence rates for quadratically regularized optimal transport with quadratic cost in the regime of small regularization $\varepsilon$. In particular, we quantify the sparsity of the support of the regularized optimizer. For smooth marginal densities in $\mathbb{R}^d$, this support lies within a distance of order $\varepsilon^{1/(d+2)}$ from the Brenier graph, and every section of the support is sandwiched between balls with radii of that order. The geometry of the support is closely linked to the dual potentials. We show that the potentials satisfy uniform two-sided Hessian bounds and converge uniformly at rate $\varepsilon^{2/(d+2)}$, while their gradients converge at rate $\varepsilon^{1/(d+2)}$. The rates are sharp, and we further identify the regularity threshold where convergence breaks down.

Self-similar solutions to the porous medium equation with dominating spatially inhomogeneous absorption $$ \partial_tu=\Delta u^m-|x|^{\sigma}u^p, \quad (x,t)\in\real^N\times(0,\infty), \quad N\geq1, $$ with exponents $1<p<m$ and $\sigma>0$, are classified. Looking for solutions in the form $$ u(x,t)=t^{-\alpha}f(|x|t^{\beta}), \quad \alpha=\frac{\sigma+2}{\sigma(m-1)+2(p-1)}, \quad \beta=\frac{m-p}{\sigma(m-1)+2(p-1)}, $$ it is shown that all their profiles satisfy the behavior at infinity given by $$ \lim\limits_{\xi\to\infty}\xi^{\sigma/(p-1)}f(\xi)=\left(\frac{1}{p-1}\right)^{1/(p-1)}, $$ but the solutions strongly differ with respect to their behavior near the origin: there exist a unique solution with $f(0)>0$, $f'(0)=0$, another unique solution such that $f$ presents a dead-core; that is, $f\equiv0$ for $\xi\in[0,\xi_0]$ for some $\xi_0>0$, and, finally, there exists $K^*\in(0,\infty)$ such that, for any $K\in(0,K^*)$, there is at least a solution such that $$ \lim\limits_{\xi\to0}\xi^{-(\sigma+2)/(m-p)}f(\xi)=K. $$ The large time behavior of general solutions, making strong use of these three types of self-similar solutions, will be addressed in a companion work.

We study a variable-exponent double-phase Neumann problem on a domain in a metric measure space, where the boundary measure satisfies an upper codimension-\(\theta\) bound for \(0\le\theta<1\). Under a local comparability condition on the double-phase growth function, we establish a modular estimate for a fractional maximal operator and use it to prove the existence and boundedness of the associated trace operator. We then establish coercivity of the associated energy functional on the normalized double-phase Newtonian space and, under reflexivity, prove the existence of minimizers. We further show that the set of minimizers is closed and convex, and that any two minimizers have the same minimal weak upper gradient almost everywhere. Finally, we prove the stability of minimizers under perturbations of the Neumann data.

This paper establishes the existence of three distinct weak solutions for a class of nonhomogeneous quasilinear inclusion systems, which are governed by locally Lipschitz functionals within Orlicz-Sobolev spaces over unbounded domains $\mathbb{R}^{N}$.By employing a new three critical points theorem established by Wu-Zhou in [29], we extend the nonsmooth critical point theory to handle scenarios with nonlinear growth conditions.Our analysis imposes a precise structural condition on the nonlinear term $H$, requiring it to exhibit superlinear growth while maintaining subcritical asymptotic behavior. This condition establishes a delicate balance between growth restrictions and functional analytic requirements. The novelties of this work include the construction of suitable energy functionals that are consistent with both the structure of Orlicz-Sobolev spaces and the characteristics of nonlinear growth. Through rigorous variational analysis and careful estimation techniques, we establish the existence of three weak solutions under these generalized conditions, thereby significantly expanding the applicability of critical point methods in nonstandard function spaces.

We analyse the essential spectrum of ${\mathcal M} = {\operatorname{curl}} {\operatorname{curl}}$ acting on divergence-free vector fields in unbounded domains of $\mathbb{R}^3$. We show that $\sigma_e({\mathcal M}) = [0,+\infty)$ in quasi-conical domains and $\sigma_e({\mathcal M}) \neq \emptyset$ in quasi-cylindrical domains. For horn-shaped domains with circular cross-section and eventually mean-convex boundary, we establish that $\sigma_e({\mathcal M}) = \emptyset$, independently of their volume. For horns with annular cross-section, the transverse normal harmonic field determines an effective one-dimensional Schrödinger operator $H_V$ with $\sigma_e({\mathcal M}) \supseteq \sigma_e(H_V)$. Finally, for a concrete family of perforated exponential horns with a double-exponential hole, we show that depending on the rate of shrinking of the hole at infinity, either $\sigma_e({\mathcal M}) = \emptyset$, or $\sigma_e({\mathcal M}) = [\gamma^2, +\infty)$, or $\sigma_e({\mathcal M}) = [0,+\infty)$.

Viscosity solutions of eikonal equations need not enjoy regularity beyond Lipschitz continuity. In contrast, differentiable solutions exhibit stronger structural properties. In the Euclidean setting, building on observations of Caffarelli and Crandall (Comm. Partial Differential Equations $\textbf{35}$ (2010), 391–414), one finds that differentiable solutions of the eikonal equation $\| Df \| \equiv 1 \,\, \text{in } \mathcal{U} \subset \mathbb{R}^d$ are of class $C^{1, 1}_{\mathrm{loc}}(\mathcal U)$. Moreover, if $\mathcal{U} = \mathbb{R}^d$, then $f$ must be affine. This manuscript revisits these phenomena in the setting of (finite– and infinite–dimensional) Banach spaces. Our results show how the geometry of the underlying Banach space determines the regularity and affineness of Gateaux–differentiable solutions of eikonal equations. As a geometric application in finite dimensions, we characterize differentiable functions whose gradient ranges are constrained to boundaries of convex bodies.

We study the simultaneous identification of the Lamé parameters $\lambda$ and $\mu$ in three-dimensional static isotropic elasticity. For smooth coefficients, we establish uniqueness from the full Dirichlet-to-Neumann map when $\mu$ is axisymmetric, admits a multiplicative or additive separation on a product domain, or is quasianalytic in one fixed direction. These results require neither smallness of $\nabla\mu$ nor real analyticity of the coefficients, and impose no corresponding structural condition on $\lambda$. In the axisymmetric and separated cases, partial boundary data suffice to determine $\mu$.

Global existence and the large-time behavior of strong solutions to the one-dimensional planar compressible magnetohydrodynamic system in unbounded domains are investigated.The longitudinal viscosity is assumed to be a positive constant plus any nonnegative power of the density, or to depend on temperature through a power law. The heat conductivity is taken to be proportional to any nonnegative power of temperature, while the transverse viscosity and magnetic diffusivity are taken to be positive constants. Global existence, uniform-in-time estimates, and convergence to equilibrium are established without imposing any smallness conditions on the initial data. In the density-dependent case, any fixed nonnegative viscosity exponent is allowed; in the temperature-dependent case, the nonnegative viscosity exponent is required to be sufficiently small in terms of the initial data. An auxiliary elliptic equation is introduced to control magnetic-pressure oscillations through a square-root-in-time estimate. Combined with the estimates appropriate to each viscosity law, this yields uniform upper and positive lower bounds for the specific volume and temperature and asymptotic stability of the global strong solution.

In this paper, we use the Green's function method to study the stability of solutions to micropolar equations in the whole space. An essential difficulty in estimating the Green's function lies in treating a coupled system of equations with variable coefficients, which is completely different from previous Green's function analyses for scalar equations. We propose a dominant factor extraction method to overcome this difficulty. Based on sharp Green's function estimates for the micropolar system, we prove that if the initial data (m_0, \omega_0) satisfy \|(m_0, \omega_{0})\|_{L^1\cap L^{\infty}}\leq c_0\mu^{3/4} for some small constant c_0 independent of the viscosity coefficient \mu, then the corresponding solution remains of order O(\mu^{3/4}) under the effect of the Couette flow. Meanwhile, we establish the optimal decay estimates of solutions in the general L^p norm. More importantly, the dominant factor extraction method developed in this work provides a general framework for the stability analysis of complex coupled fluid systems.

We establish interior $C^{1,\alpha}$ regularity estimates, for some $\alpha > 0$, for fractional $p$-caloric functions when $p$ is in the range $p \in (2,2/(1-s))$. As a consequence, we deduce improved regularity in time, which yields interior continuous differentiability in time for $p \in [1/(1-s), 2/(1-s))$.

In this paper, we study the existence of nonnegative bound states for the higher-order logarithmic Schrödinger equation $$ -\alpha\Delta u+V(x)u-\beta u\ln|u| = \gamma u(\ln|u|)^m, \qquad u\in H^1(\mathbb R^N), $$ where $\alpha,\beta,\gamma>0$, $m$ is an odd positive integer and $V$ is a positive bounded potential converging to a constant at infinity. The simultaneous presence of the first- and higher-order logarithmic terms leads to a nonsmooth variational structure and additional compactness difficulties. We introduce a family of power-law approximations and derive estimates that are uniform as the approximation exponent tends to the logarithmic limit. Under a structural condition on the autonomous nonlinearity, we obtain uniqueness, up to translations, of the nonnegative autonomous profile with connected positivity set and a corresponding energy gap below the two-profile threshold. The small-amplitude behavior exhibits a maximum-principle/compact-support dichotomy: the case $m=1$ yields positivity, whereas the higher odd orders fall into the compact-support regime for the autonomous profile. We then establish a profile decomposition for constrained Palais-Smale sequences and construct a barycenter-based min-max level below the splitting threshold. This prevents loss of mass through multiple profiles and allows us to pass to the logarithmic limit. Under the stated structural and energy conditions, we obtain a nontrivial nonnegative bound state of the original equation.

We analyze a nonlocal operator in space and time, called the multidimensional fractional material derivative. We derive its pointwise representation, which allows us to study its other properties. We define an inverse operator, called fractional material integral, and derive its pointwise representation. Furthermore, we analyze a class of linear partial differential equations, which corresponds to deterministic descriptions of the scaling limits of multidimensional Lévy walks, in which transport is driven by a multidimensional fractional material derivative with a speed vector integrated with respect to a suitable probability measure and a distributional source term. Using Fourier-Laplace transform techniques and a direct convolution-kernel construction, we prove the existence and uniqueness of exponentially bounded measure solutions for measure data. Moreover, we identify a necessary and sufficient condition on the source term for conservation of unit mass and provide separate sufficient conditions for non-negativity and weak convergence to $\delta_0$.

The paper studies the weighted dual Minkowski problem for the weighted $q$th dual curvature measure of convex bodies in $\mathbb R^n$ first posed by Huang, Lutwak, Yang, and Zhang. Most of the results are in the G-invariant setting, i.e. when the convex bodies are invariant under a closed subgroup $G$ of $O(n)$. They generalize previous results on the existence of solutions to the dual Minkowski problem for origin-symmetric convex bodies. It is proved: (i) a full characterization of the G-invariant weighted dual curvature measure for q\in(0,1], (ii) a necessary and sufficient condition for the existence of solutions to the G-invariant weighted dual Minkowski problem for q\in(1,n), (iii) a sufficient condition for the G-invariant weighted dual curvature measure for q>1, and (iv) an extension of Henk-Pollehn's dual curvature measure concentration property.

Blow-up in the self-dual Chern–Simons–Higgs equation has a rich multiscale structure, but the detailed theory has largely focused on the simple regime, where one entire profile resolves each concentration point. By contrast, non-simple blow-up is analytically more delicate: several interacting regular cores must be resolved simultaneously as they collapse toward a single vortex while remaining separated on their own scales. Because of this difficulty, the existence of such solutions has remained a long-standing open problem. To the best of our knowledge, we prove the first existence result for non-simple blow-up solutions of this equation, covering in particular the previously unresolved finite-height Chern–Simons regime. We also show that non-simple clusters have equal limiting core masses, a common profile type, and regular-polygon arrangements. We determine their possible total masses and derive a mass-gap criterion that rules out such blow-up for certain vortex configurations. Our quantitative profile and moment estimates for general finite-height clusters guide the design of the approximate solution and clarify the core interactions underlying the finite-height construction. A separate construction on the unit disk produces two mean-field cores along a sequence with varying Dirichlet traces.

We give an alternative proof of a theorem of Brena, De Lellis and Franceschini (arXiv:2503.00649): the support of a stationary integral $d$ dimensional varifold in an open subset of $\mathbf{R}^n$ is $(\mathscr{H}^d, d)$ rectifiable of class $(k, \alpha)$ for every positive integer $k$ and every $0 < \alpha < 1$, and in particular of class $\mathscr{C}^{\infty}$. Our proof is independent of theirs and proceeds by different means. It compares the varifold, at every scale and at $\mathscr{H}^d$ almost every point, with the graph of a solution of the Euler–Lagrange system of a smoothed area integrand, and thereby derives an excess-decay estimate that is faster than any power of the scale; the conclusion then follows from Santilli's characterisation of higher order rectifiability together with Whitney's extension theorem.

In this article, we investigate the inverse problem of determining time-dependent first- and zeroth-order coefficients for a wave equation with nonlocal-in-time attenuation on a Riemannian manifold, using boundary measurements. This problem is motivated by applications in imaging and viscoelasticity, where such equations arise naturally, and we aim to characterize the properties of the underlying medium by recovering these coefficients. Our main objective is to exploit the memory properties of solutions induced by nonlocal attenuation, establishing the recovery of a general class of lower-order coefficients from boundary measurements collected over an arbitrarily small time interval. As a byproduct of our analysis, we also prove recovery results for time-dependent coefficients using measurements supported on disjoint time intervals. While such data restrictions are unavailable, and generally impossible, for the classical wave equation, we demonstrate that they become feasible under standard assumptions due to the presence of nonlocal-in-time attenuation. Our analysis combines techniques from differential geometry, the theory of partial differential equations with nonlocal terms, complex analysis, and the theory of nonlocal operators.

In this paper, we prove the existence and far-field behavior of an axisymmetric solution $(\rho,\mathbf{u},p)$ to the steady Euler system in an infinitely long cylindrical nozzle with a perturbed wall $r=R(x_1)$, containing a contact discontinuity $r=g_D(x_1)$. The background flow carries a nonzero swirl component $\beta_0\mathbf{e}_\theta$, where $\beta_0$ is not necessarily small. Consequently, the background entropy, Bernoulli function, and angular momentum density depend on the radial variable, and their radial derivatives are not necessarily small.

In this paper, we consider the inequality $\Delta_pu+\Delta_qu+u^s|\nabla u|^t\leq 0$ on geodesically complete noncompact Riemannian manifolds. By a test function argument, we establish Liouville-type theorems under the upper bound of volume of geodesic ball. In the Euclidean space $\mathbb{R}^n$, we obtain new nonexistence results which extend the result of Bhakta-Biswas-Filippucci \cite{BBF}. In particular, we have addressed the influence of the higher order term for $s<0$. At last, we present some examples to illustrate sharpness in some cases.

Locally commuting multiple radial Schramm-Loewner evolutions ($\mathrm{SLE}_{\kappa}$) are encoded by partition functions satisfying the radial Belavin-Polyakov-Zamolodchikov (BPZ) equations and a conformal Ward identity with spectral parameters $\lambda,\nu\in\mathbb{R}$. For $\kappa>0$ and $\lambda\in\mathbb{R}$, we show that the solution space of the radial BPZ system has dimension $2^n$, where $n$ is the number of variables. We then determine all admissible Ward parameters $\nu$ and the exact dimensions of the subspaces selected by the conformal Ward identity, covering both generic and degenerate cases. The classification reveals a parity difference: nonzero rotation-invariant solutions exist for every $\lambda$ when $n$ is even, but only at finitely many exceptional values when $n$ is odd. When $0<\kappa\leq4$, $\lambda>0$ for odd $n$ or $\lambda>-3/2$ for even $n$, we construct a basis of positive solutions using multiple SLE, providing global realizations of the locally commuting SLEs. At $\kappa=4$, we identify a family of explicit solutions as partition functions for level lines of a Gaussian free field with suitable boundary data and interior singularities.

In this paper, we establish existence and pointwise Wolff potential bounds for nonnegative solutions of $-\Delta_{\mathbb{G},p}u=H_l(\alpha u^\beta)+\mu$ on Carnot groups, where $H_l(t)=e^t-\sum_{j=0}^{l-1}t^j/j!$ and $\mu$ is a nonnegative Radon measure. Under suitable conditions on the reaction parameters, we treat arbitrary open sets in the subcritical range $1<p<Q$ under a maximal potential condition, and bounded domains in the critical case $p=Q$ under a small total mass condition.

In this paper, we study the fixed angle scattering problems raised by Rakesh and Salo. We show that a compactly supported potential is uniquely determined by the far field pattern at a fixed angle, which is an open problem. Our method is based on a novel Carleman estimate and the ideas introduced by Bukhgeim and Klibanov on the use of Carleman estimates for inverse problems.

It is shown that if $u$ solves the wave equation in $\mathbb{R}^{4+1}$ with initial data $u(\cdot,0) = u_0(\cdot) \in H^s$ and $\partial_tu(\cdot,0) = u_1(\cdot ) \in H^{s-1}$, where $0.5 < s \leq 0.55$, then $u(x,t) \to u_0(x)$ and $\partial_tu(x,t) \to u_1(x)$ pointwise as $t \to 0$, for all $x$ outside an exceptional set of Hausdorff dimension at most $6-4s$. In a very small range of $s$, this verifies a conjecture of Barceló, Bennett, Carbery, and Rogers. More generally, a partial improvement to the exceptional set bound in $\mathbb{R}^{n+1}$ is obtained for $n \geq 4$ and $1/2 < s < n/4$.

In this article, we give various real-variable properties of local Muckenhoupt matrix weights $W$ and establish the quantitative boundedness of several operators on local matrix-weighted Lebesgue spaces $L^p(W)$, including local (fractional) maximal operators, local fractional integral operators, local Haar square functions, and Calderón–Zygmund operators with exponential decay. For the local maximal operators, we obtain the sharp quantitative bounds when $p\in(1,2]$, while, for local fractional integral operators, we obtain the quantitative bounds matching the global best known exponents, whose scalar case is known to be sharp. The key used strategies include giving a new extension property (which can clarify their relationships with global ones) and an optimal scale lifting property (which can balance the locality of weights and operators under consideration) of local matrix weights. As an application, we establish the quantitative boundedness on $L^p(W)$ of the Riesz transform associated with Schrödinger operators $-\Delta+m^2I$ with $m\in(0,\infty)$ being large enough, whose quantitative bound when $p=2$ is precisely $[W]_{\mathscr{A}^{\operatorname{loc}}_{2}(r)}^{\frac 32}$.

We consider the elastic wave equation in three-dimensional isotropic medium with up to quadratic nonlinear terms. We show that five out of the six elastic parameters can be uniquely determined by boundary measurements under the assumption that the wavespeeds are non-trapping, while the last one can also be determined in a generic setting.

We study global energy minimization for matrix-valued heteroclinic connections whose vacuum sets are continuous conjugacy orbits rather than isolated points. For two distinguished parameter regimes of a one-dimensional $\mathrm{SU}(5)\times\mathbb Z_2$ matrix field model, we first identify the full zero set of the quartic potential as the disjoint union of two compact homogeneous vacuum orbits. We then construct auxiliary potentials controlled by the Frobenius distance to these orbits. Pointwise comparison with the original potential, combined with a calibration inequality and the one-dimensional coarea formula, gives a sharp lower bound for every finite-energy connection joining the two vacuum components; no diagonality, commutativity, or symmetry-reduced ansatz is imposed on the competitors. Two explicit non-Abelian kinks attain the bound and are therefore global minimizers in the full orbit-to-orbit heteroclinic classes. Their exact wall tensions are $4\sqrt2\,\mu^3/(3\lambda)$ and $9\sqrt2\,\mu^3/(10h)$, respectively. Equivalently, the corresponding line-segment paths realize the degenerate weighted distance between the vacuum manifolds. We also resolve every fixed-endpoint sector: for arbitrary prescribed representatives on the two vacuum orbits, the energy infimum is the same wall tension, and it is attained if and only if the prescribed representatives form a closest pair of the two orbits. Nonclosest endpoint sectors therefore exhibit non-attainment through increasingly slow tangential motion along the vacuum manifolds. The proof separates a model-independent distance–coarea criterion from the model-specific orbit geometry and polynomial positivity estimates, providing a reusable mechanism for matrix-valued multiwell potentials with symmetry-generated vacua.

We develop a Banach-valued lateral Stieltjes calculus adapted to linear evolution equations on a Gelfand triple \[ V\hookrightarrow H\simeq H^*\hookrightarrow V^*. \] The evolution problem is formulated with the posterior state, \[ D_g u(t)+A(t) S_g u(t)=f(t),\qquad u(a)=u_0. \] This choice is decisive at atoms of the Stieltjes measure: if $\delta=\Delta_g(r)>0$, the equation yields the implicit variational resolvent \[ (I+\delta A(r))u(r^+)=u(r)+\delta f(r). \] We prove a Banach-valued lateral product formula, a weak–strong reconstruction theorem, and a Stieltjes analogue of the Lions–Magenes energy identity. In the separable Hilbert setting considered here, the stable projection chain used in the proof is shown to exist automatically by a spectral-band construction. The latter contains the exact positive jump contribution \[ \frac12\sum_r\|u(r^+)-u(r)\|_H^2. \] Existence is constructed independently by Galerkin approximation and by a Rothe scheme adapted to the Stieltjes mass and its atoms. The Rothe construction does not use spatial projections. We also explain why the available Aubin–Lions theorem for anterior Stieltjes–Bochner evolution graphs is not automatically applicable to the posterior energy space and, in the present linear coercive setting, is not needed. A heat equation with Stieltjes time is included as a model application. The claims are positioned relative to earlier Stieltjes parabolic equations, general measure-driven evolution, and operator-valued-measure parabolic systems; no priority is claimed for measure-driven parabolic theory or Galerkin approximation in general.