Papers

In this paper, we prove a global $C^{2,\alpha}$ estimate for the complex Monge–Ampère equation with a positive $C^\alpha$ right-hand side on a compact Kähler manifold.

We establish a priori estimates and compactness results for positive solutions of the CR Nirenberg problem under natural weighted flatness conditions. The proofs combine refined blow-up analysis, Pohozaev identities, and the complete classification of positive entire solutions to the critical CR Yamabe equation recently obtained by Li–Liu–Ma \cite{LLM2026} and Li \cite{L2026}. Existence and degree results are established in our subsequent paper \cite{QTWPartII}.

We study a class of periodic NLS-Dysthe (NLS-Dy) equations, also referred to as higher-order nonlinear Schrödinger equations (HNLS), of the form $i\partial_t u+a\partial_x^2u+b|u|^2u+i\varepsilon(c\partial_x^3u+d|u|^2\partial_xu+eu^2\partial_x\overline{u})=0$ with $ x\in\mathbb T$, $a,b,c,d,e\in\mathbb R$, $abc\neq0$, and $\varepsilon>0$. For $0<\varepsilon\ll1$, equations of this type arise as higher-order modulation models for some nonlinear dispersive PDEs, including the Klein-Gordon equation and the Serre-Green-Naghdi equation modeling the evolution of nonlinear shallow water waves. We prove local well-posedness in $H^s(\mathbb T)$ for $s>3/2$ for arbitrary admissible coefficients, and establish an $H^1$ continuation criterion for $s\ge2$. In the parameter regime $e\neq0$, a lower-order conservation law provides uniform $H^1$ control and, together with the continuation criterion, yields global well-posedness for $s\ge2$. This result does not require the non-resonance condition appearing in Takaoka's periodic theory, while the condition $e\neq0$ naturally arises in the modulation setting considered here. Finally, motivated by the long-time regime relevant for modulation, we establish uniform long-time control without imposing a smallness assumption on the initial datum. Using the conservation laws of cubic NLS and their invariance under the commuting complex mKdV flow, we obtain uniform $H^s$ bounds on time intervals of order $\varepsilon^{-1}$ for integer $s\ge2$. For non-integer real $s\ge2$, the maximal $H^s$ solution exists on the same time scale, while uniform control is obtained at a lower regularity, with a loss of less than one derivative.

Given a smooth domain $D\subset\mathbb S^{N-1}$, $N\geq 3$, and a positive constant $H$, we study radial graphs over $D$ with constant mean curvature $H$ which meet orthogonally the boundary of the cone spanned by $D$. The corresponding problem is given by a quasilinear elliptic equation on $D$ with homogeneous Neumann boundary condition. We investigate the structure of the solution set under smooth perturbations of the domain which are parametrized by an open set of $C^{3,\alpha}$-diffeomorphisms of the sphere. After pulling the equation back to a fixed reference domain, we apply a classical transversality theorem and prove that, in the class of domains for which $N-1$ does not belong to the Neumann spectrum of the Laplace-Beltrami operator, there is a residual set for which every solution is nondegenerate. A consequence of our result is that, for every such domain and every $R>0$, the set of solutions whose $C^{2,\alpha}$-norm is bounded by $R$ is finite.

In this paper, we first apply the implicit function theorem to find nontrivial solutions to the following Bernoulli's overdetermined problem \begin{equation} -\Delta u=\gamma\,\, \text{in}\,\,\Omega \setminus\overline{D}, \,\, u=1,\,\,\partial_\mathbf{n} u=Q\,\,\text{on}\,\,\partial D, \,\,u=0\,\,\text{on}\,\,\partial\Omega \nonumber \end{equation} for a fixed boundary $\partial\Omega$ and a free boundary $\partial D$ in $\mathbb{R}^{N}$ with $N\geq2$, where $\gamma$ is a sufficiently small constant, the Bernoulli value $Q$ is determined by the construction, and $\Omega \setminus\overline{D}$ is a bounded ring-shaped domain. In dimension $N=n+1$ with $n\geq2$, we further study the existence of a nontrivial unbounded domain $\Omega \setminus\overline{D}$ in $\mathbb{R}^N$ such that the Bernoulli's overdetermined problem admits a solution. This unbounded domain we construct is periodic in one variable and radial in the other variables by bifurcation analysis.

We prove that the full Dirichlet-to-Neumann map for static isotropic elasticity uniquely determines both smooth Lamé parameters in a bounded smooth domain of \(\mathbb R^3\) under strong convexity. The proof uses an exact four-component Laplace-type reduction, but its decisive identities are obtained from physical elastic solutions. The characteristic quotient of the reduced system has rank three, whereas the leading physical adjoint lift supplies only two directions. We construct the missing polarization from the first nonvanishing correction of a reduced amplitude in the kernel of that lift. Paired with fixed primal and adjoint quotient frames, the resulting complex geometrical optics solutions yield a complete family of holomorphically weighted \(3\times3\) matrix moments through the physical Betti identity. Cauchy–Plemelj factorization converts these moments into an invertible relative transport normalized in the common exterior. Holomorphic dependence on the projective null direction and transport rigidity then determine a material ratio and the shear modulus, hence both Lamé parameters. No smallness, symmetry, separability, or analyticity hypothesis is imposed.

We reduce the problem of establishing pointwise upper decay estimates in thin domains for viscosity subsolutions of fully nonlinear partial differential equations to solving a one-dimensional differential inequality. The resulting estimates make explicit the interaction between the transverse geometry of the domain and the ellipticity of the equation, and are shown to be sharp. Our framework allows the ellipticity to degenerate at infinity and accommodates nonhomogeneous equations with lower-order terms. As an application, we obtain decay estimates for $p$-subharmonic functions when $1<p\leq 2$

We study stable solutions of the mean curvature equation $$ -\operatorname{div}\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right) =|u|^{p-1}u $$ in domains of $\mathbb R^n$. We establish Liouville-type results in three settings. First, for globally stable solutions in $\mathbb R^n$, we prove that $u\equiv0$ if $1<p<p_{JL}$, where $p_{JL}$ is the Joseph–Lundgren exponent. Second, for solutions in $\mathbb R^n$ that are stable outside a compact set, we show that if $n>2$ and $p=p_S$, where $p_S$ is the Sobolev critical exponent, then $u\equiv0$. Third, for solutions of the Dirichlet problem on connected unbounded domains with smooth boundary satisfying an epigraph-type geometric condition, we prove that if $n>2$, $1<p\le p_S$, and the solution is stable outside a compact set, then $u\equiv0$. The proofs are based on a gradient estimate for stable solutions, an integrability result, conservation laws and Pohozaev-type identities, and a unique continuation argument. These results extend Liouville-type theorems for stable solutions from the semilinear Lane–Emden equation to the mean curvature equation.

The focus of this article is to establish the existence portion of the Strauss conjecture for wave equations with spatially dependent scattering damping. In the current setting, the damping is not sufficiently strong to alter the critical Strauss exponent. In a subsequent study, the authors will consider the case of damping coefficients with critical decay where a shift of the Strauss exponent is expected. While several recent studies of the Strauss conjecture have relied upon weighted Strichartz estimates, here, instead, an approach that is based only on integrated local energy estimates, in particular those with both $r^p$ and ghost weights, is developed. This technique also applies to the problem for the free equation. Space-time Klainerman-Sobolev estimates provide the necessary decay to close the argument.

The Strauss conjecture concerns wave equations with power-type nonlinearities and seeks the optimal range of powers that admit global solutions for all sufficiently nice and sufficiently small initial data. In three spatial dimensions, it is well known that small data global existence is available provided that the power $p>p_c(3)=1+\sqrt{2}$, where $p_c(n)$ denotes the critical power in $n$-spatial dimensions. In the current article, the analogous problem is considered for damped wave equations where the damping coefficient is at the critical scaling. When $a$ denotes the constant multiple applied to the damping term, it is shown that the corresponding critical power is $p_c(3+a)$ provided that $a$ is sufficiently small.

We consider an elliptic Cauchy problem for the Poisson equation in which both Dirichlet and Neumann boundary data are prescribed only on a portion of the boundary, while no boundary conditions are available on the remaining part of the boundary. Such problems are severely ill-posed. In this work, we study the convergence of an iterative method based on the alternating imposition of Dirichlet and Neumann boundary conditions. We formulate the scheme within a Hilbert space framework, where a yet another alternation becomes apparent: each iteration can now be interpreted as an orthogonal projection onto an appropriate closed subspace. This perspective provides a transparent geometric interpretation of the algorithm and yields a concise convergence proof based on von Neumann's classical result on alternating (orthogonal) projections. In addition to elucidating the convergence of the method, this alternating projections perspective also offers a general rate of convergence in terms of the Friedrichs angle (between two closed subspaces). We point out by means of an example that such convergence rate estimate degenerates for the elliptic Cauchy problem under consideration.

We study the global geometry and endpoint dynamics of positive solutions of $-\Delta u+|y|^2u+\lambda u=g(u)$, $(y,z)\in\mathbb R^d\times\mathbb R^m$, $m=N-d$. Despite the loss of coercivity at $\lambda=-d$, we construct a connected component of centered positive solutions with frequency projection $(-d,\infty)$, assuming subcritical asymptotics of $g$ at zero and infinity, a zero-mass Liouville condition, and a strict Hamiltonian sign condition. At the lower edge, $\lambda+d\rightarrow 0^+$, anisotropic rescaling selects the lowest confined harmonic mode coupled to an $m$-dimensional scalar ground state. As $\lambda\rightarrow\infty$, isotropic blow-up yields the $N$-dimensional scalar ground state, revealing a dimensional crossover. The endpoint profiles and mass laws hold for every sequence of centered positive solutions, not merely along this component. The proof combines Gaussian–exponential localization on compact frequency strips, Lyapunov–Schmidt reduction, compact spectral perturbation, and Leray–Schauder continuation. At both ends, we establish uniqueness and differentiability of the centered branch, identify the Morse indices and symmetry kernels of the real and imaginary linearizations in the full energy space, and derive sharp mass-slope asymptotics. Under standard Hamiltonian-flow hypotheses, the resulting slope criterion classifies orbital stability and instability away from the mass-critical powers. The mass map on this component, combined with endpoint control on the full centered solution set, yields existence, nonexistence, and multiplicity for normalized standing waves, revealing the roles of both critical exponents $2+4/m$ and $2+4/N$.

We study the three-dimensional primitive equations with full viscosity and horizontal thermal diffusion in bounded domains. We establish global well-posedness for admissible $H^1$ initial data and persistence of $H^2$ regularity on the flat-bottom cylinder. The higher regularity analysis combines fractional hydrostatic Stokes regularity with anisotropic estimates of the vertical transport. We also consider variable-depth domains with positive $h\in C^3(\overline M)$ and a diffusion operator adapted to the topography compatible with layers $z/h(x,y)=\mathrm{constant}$. This operator reduces to $\Delta_H$ at constant depth and generally differs from the Cartesian horizontal Laplacian over a varying bottom. Its weighted formulation yields global strong solutions for admissible $H^1$ data. Persistence of $H^2$ regularity is proved under the additional assumption $\partial_\nu h=0$ on $\partial M$, where $\nu$ is the outward horizontal unit normal.

This paper investigates intrinsic critical Sobolev-curl constants on bounded domains and the structure of finite-energy solutions to the critical curl-curl equation in R^3. First, for 1< p < 3, we prove that the intrinsic critical p-curl constant on (0, a)\times D is bounded above by a^{p/3} times a Sobolev quotient on D. The proof uses an aligned test field whose optimal curl-free correction vanishes. Consequently, these constants tend to zero along thin convex boxes. For p = 2, we construct explicit convex and nonconvex domains with intrinsic constant below the sharp scalar Sobolev constant. The nonconvex example gives an affirmative answer to the domain-existence question in Problem 5 of Mederski and Szulkin(Arch. Ration. Mech. Anal. 241 (2021), 1815-1842), while the convex example disproves the lower bound stated there for convex domains. Concerning their Problem 2, we obtain an implicit Newtonian representation of finite-energy solutions and exclude two special forms of solutions. These structural results do not yet provide an explicit ground state or an unconditional ground-state decay rate as asked in Problem 2.

We study a density heat flow associated with normalized Liouville systems on a closed connected Riemannian surface, and we show that this flow leads to the existence of solutions for the corresponding subcritical Liouville systems. The coupling matrix is symmetric and entrywise nonnegative, with no invertibility, irreducibility or inverse-matrix sign assumption. Under the strict subcriticality conditions $8\pi\sum_{i\in J}\rho_i>\sum_{i,j\in J}a_{ij}\rho_i\rho_j$ for every nonempty set of components $J$, every smooth positive initial density of masses $\rho_i>0$ generates a unique global smooth positive solution converging in every $C^m$ norm to a stationary density. The limiting potentials solve the stationary Liouville system. Our proof uses entropy coercivity and a gradient inequality in the density variables, which also yields algebraic or exponential convergence rates.

We prove the persistence of conormal analyticity through the $H^r$ norm, with $r \geq 4$, for the 3D incompressible Euler equations in the half-space $\Omega = \mathbb{R}_+^3$. We also establish a lower bound on the radius of conormal analyticity which depends on $\|u\|_{H^r}$ and initial data. In particular, persistence is established in both of the spaces $X_\tau^{\rm co}$ and $A_\tau^{\rm co}$, the former being based on the conormal operator $Z_k = \phi^k \partial_3^k$ and the latter being based on $L^k = (\phi \partial_3)^k$. Finally, we show that the norms $\|\cdot \|_{X_\tau^{\rm co}}$ and $\|\cdot \|_{A_\tau^{\rm co}}$ are equivalent up to a small loss in radius, the proof of which uses explicit identities relating the two conormal operators.

In this paper, we establish the stability of general entropy wave solutions of the compressible Euler equations (that is, profiles with constant velocity and pressure) in the class of vanishing dissipation limits from the Navier–Stokes–Fourier system. If the initial data of our sequence of solutions to the Navier–Stokes–Fourier system converges to the initial data of the entropy wave solution, we further have convergence of the solutions to the entropy wave. This gives uniqueness and stability of the entropy wave in this class. The proof is based on the $a$-contraction theory.

The interplay between a weighted volume measure and its naturally associated weighted Laplace operator is fundamental to the geometric analysis of weighted structures governed by gradient vector fields. Beyond the gradient setting, however, the theory has remained considerably less developed, largely because this natural correspondence is lost. In this paper, we identify invariant measures as the appropriate framework for extending the theory to general drifted Laplace operators. This approach yields a systematic treatment of parabolicity in the non-gradient setting, reveals its connection with volume growth, and leads to new volume finiteness results for non-gradient geometric structures.

We prove sharp improved Sobolev inequalities on $\mathbb S^n$, $n\ge3$, for antipodally even functions and for axially symmetric functions whose critical density $|u|^{2n/(n-2)}$ has zero barycenter. In both cases, the Dirichlet coefficient improves by the optimal factor $2^{-2/n}$, while the coefficient of the $L^2$ term remains one. Equality holds precisely for constant functions. The even case resolves a conjecture of Dolbeault (2021).

We consider functions $u : \R^2 \rightarrow \R^N$ from the class $\Bloc$ which locally minimize the relaxed version of a variational integral with a strictly convex density being in addition of linear growth. It turns out that the boundedness of entire local minimizers $u$ implies their constancy.

We provide some extensions of the classical Calabi-Bernstein theorem to the case of entire solutions of certain nonlinear elliptic equations in two variables. Using an appropriate version of the Calabi correspondence, which formulates a duality between minimal graphs in $\mathbb{R}^3$ and maximal spacelike graphs in $\mathbb{L}^3$, our results are a consequence of recent Bernstein type results.

Deterministic flows can contract Wasserstein distances between probability laws without being contractive in point-state space. We develop a differential contraction theory for pushforward flows on forward-invariant subsets of the 2-Wasserstein space that quotients out the gauge freedom of the continuity equation. Weighted Helmholtz-Hodge decomposition and the Wasserstein contingent cone define the weighted horizontal tangent cone (whtc). Requiring the contraction quadratic form to be nonpositive only on the whtc yields the Intrinsic Wasserstein Contraction Condition (IWCC), which generalizes the classical contraction LMI and recovers it on Dirac measures. An almost-everywhere Euler approximation based on the superposition principle and Banach-valued Lebesgue differentiation theorem proves that IWCC is sufficient for intrinsic Wasserstein contraction. Law contraction without point-state contraction is illustrated by a rotationally symmetric shear example. Finally, a new comparison lemma yields an incremental Wasserstein ISS theorem; for Brockett's integrator, we prove exponential stabilization of a class of axisymmetric laws to the uniform law on a circle via a smooth feedback.

We develop and analyze a projected extragradient method, preconditioned by a Riesz map, for a first-order price-formation mean field game in one space dimension. The model couples Hamilton–Jacobi–Bellman and transport equations through a global market-clearing constraint. Finite propagation justifies truncating the first-order transport equation to a bounded interval. A monotone weak formulation incorporates the mixed initial–terminal data. Its Riesz map uses $L^2$ norms for density and price directions and a space-time $H^1$ norm for value-function directions, preconditioning the residual without adding higher-order regularization terms to the MFG equations. A density-only argument rules out cocoercivity on the localized admissible set in every equivalent Hilbert metric, motivating extragradient updates. For $H(x,p)=p^2/2$, we use a monotone upwind discretization with adjoint-consistent transport and a state-constraint boundary closure. We establish a mesh-independent Lipschitz bound on bounded convex admissible sets with uniform density, slope, and price bounds. With exact metric projection onto these sets, the iteration converges to a solution of the discrete variational inequality on each fixed grid, with an admissible step-size threshold independent of both mesh sizes. Convergence under mesh refinement is not proved. On a quadratic benchmark, all reported runs reach the stopping tolerance with one fixed step size across a factor-of-16 reduction in both mesh sizes. Price errors and value-function errors measured away from low-density regions decay approximately at first order under refinement, while density errors decrease more slowly.

We give an exposition of a proof, discovered by Astra, of an Aleksandrov estimate for kinetic equations with rough coefficients. We explain this method in the simpler elliptic and parabolic settings as well.

In every sub-Riemannian Heisenberg group $\mathbb H^n$, with $n\geq 2$, we prove the existence of a global perimeter minimizer whose boundary is a smooth, non-characteristic, entire intrinsic graph but it is not a vertical hyperplane.

We consider solutions to the two-dimensional Navier-Stokes and Euler equations with the velocity in $L^{\infty}(\mathbb{R}^2)$ and the vorticity in the uniformly local Yudovich space $Y^{\theta}_{ul}(\mathbb{R}^2)$. We show that if $\theta$ grows sufficiently slowly, then for every $T>0$, solutions of the Navier-Stokes equation converge to solutions of the Euler equation in $L^{\infty}([0,T];L^2_{ul}(\mathbb{R}^2))$ as $\nu \to 0^+$, provided they are generated from the same initial data.

We study the logarithmic Schrödinger equation with an arbitrary positive semidefinite quadratic potential and a periodic potential in the unconfined directions. The main variational difficulty is to turn a constrained minimizer into a critical point when the energy is extended-valued and not of class $C^1$ on the natural Hilbert space. We establish an abstract principle for convex lower semicontinuous energies with a smooth perturbation, allowing a scaling law for the full energy without requiring homogeneity of its convex part. Combined with a profile decomposition under integer translations, this principle yields a positive ground state for every rank of the quadratic form. When the periodic potential vanishes, we prove a sharp anisotropic logarithmic Sobolev inequality, determine its optimal constant and all equality cases, and classify positive ground states as explicit anisotropic Gaussons, unique up to translations along the kernel of the quadratic form. When the periodic component is constant, we also establish existence and classify all normalized ground states at every prescribed positive mass, with explicit formulas for the minimum energy and the associated Lagrange multiplier.

We study a continuum model of frustrated magnetism that supports particle-like field configurations with nontrivial Hopf index. The energy functional combines competing second and first order exchange interactions with a stabilizing external magnetic field and gives rise to a variational problem for maps from three-dimensional space into the two-sphere. We prove the existence of energy minimizers in infinitely many nontrivial homotopy classes. We further establish the existence of a unit-charge Hopfion in the regime of strong external fields. Finally, we analyze the corresponding large-field limit, identify the natural concentration scale, and prove strong convergence of suitably rescaled minimizers to a minimizer of a limiting variational problem.

In this paper we characterize the regularity of a domain in terms of the behavior of a certain spherical function. More precisely, given two domains we show that a certain spherical spectral function (first defined in \cite{FTV23}, see also \cite{AKN21}) has Hölder decay if and only if the domains are complementary and separated by a $C^{1,\beta}$-manifold. This extends the analysis in \cite{Ca24} to dimensions $\geq 3$ and in \cite{FTV23, CTV25} to regularity above the continuous threshold. Along the way, we introduce a spherical analogue of the Jones $\beta$-numbers and show that the decay of these numbers implies geometric regularity of the domain on which they are defined. We believe this result is of independent interest and partially addresses a question of Fleschler-Tolsa-Villa \cite[Question 1]{FTV23}.

Standing waves $e^{i\omega t}u(x)$ of nonlinear dispersive equations, such as the nonlinear Schrödinger and Klein–Gordon equations, have a profile $u$ that solves a nonlinear elliptic ODE of the form $\Delta u+f(u)=0$. We prove the existence of radial bound states with a prescribed number of nodal surfaces for this equation, on a large class of spherically symmetric manifolds and nonlinearities $f$. Under mild assumptions, we prove uniqueness of such bound states for the model nonlinearity $f(u)=|u|^{p-1}u-u$. Both existence and uniqueness results apply to some manifolds with a singularity at the origin.