Papers

We prove the strict Pólya inequality for Neumann eigenvalues of spherical shells in every dimension $d\ge2$, for every radius ratio and every positive energy. The proof also covers balls. For shells, the excess over the leading Weyl term has a …

We prove a sharp linear upper bound for the interior nodal length of Steklov eigenfunctions on smooth compact Riemannian surfaces with boundary, establishing the upper bound predicted by the Steklov analogue of Yau's conjecture in dimension two. On each fixed …

We establish strict Dirichlet Pólya inequalities for spherical shells in every dimension $d\ge2$ and for every radius ratio. The inclusive eigenvalue counting function lies strictly below the leading Weyl term at every positive energy; consequently, each eigenvalue strictly exceeds its …

For the complex Monge--Ampère equation on $\mathbb{C}^{n}$, we prove the Liouville theorem for solutions with quadratic growth.

We consider the three-dimensional incompressible Navier--Stokes equation on a bounded smooth domain with no-slip boundary condition and finitely many smooth, divergence-free, boundary-tangent transport noises. We prove local pathwise well-posedness under the stochastic parabolicity condition $\nu I-\frac12\sum_k b_k\otimes b_k\geq\delta I>0$, without …

We prove the planar Pólya–Szegő conjecture for the first clamped buckling eigenvalue. Our main result establishes the sharp comparison Λ₁(Ω) ≥ (j₁,₁²/j₀,₁²) λ₁ᴰ(Ω) for every planar open set Ω of finite positive area, where Λ₁ denotes the first buckling eigenvalue …

Complete proof of the unchanged all-graph real symmetric delta target, with exact copied source closure and kernel trust. Independent full source/package reviews and actual isolated Linux Comparator/kernel verification are complete.

We give a computer-assisted proof that every finite simplicial triangulation of real projective five-space has at least twenty-four vertices. Together with the construction of Guyer, Steinerberger, and Yang, this determines the minimum. A cup-product argument produces an induced subcomplex on …

The finite element scheme for relaxed minimal-deformation surface evolution prescribes normal velocity while allowing tangential mesh motion. We establish contrasting results for its original exact-integral formulation with the discontinuous geometric unit normals of the current elements. For linear elements, one …

We prove that for every choice of five positive masses the planar five-body problem has finitely many central configurations up to similarity. This settles Smale's sixth problem for five bodies in the plane. Albouy and Kaloshin proved it for masses …

We prove finite propagation for a fourth-order aggregation--diffusion equation with linear mobility and quadratic attraction on $\R^d$. For every $d\ge2$, every positive attraction coefficient and every compactly supported nonnegative $H^1$ probability density, there is one global weak solution whose support …

We prove almost-sure global regularity and pathwise uniqueness for the critical surface quasi-geostrophic equation on the two-dimensional torus with a finite collection of smooth nonlinear state-dependent Itô noise coefficients. For every deterministic smooth datum, globally scalar-Lipschitz coefficients with bounded positive-order …

We prove local well-posedness of the three-dimensional incompressible Navier--Stokes equation with prescribed smooth, compactly supported, divergence-free Ito transport noise and deterministic initial velocity in the inhomogeneous space $B^{-1/2}_{6,4}(\R^3)$. Under strict stochastic parabolicity, with no further smallness assumption, the solution is …

We consider scalar divergence-form parabolic Itô equations on the torus, with bounded measurable diffusion and progressively measurable transport coefficients that are smooth in space with deterministic derivative bounds. Under stochastic parabolicity, we prove that every variational solution with deterministic smooth …

We prove a global $C^{1,s}$ estimate for the finite-energy obstacle problem of a symmetric stable operator whose angular density is only integrable. The constant depends on the order, dimension, angular mass bound, and quadratic nondegeneracy bound, without stronger angular integrability …

Let a smooth strongly convex norm on the plane define both a billiard table and the length of its chords. We prove that if the entire billiard phase annulus is foliated by continuous essential invariant curves, then the norm comes …

We construct two global renormalized solutions with the same nonnegative finite-entropy initial data for the reversible cubic reaction $A+2B\rightleftharpoons3B$ on a three-dimensional ball. The two diffusion coefficients are equal. One species starts from zero, while the other has an integrable …

Let \(A\) generate a strongly continuous semigroup on an arbitrary Banach space \(X\), and let \(B\) be a bounded operator from an arbitrary Banach input space into the extrapolation space \(X_{-1}\). We prove that \(L^\infty\)-admissibility for all strongly measurable inputs …

Computing a few singular vectors of a bidiagonal matrix requires control of orthogonality as well as residuals when singular values cluster. We give a rational-filter construction for any requested set of $k$ ordered singular triplets of a real $n\times n$ …

We consider the one-dimensional Schrödinger equation on an interval of length $L$, driven by an $L^2$ Dirichlet control at one endpoint, with initial data measured in $H^{-1}$. We prove that its null-control norm cost satisfies $C_S(T,L)\ge c_L T^{7/2}\exp(L^2/(\pi T))$ for …

We prove a logarithmic convexity estimate for a pair of coupled Caputo subdiffusion equations with different time orders. For every bounded smooth domain, every $0<\alpha_1<\alpha_2<1$, and every final time $T>0$, the product $L^2$ norm at time $t$ is bounded by …

We investigate interior regularity for stable solutions of the semilinear elliptic equation $-Δu=f(u)$ in an open set $Ω\subseteq \mathbb R^n$, where $f$ is locally Lipschitz and may change sign. We establish sharp interior Hölder estimates for stable classical solutions in …

We prove mass conservation for the spectral fractional porous-medium equation on complete noncompact Riemannian manifolds with Ricci curvature bounded below and a Euclidean-dimensional Faber--Krahn inequality. The result holds for every nonnegative weak dual solution that is spatially integrable at each …

We construct solutions of the spectral fractional porous medium equation starting from a point mass on a complete noncompact Riemannian manifold. The assumptions are a lower Ricci curvature bound and a uniform Euclidean Faber--Krahn inequality. Every exponent $m>1$, fractional order …

We prove uniqueness, up to a positive factor, of positive harmonic functions outside a proper closed convex cone with nonempty interior for every nondegenerate symmetric stable operator. The spectral measure may be singular or supported on finitely many directions. Each …

We prove convergence from regional nonlocal diffusion to the local Shigesada--Kawasaki--Teramoto system with superlinear transition rates. For any number of species and any fixed exponent greater than one, the given nonnegative strong nonlocal solutions admit a subsequence converging strongly in …

We consider the infinity fractional evolution in dimension $n\ge2$, with $1/2<s<1$. At a zero spatial test gradient, the viscosity subsolution condition uses the supremum of symmetric directional integrals, and the supersolution condition uses their infimum. For this convention, comparison fails …

We construct a positive monostable reaction and two disjoint disks for which an exterior reaction--diffusion equation with zero Dirichlet boundary data has two distinct bounded positive equilibria. This gives a negative answer to the universal two-obstacle uniqueness question posed by …

We consider $u_t=\nabla\cdot(u^{m-1}\nabla(-\Delta)^{-s}u)$ on $\R^N$. For $N\ge2$, $m>2$ and $1/2<s<1$, we prove that a global bounded energy weak solution with nonzero compactly supported bounded integrable data cannot have a jointly continuous representative at all positive times. The pressure gradient is …

We construct a connected unbounded uniformly smooth domain in $\R^{21}$ whose bottom Dirichlet eigenvalue is $1$, but on which $-\Delta u=u-u^2$ has a bounded positive solution with zero boundary values. Thus equality between the spectral threshold and the derivative of …

We consider interior full-order Sobolev estimates for concave, translation-invariant nonlocal equations with even kernels comparable to that of the fractional Laplacian. Such estimates are required to be uniform over the kernel family and to depend on the forcing only through …

We construct a connected unbounded domain in $\R^{21}$ with uniformly $C^{2,1/2}$ boundary on which the Dirichlet Fisher--KPP equation has two distinct bounded positive equilibria. The boundary consists of two smooth global graphs whose separation tends to the critical slab width. …

We prove continuity of the finite-cluster mass at the gelation time for every weak Flory solution with a continuous symmetric kernel comparable to $x^\alpha y+xy^\alpha$, where $0<\alpha<1$, and a finite initial measure of unit first moment. No higher initial moment …

We construct two nonnegative Schrödinger operators on the line with smooth bounded potentials and spectral bottom zero. One is subcritical and the other null-critical, while the quotient of their diagonal heat kernels has infinite limsup. This gives a negative answer …

We study renewal pinning with independent standard Gaussian disorder and gap law $K(n)=n^{-5/4}/\zeta(5/4)$. For every inverse temperature $\beta>0$, we prove that almost-sure extinction of the normalized free partition function at the annealed field $-\beta^2/2$ implies a strictly larger quenched critical …

Does Fitch parsimony on a tree whose leaves are all the same distance from the root reconstruct the root state at least as accurately as any single leaf? We give a negative answer for the symmetric twenty-state model. A rooted …

We consider a catalytic surface model on \(\Z^2\) with CO adsorption, diatomic oxygen adsorption, finite-rate reaction and finite-rate stirring. If the CO arrival rate \(p\) exceeds the oxygen parameter \(q\), we prove that every site is eventually occupied by CO …

For the directed polymer based on simple random walk in $\Z^3$ with iid standard Gaussian disorder, let $p^*(\beta)$ be the supremum of the orders for which the normalized partition function has moments bounded uniformly in the polymer length. We prove …

We prove a Weyl formula with remainder $O(r^{d-1})$ for the interior transmission problem with a constant refractive index $n>1$ in an arbitrary bounded smooth domain in $\mathbb R^d$, $d\ge2$. The count includes every nonzero complex energy of modulus at most …

Small-time local controllability asks whether a system can reach a neighborhood of an equilibrium in arbitrarily short time. We give a negative answer to the finite-jet determination question for real-analytic systems with bounded measurable controls. Starting from a previously constructed …

We construct a spatially homogeneous, radial, nonnegative smooth solution of the full-sphere hard-sphere Boltzmann equation whose initial value is a Gaussian but whose velocity tail is a positive multiple of \(r^{-6}\) at every positive time. Mass is one, and energy …

We prove that the separated spherical-strip operators with PT-symmetric Robin conditions have real spectrum for every half-width $0<a<\pi/2$, every real boundary parameter and every integer Fourier mode. The proof establishes strict interlacing between the associated real Robin and Dirichlet spectra …

We study the zero center-of-mass diffusion limit for the full kinetic FENE dumbbell model on the three-dimensional torus. For every solvent viscosity $\nu>0$, FENE exponent $k>2$, smooth divergence-free initial velocity, and smooth normalized initial density with bounded positive Maxwellian quotient, …

We construct a finite-time singularity for a three-dimensional focusing Schrödinger envelope coupled to an ionization density. For one strictly positive ionization coefficient, with cubic absorption and no avalanche term, a nonzero compactly supported smooth optical datum and zero initial density …

For every smooth compact connected embedded boundary of positive dimension, the first positive Steklov eigenvalue of its smooth embedded fillings is unbounded, provided the ambient codimension is at least two and a filling exists. The boundary remains fixed pointwise, and …

Let \(0<\lambda<1\) and let \(u\) be a convex Alexandrov solution on a bounded convex planar domain, with Monge--Ampère density between \(\lambda\) and \(\lambda^{-1}\). We prove that its full distributional Hessian belongs locally to strong \(L^{p_\lambda}\), where \(p_\lambda=(1+\lambda^2)/(1-\lambda^2)\). This implies the …

We give a negative answer to a proposed universal error estimate for approximate small-BV hyperbolic flows. For a smooth strictly hyperbolic two-component system with both fields genuinely nonlinear and a uniformly strictly convex entropy, we construct one fixed initial datum …

We prove norm inflation in the homogeneous Sobolev space $\dot H^{1/4}(\R^3)$ for the defocusing quintic Gabitov--Turitsyn equation with averaging interval $[0,1]$. A family of Schwartz initial data has arbitrarily small initial norm, while its global smooth solutions attain arbitrarily large …

We prove uniform vanishing-viscosity convergence for the local mean field game system with quadratic Hamiltonian and linear density coupling on a flat torus in dimensions one, two and three. For every smooth nonnegative probability initial density, smooth fixed terminal cost …