Papers

In this paper, we construct weak Lyapunov functionals for kinetic Fokker-Planck equations with weak external confinements. This problem has remained widely open in the existing literature. Apart from showing interesting structures from the models, deriving these Foster-Lyapunov weights allows us to use Harris type arguments to show existence of steady states for these equations, and convergence to equilibrium with an explicit stretched exponential rate.

We study Robin and Neumann problems for the scalar curvature equation of admissible graphs over bounded uniformly convex domains in three dimensions. Under a small-volume assumption, we prove existence and uniqueness for the Robin problem and obtain a classical Neumann solution as the Robin parameter tends to zero. The volume threshold is optimal among conditions depending only on the volume. The main step is a boundary second-derivative estimate uniform in the Robin parameter; known interior and global-to-boundary curvature estimates then give the global bound.

Let $(\mathcal{M},g)$ be a compact connected $W^{2,\infty}$ Riemannian manifold with nonempty boundary and a uniformly elliptic Lipschitz metric. We establish quantitative boundary spectral inequalities for both Dirichlet and Neumann eigenfunctions from arbitrary measurable subsets of $\partial\mathcal{M}\times(-s_0,s_0)$ of positive surface measure. Moreover, we apply the Dirichlet spectral inequality to establish the boundary observability estimate \[ \|u(T)\|_{L^2(\mathcal{M})}^2 \le C \int_J \|\partial_{\nu_g}u(x,t)\|^2 \,\mathrm{d}S_g\,\mathrm{d}t . \] where $J\subset\partial\mathcal{M}\times(0,T)$ is measurable and has positive surface–time measure. A localized theorem requiring $W^{2,\infty}$ regularity only near the observed boundary patch is retained as a separate consequence. The proof of the observability inequality combines quantitative continuation from positive-measure boundary sets with low-frequency spectral concentration. This result shows that boundary observability of solutions to the heat equation can also be achieved in $W^{2,\infty}$ domains. Finally, as an application, we prove that measurements of the boundary normal derivative on $J$ uniquely determine the initial state of heat equation. Under an a priori bound in $D(-\Delta_{g, D}^{s/2})$, the recovery satisfies a logarithmic stability estimate of optimal order $s/2$.

We construct infinitely many weak singular solutions with zero initial data and compact time support for third- and fifth-order KP-I, KP-II, and their modified counterparts on $\mathbb T^2$ and $\mathbb R^2$. Their nonlinearities are cutoff-independent, absolutely convergent Fourier convolutions. Quadratic solutions belong to $C_tL^p$ for $p<2$ and $C_t(H^{-\sigma,0}\cap H^{-\sigma})$ for $\sigma>0$. Modified solutions belong to $C_tL^p$ for $p<3$. One cubic family lies in $C_tH^\alpha$ for $\alpha<1/3$; another has parabolic Fourier support and lies in $C_tH^{s,0}$ for $s<1/2$ and $C_tH^\alpha$ for $\alpha<1/4$. The exponents $1/3$ and $1/2$ are sharp at the $L^3$ product threshold. For quadratic fifth-order KP on $\mathbb R^2$, the nonuniqueness range is almost sharp. We also construct periodic stationary KP-I and KP-II solutions and prove that $L^2$ is the sharp threshold between singular stationary KP-I solutions and smoothness.

We develop a concentration-compactness theory for geometric evolution equations of arbitrarily high order, using the geometric polyharmonic heat flow of closed immersed surfaces in \(\R^3\) as the model case. The flow is the \((2p+2)\)-order normal evolution \[ \partial_t f=(-1)^{p+1}\Delta^p H\,\nu,\qquad p\geq1, \] which includes the surface diffusion flow when \(p=1\). We prove localised energy estimates with sharp cut-off bookkeeping, interior estimates, a lifespan/concentration alternative, tracefree-curvature \(\varepsilon\)-regularity estimates, and a gap theorem for stationary solutions. These tools are then combined with a blowup argument, the preservation of signed enclosed volume, and the monotonicity of area to rule out singularities below a small tracefree-curvature threshold. Consequently, for connected initial immersions satisfying \(\|A^o\|_2^2<\varepsilon\), where \(\varepsilon>0\) depends only on the order of the flow, the solution exists for all time and converges exponentially in \(C^\infty\) to a round sphere with the preserved enclosed volume.

We prove a surprisingly simple pointwise formula for the Frobenius norm of the tensor of $n$-th order partial derivatives of a radially symmetric function $f(x)=g(r(x))$. Using the iterations of the differential operator $\mathcal{D} g(r)=g'(r)/r$, we avoid technical difficulties usually caused by higher-order radial derivatives. As a consequence, we obtain a complete characterization of the subspace of radially symmetric functions in both inhomogeneous and homogeneous Sobolev spaces of arbitrarily high order and all integrability parameters $p\in [1,\infty).$ Furthermore, the pointwise nature of our approach allows us to obtain similar results also for Sobolev-type spaces built upon more general Banach lattices.

We prove strict convexity of the square-root transformation $v=-\sqrt{-u}$ for admissible solutions of a graphical $\sigma_2$-curvature Dirichlet problem on smooth uniformly convex domains. A key ingredient is a constant-rank theorem for $D^2v$, proved by a direct Ma-Xu type argument in dimension three and by the Bian-Guan microscopic convexity principle together with inverse-convexity methods in arbitrary dimensions. Combined with boundary strict convexity and a domain-deformation argument, the constant-rank theorem yields $D^2v>0$ throughout the domain.

Concentrating (that is, non-equi-integrable) sequences of functions satisfying linear PDE constraints arise in a wide variety of problems in PDE, the calculus of variations, and geometric measure theory. In 2001, Bouchitté conjectured that such concentrations can always be represented as superpositions of “simple” concentrations, a claim he termed the Vanishing Mass Conjecture. We fully resolve this conjecture for all first-order constant-coefficient linear differential operators, proving that the value-distribution (Young) measure of any such sequence admits a Choquet-type decomposition into probability measures whose barycenters lie in the associated Tartar wave cone. In fact, we prove a significantly stronger statement than originally conjectured, namely that the constituent measures are themselves generated by concentrating sequences satisfying the same PDE constraint. This provides a complete structural description of concentrating sequences and yields several applications, including a new proof of a theorem of De~Philippis and the second author on the singular polar of PDE-constrained measures, two types of compensated compactness results, and a surprising result on the support cardinality of extremal concentration Young measures. We also place our results in the context of Morrey's Conjecture, showing that its analogue for pure concentrations is false. Our proofs introduce several new techniques, most notably convexity arguments involving “barrier functions” and a careful smoothing procedure via the heat flow that replaces classical Fourier methods.

We study the propagation direction of bistable traveling fronts in the two-species Lotka–Volterra competition–diffusion system under strong competition. A complete characterization of the sign of the wave speed has remained a long-standing unsolved problem. We establish the first global necessary and sufficient criterion for zero wave speed by combining a Maxwell-type identity with a phase-plane rigidity argument. This criterion yields a unique zero-speed threshold, identifies its value in the symmetric case and its limiting values in two extreme regimes, and consequently determines the propagation direction throughout the strong-competition parameter region.

We prove that any sufficiently slow uniform rotation stabilizes the supermassive star under axi-symmetric perturbations, and the stability persists for every adiabatic exponent in a left neighborhood of the mass-critical one. The key new ingredient of the linear analysis is a structural property of the self-similar family inherited from the mass-critical scaling: the associated self-similar direction becomes a reduced kernel direction of a modified linearized operator.

For origin-symmetric convex bodies in $\R^n$ invariant under $n-2$ orthogonal hyperplanes reflections, we establish the logarithmic Brunn–Minkowski inequality and classify all equality cases without regularity assumptions. We derive the logarithmic Minkowski inequality and rigidity for cone-volume measures, including an explicit classification for bodies of revolution and uniqueness for simplicial polytopes in the prescribed symmetry class.

We develop a parameterization of small amplitude finite gap solutions to the KP equation with prescribed spatial wavenumber vectors using classical Schottky uniformization. This parameterization is suitable for a Lyapunov-Schmidt reduction. Using this parameterization, we prove the persistence, under Hamiltonian perturbations, of periodic one-gap solutions for both KP-I and KP-II and of bi-periodic two-gap solutions for KP-I.

We consider the isotropic-nematic interface problem for liquid crystals based on the Landau-de Gennes model. We show that when the anisotropic elastic constant $L_2$ is negative, then the surface tension strength of the isotropic-nematic interface is proportional to $\sqrt{L_1+\frac{2}{3}{L}_2}$, and the homeotropic alignment is more favored near the interface. This result gives a rigorous confirmation for de Gennes' formal derivation in \cite{deGen1971} on the isotropic-nematic tension strength and the anchoring alignment condition in the case of $L_2<0$.

We prove Lipschitz, sub-Hölder and Hölder stability estimates for recovering the low Fourier modes of an unknown potential from the Dirichlet-to-Neumann (DN) map. We study three different cases, depending on the regularity of the difference $q_1-q_2$. First, we consider \[ (-\Delta - \lambda^2 + q)u=0 \quad \text{in} \quad \Omega\subset \mathbb{R}^n. \] We show that the difference $q_1-q_2$, assumed to be $M$-bandlimited, can be recovered in a Lipschitz stable way from the difference of the corresponding DN maps. This holds whenever $\lambda$ is sufficiently large relative to $M^{n/2}$. The proof involves real geometrical optics solutions. Secondly, we consider \[ (-\Delta + q)u=0 \quad \text{in} \quad (-\pi,\pi)^n. \] We show that the low Fourier coefficients of the difference $q_1-q_2$, assumed to be real-analytic and periodic, can be recovered in a sub-Hölder stable way from the difference of the corresponding DN maps. The number of recoverable Fourier modes grows as the DN maps become closer. Finally, we consider the case where the Fourier coefficients of the difference $q_1-q_2$ decay at a super-exponential rate $e^{-c|k|^{n/2}}$. We prove that the low Fourier modes can be recovered with Hölder stability, with the number of recoverable modes tending to infinity as the DN maps become closer. In all cases the $L^\infty$ potentials themselves do not need to satisfy additional assumptions or belong to a finite dimensional space. The constants in the stability estimates are uniform in the number of recovered Fourier modes.

We consider the two-dimensional Dirac operator with constant magnetic field that is perturbed by a combination of electrostatic and Lorentz-scalar delta interactions with variable coefficients supported on a smooth closed curve. Self-adjointness is studied in the so called non critical and critical cases. In the non-critical case the essential spectrum is unchanged - it remains to be the set of the Landau-Dirac levels, the eigenvalues of infinite multiplicity of the unperturbed operator - while in the critical case an additional interval of essential spectrum emerges in the spectral gap containing zero. Our main result concerns the discrete spectrum in the non-critical case: using the pseudodifferential properties of the involved boundary integral operators, we show that the eigenvalues accumulate at each Landau-Dirac level at a rate governed by the logarithmic capacity of the curve. A novel and surprising phenomenon is the change in the side of the accumulation depending on the position relative to the critical value. As a byproduct, clusters of eigenvalues for a family of exterior boundary value problems are obtained via confining couplings; the infinite-mass boundary condition arises as a special case.

We consider the Cauchy problem for semilinear classical damped wave equations with general nonnegative nondecreasing nonlinearities, without imposing a modulus-of-continuity structure or convexity. A Dini-type condition expressed directly in terms of the nonlinearity gives small data global in-time existence and extends the nonlinear framework of Ebert–Girardi–Reissig [Math. Ann. 378 (2020), 1311–1326]. When this integral diverges and the effective total mass of the initial data is positive, convergence of the Keller–Osserman tail integral is necessary and sufficient for finite-time Sobolev blow-up. Its divergence gives global grow-up under the same near-zero and mass assumptions.

In this paper we develop a Li-Yau-Hamilton (LYH) type differential Harnack estimate for positive solutions to the FitzHugh-Nagumo equation on $\mathbb{R}^n$. We then use our LYH-differential Harnack inequality to prove several properties about positive solutions to the equation, including a classical Harnack inequality and a lower bound for the speed of traveling wave solutions.

We study the Schrödinger evolution generated by the isotropic Ornstein–Uhlenbeck operator \[ \mathscr L=\Delta-\langle x,\nabla\rangle \] in $L^2(\mathbb R^d,d\gamma)$, where $d\gamma$ is the invariant Gaussian measure. Combining the classical lens correspondence between the free Schrödinger equation and the harmonic oscillator with Gaussian conjugation, we identify the free Schrödinger structure hidden in the Ornstein–Uhlenbeck evolution. The main consequence is a dynamic restriction theorem of spacetime type. After an explicit transformation of spacetime variables, the adjoint Ornstein–Uhlenbeck extension operator becomes an ordinary spacetime Fourier transform restricted to the classical Schrödinger paraboloid. On the Ornstein–Uhlenbeck side this corresponds to an explicitly time-dependent Gaussian spacetime measure. Thus a restriction geometry which is neither suggested by the symbol of $\mathscr L$ nor selected by a scaling symmetry of the Ornstein–Uhlenbeck equation emerges dynamically from the free Schrödinger representation. The same correspondence identifies the weighted Gaussian spaces in the homogeneous and inhomogeneous Ornstein–Uhlenbeck Strichartz estimates and transports the mass-critical nonlinear Schrödinger equation to \[ i\partial_tu+\mathscr Lu = \mu e^{-|x|^2/d}|u|^{4/d}u. \] We also obtain a global description of the unitary Ornstein–Uhlenbeck group, including its oscillatory representation away from the caustics and its exact values at the caustic times, together with the associated sharp weighted dispersive estimate and Hardy-type uncertainty principles.

In this paper we prove matrix weighted Sobolev and Poincaré inequalities using techniques derived from the theory of Rubio de Francia extrapolation. Given weights $w,\,v$ and a symmetric non-negative definite matrix valued function $Q$ defined on a connected open subset $\Omega$ of $\mathbb{f}R^n$ that satisfies the lower ellipticity condition \[ w(x)^p \leq |\sqrt{Q(x)}\xi|^p,\quad \xi\in \mathbb{R}^n, \] we give Lebesgue integrability conditions on the weights $w,v$ that ensure there exists $\tau\geq 1$ so that Sobolev and Poincaré inequalities of the form \[\bigg(\int_\Omega |u|^{\tau p} \,vdx\bigg)^{\frac{1}{\tau p}} \leq C(v,w) \bigg(\int_\Omega |\sqrt{Q}\nabla u|^p\,dx\bigg)^{\frac{1}{ p}},\textrm{ and}\] \[\bigg(\int_\Omega |u-\langle u\rangle_{\Omega,v}|^{\tau p} \,v dx\bigg)^\frac{1}{\tau p} \leq C(v,w)\bigg(\int_\Omega|\sqrt{Q}\nabla u|^p \, dx\bigg)^{\frac{1}{p}}\] hold for smooth $u$. We explore these and related results in the context of several examples that include John domains, the Heisenberg group, and CR manifolds.

Let $\Omega\subset\mathbb{R}^d$, $d\ge2$, be a bounded connected domain with boundary of class $C^{1,\alpha}$, where $0<\alpha<1$. For the adjoint Neumann–Poincaré operator $K^*_{\partial\Omega}$, normalised so that its distinguished eigenvalue is $1/2$, let $\lambda_j^+(\Omega)$ denote the upper min–max values on the mean-zero energy space. We prove $\sum_{j=1}^{d}\lambda_j^+(\Omega)\ge \frac{d-2}{2}.$ It follows that $\lambda_1^+(\Omega)\ge \frac{d-2}{2d},$ with equality if and only if $\Omega$ is a ball. In dimension three this proves the $1/6$-conjecture of Miyanishi and Suzuki. The proof uses the coordinate boundary-charge densities induced by uniform applied fields. Their energy Gram matrix is the perfect-conductor polarization tensor $M_\infty$. Positivity of a $2d\times2d$ Gram matrix yields the endpoint Hashin–Shtrikman inequality $|\Omega|\mathrm{Tr}(M_\infty^{-1})\le1,$ and bounds the trace of the compression of $K^*_{\partial\Omega}$ to the applied-field space. Equality in the inverse-trace inequality makes the interior Newtonian potential quadratic, so a converse to Newton's theorem identifies $\Omega$ as an ellipsoid. Equality in the spectral estimate also makes the Hessian of this potential isotropic, which forces the ellipsoid to be a ball.

We study $G$-equivariant Hörmander pseudodifferential operators on a noncompact symmetric space $G/K$. We define a notion of a complete symbol function, called the Harish-Chandra symbol function, on the spherical tempered dual of $G$, for operators that satisfy a rapid off-diagonal decay condition on their Schwartz kernels, and we characterize the class of such operators in terms of their Harish-Chandra symbol function.

We consider a reference wave, a radiation solution, and the sum of these solutions (total solution) for the Helmholtz equation in an exterior region. We give two-point formulas for approximate phase recovering of the radiation solution from the intensity of the total solution for the case of spherical reference wave. We show that these formulas can be used, in particular, for approximate phase recovering from holographic data on a single plane. By these formulas, we continue previous studies with plane wave reference.

In this paper, we show that the codimension of the singular set of a stable stationary harmonic map to a round $k$-sphere is at least $k+1$ when $k$ is between 3 and 6, and is at least 7 when $k$ is at least 7. The result is sharp in the sense that there exist energy minimizing 0-homogeneous maps in the aforementioned critical dimensions. We also establish non-trivial index bounds for the non-constant harmonic maps from $n$-spheres to $k$-spheres, provided that $n$ is less than $k$.

We study the role of Banach space geometry in deterministic and stochastic maximal regularity. We first construct an example showing that the UMD assumption in the characterisation of maximal $L^p$-regularity in terms of $R$-sectoriality cannot be omitted. Combining this construction with an equivalence between stochastic maximal regularity and deterministic maximal regularity on the $2$-concavification of the underlying space, we obtain an operator on a UMD Banach function space of type $2$ that has a bounded $H^\infty$-calculus of angle zero, but fails stochastic maximal $L^p$-regularity (SMR$_p$) for every $p\in[2,\infty)$. Motivated by this example, we study the Banach space geometry hypothesis underlying SMR$_p$ more closely. This is an $R$-boundedness condition $(S_p)$ for stochastic convolution operators. For UMD spaces $X$ of type $2$, we show that this condition is not only sufficient, but also necessary for two canonical test operators: a diagonal multiplier on a Rademacher space and, for $q>2$, the Laplacian on $L^q(\mathbb R^d;X)$. Finally, we prove that its interval-kernel and exponential-kernel formulations are equivalent and that, at the endpoint $p=2$, condition $(S_2)$ holds if and only if $X$ is isomorphic to a Hilbert space.

A notable feature of confined active matter systems is the tendency for motile particles to accumulate near solid boundaries. In various linear models with no-flux boundary conditions, this accumulation is realized through the development of sharp boundary layers at small particle diffusivity $\kappa$. In this paper, we present the first rigorous investigation of nonlinear boundary layers in the context of confined active matter. Specifically, we consider a family of 1D run-and-tumble models with nonlinear advection and tumbling on the half-line $\mathbb{R}_+$. We rigorously prove the vanishing diffusivity limit with quantitative convergence rates. In the limiting system, the boundary mass enters as a new variable which solves a nonlinear ODE, coupled to the PDE through a dynamic boundary condition. Interestingly, the nonlinearity on the boundary at $\kappa = 0$ cannot be obtained without reference to the boundary layer analysis at $\kappa \ll 1$. Numerically, these models exhibit rich behavior, including phase transition and hysteresis in the boundary layer.

We present a one-dimensional model for water infiltration coupled with a hydration reaction, relevant to coupled transport and chemical processes in the Earth's subsurface. In this model the sharp interface separating the saturated and dry regions evolves over time, leading to a moving free boundary problem. Consistent with recently presented numerical treatments in the literature, this solution indicates an interesting dynamic for the free boundary. At early time, for a given domain porosity, the infiltration front advances with a specific square-root-in-time behavior. At later time, depending on the consumption of the hydration reaction, the front advance can exhibit square-root, linear, or exponential forms. The presented closed solution provides an analytical tool that can be used to quantify important behavior in coupled transport and reaction systems.

In this note, we prove a priori interior curvature bounds for strictly convex solutions to elliptic Weingarten curvature quotient equations. The proof does not employ the advanced methods involving integral estimates or compactness arguments. Instead, it relies on a concavity inequality for the equation operator and a novel choice of the auxiliary function to carry out an elementary maximum-principle argument. Interior curvature estimates for strictly convex solutions to the special Lagrangian curvature equation in low dimensions also follow as a consequence.

In this article, we have studied a particular criterion to establish if a particular class of nonlinear PDEs $\mathcal{O}_t u=\mathcal{O}_x u $ in unidimensional space-time $(x,t)$ admits elementary solutions respect to the spatial coordinate $x$, through the nested derivative method. Where a nested derivative is a generalization of the usual partial derivative of a order higher than one and, in terms of differentials operators, is a generalization of the Laguerre operator. This method, integrated with Liouville's theorem and Galois theory and method by integrate by parts, establishes if solution of nested derivative equations admits elementary solutions respect with $x$. These solutions could help to study better evolutions of possible physical systems in space-time.

We construct a random, time-dependent divergence-free velocity field on $\mathbb{T}^3$—refreshing iid on finite time blocks and obeying deterministic $C^\infty_{t,x}$ bounds—that exhibits fast dynamo behavior. That is, for every fixed, sufficiently small resistivity, the almost sure exponential growth rate of the magnetic field solving the associated linear resistive induction equation is at least $1/2$; the exceptional null set may depend on the resistivity. We in fact get a time-uniform lower bound—with a random prefactor obeying a uniform-in-$\kappa$ inverse moment bound. The argument relies on a particular algebraic structure in Fourier space of the induction equation solution operator that allows us to propagate expected growth of the logarithmic size of three specially chosen Fourier modes. This allows us to reduce to a simple recursion, avoiding the complicated infinite-dimensional dynamics typical to the dynamo problem. The central proof idea was generated autonomously by ChatGPT 5.6 Sol Ultra; the manuscript was written (and verified) by the author.

Backstepping control provides local stability guarantees for nonlinear Saint-Venant systems, but its regulation performance may degrade when the system operates far from the nominal equilibrium. This letter proposes a backstepping-guided soft actor-critic (SAC) controller framework that incorporates model-based control knowledge into reinforcement learning (RL). The nominal backstepping control law is first learned by deep operator network (DeepONet) and embedded into the actor and critic networks as prior informed feature representations. The learned prior is further combined with the SAC policy to generate the final control input, while a transfer-learning strategy preserves the useful backstepping knowledge during adaptation to the nonlinear dynamics. Simulation results on the Sambre River model demonstrate that the proposed method improves learning efficiency and maintains effective regulation over larger initial deviations than backstepping control.