Papers

We formulate an improved version of the monotonicity method for shape reconstruction in the inverse problem of linear elastostatics. We show that this method can reconstruct inclusions in the second Lamé parameter, i.e., the shear modulus, independently of inhomogeneities in the first Lamé parameter. This improves on earlier methods that only recover inclusions in the bulk modulus under certain assumptions on the local interplay between the Lamé parameters. Our proofs are based on linearization and divergence-free localization of energy using harmonic vector fields.

We study the stability of one-dimensional planar isotropic–nematic interfaces in the Landau–de Gennes model with anisotropic elastic constant $L$. Earlier work proved the instability for $L<0$ only under an extra condition. We remove this condition and prove that any non-negative minimizer of the reduced energy within the diagonal planar class is unstable under general one-dimensional perturbations throughout $-\frac{3}{2}<L<0$. This result shows that $L=0$ is the sharp endpoint of the instability range from the negative-$L$ side: the critical operator is non-negative at $L=0$, while instability holds over the full negative-$L$ range of the reduced problem. We also define the optimal stability index and correct a factor-of-two normalization inconsistency in a previously stated interface profile.

We study an abstract class of bilinear infinite-dimensional systems that arises in various applications, including district heating systems and quantum control. First, we analyze the existence of mild and classical solutions for abstract bilinear systems with admissible input operators in Banach spaces and study continuous dependence on the data. When the underlying space is a Hilbert space, these results are used to show passivity for a suitably defined co-located output and for mild solutions. The results are applied to the bilinear Schrödinger equation, the Fokker–Planck equation, and a district heating or cooling pipe.

For the first time, we establish higher integrability of the gradient and local $L^\infty$ estimates of weak solutions to the $p(x)$-Laplacian without assuming $\log$-Hölder continuity of $p$. Instead, we establish these results for exponents satisfying a generalized Muckenhoupt condition. This admits discontinuous exponents acting as pointwise multipliers of the BMO class. Our framework bridges the gap between classical $p(x)$-regularity and weighted Muckenhoupt theory, providing a new foundation for the analysis of differential equations with highly irregular non-standard growth.

In this paper, we establish variational formulas and a rearrangement inequality for principal eigenvalues of asymmetric second-order difference operators on periodic lattices. For a general irreducible nearest-neighbor operator, positive right and left eigenvectors give an explicit saddle point and hence equal minimum–maximum and maximum–minimum formulas over positive profiles and periodic logarithmic correctors. The corrector is unique and satisfies a nonlinear discrete flux-conservation law. For exponentially tilted symmetric diffusion, the formula separates the discrete Dirichlet and potential terms from an exact nonlinear periodic correction. In one dimension the flux is constant, and an elementary scalar-flux representation yields a bell-shaped cyclic rearrangement that maximizes the tilted principal eigenvalue for every tilt.

The weak-strong uniqueness of bounded solutions is established for a broad class of cross-diffusion systems, with or without volume-filling effects, and with or without full coercivity. Compared to existing results, the regularity and positivity assumptions on the strong solution are significantly relaxed. The analysis also extends to non-coercive systems, with the volume-filling constraint compensating for the lack of coercivity. The proof is based on two key ideas: The positivity condition is eliminated through the construction of a glued entropy, while the regularity requirement can be relaxed by using the gradient estimate and the Gagliardo-Nirenberg inequality.

We study the one-dimensional strong-weak Lotka-Volterra competition-diffusion system \[ u_t=u_{xx}+u(1-u-av),\qquad v_t=dv_{xx}+rv(1-v-bu), \] with compactly supported initial data under the parameter condition $0<a<1<b$. Existing literature only gives leading-order spreading speed asymptotics without refined logarithmic corrections for wave fronts over the full parameter space. We convert the competitive system into an equivalent cooperative parabolic system and establish moving-domain comparison principles. Based on heat-kernel estimates, we derive sharp logarithmic asymptotic expansions for the rightmost level set of the stronger species $u$. According to the magnitudes of three characteristic speeds, five long-time dynamical regimes are classified, including pulled, nonlocally pulled, pushed and critical transition fronts, with explicit logarithmic and double-logarithmic phase corrections. Our results capture delicate long-time phase offsets of wave profiles neglected in previous leading-order theories.

We prove a nonlinear parabolic characterization of stochastic completeness at infinity for weighted graphs. For the filtration equation \[ (\partial_t + \Delta \Phi)u =0 \] where $\Delta$ is the non-negative formal graph Laplacian and $\Phi u =\phi \circ u$ with $\phi \colon \R\to\R$ nonconstant, continuous and increasing, stochastic completeness at infinity is equivalent to uniqueness of bounded pointwise solutions for every bounded initial datum. For \(\Phi=\id\), this recovers the classical heat equation characterization of stochastic completeness at infinity, and of stochastic completeness when the killing term is trivial, i.e., when \(\kappa=0\). If stochastic completeness at infinity fails, then every bounded initial datum admits infinitely many bounded pointwise solutions of the filtration equation. Admissible nonlinearities include the signed porous medium and fast diffusion powers $\phi(s)=s|s|^{m-1}$ for all $m>0$, as well as many others. Stochastic completeness at infinity is further characterized by a generalized mass balance: the total mass of a positive pointwise solution at time $t$, augmented by the mass $\int_0^t\sum_{x}\kappa(x)\phi(u(s,x)) \dd s$ dissipated by the killing term $\kappa$, equals the initial mass. This balance holds for every bounded positive solution on graphs of finite measure and for bounded finite-mass data on graphs of arbitrary measure under the sharp condition $\limsup_{r\to0^+}\phi(r)/r<\infty$. It also extends to positive pointwise solutions in $\ell^1$ that are bounded on every positive time interval. When the killing term is trivial, stochastic completeness at infinity reduces to stochastic completeness and generalized balance to conservation of mass.

In this paper, we establish the Weinstock inequality for the first non-zero Steklov eigenvalue on star-shaped mean convex domains in hyperbolic space $\mathbb{H}^n$ for $n\geq 3$. We note that when $n\geq 4$, the result was obtained in our previous paper [23]. In particular, when the domain is convex, our result gives an affirmative answer to Open Question 4.27 in [13] for the hyperbolic case.

We consider the fractional Lane-Emden equation with a nonlocal Neumann condition in a half-space. We establish the existence of non-constant solutions for the critical problem in any dimension. On the other hand, we show that, in dimension $n=1$, the subcritical problem admits only the trivial solution. This result follows from a new Pohozaev-type identity, which we obtain by using suitable decay estimates for the solutions.

We consider the three-dimensional defocusing cubic nonlinear Schrödinger equation with variable coefficients, a magnetic potential, and a non-negative localized damping term, \[ i\partial_tu+(\nabla-iA)\cdot G(\nabla-iA)u+ia(x)u=|u|^2u, \qquad t>0,\quad x\in\mathbb R^3. \] No non-trapping condition is imposed on the metric $G$. Instead, the variable-coefficient region is assumed to be contained in the effective damping region. Under a one-centre condition on the tangential magnetic field, we prove global well-posedness for initial data in $H^{1+\varepsilon}$, uniform mass and energy bounds, and show the local energy decay. To obtain scattering, we impose a support condition on the full magnetic field inside the damping region. Under these stronger assumptions, the solution scatters to a free Schrödinger evolution in $H^s$ for every $0\le s<1$. The appendix discusses a separate constant-damping framework for abstract Hamiltonians.

We determine the existence exponent for the global small data solutions for a class of evolution equations with power nonlinearity under the effect of a dissipation that dampen the oscillations at low frequencies and produce overdamping at high frequencies. In the case of the wave equation with viscoelastic dissipation, the counterpart of blow-up in space dimension $n=3$ has been very recently proved by Wenhui Chen, showing that the existence exponent is optimal.

We construct the local-in-time solution of the generalised anisotropic direction-dependent curvature flow in dimension $d \geq 2$ forced by a white-in-time and smooth-in-space Gaussian noise. This seems to be the first construction with a white-in-time noise which also allows spatial dependence, even in the simpler case of isotropic stochastic mean curvature flow. The main difficulty is that the stochastic PDE describing the flow has a multiplicative noise depending nonlinearly on both the solution and its gradient. The key technique is a transform developed in \cite{BKMZ20} based on rough characteristics that removes this rough multiplicative term. We also illustrate the relationship of this transform with previously known special situations. As an application of the construction, we show that in a short time interval, the sharp interface limit of the stochastic Allen-Cahn equation with nonlinear diffusion and the same noise (slightly smoothened in time) is given by the above direction-dependent stochastic curvature flow.

We prove that the positive ground state $u$ of the fractional Laplacian $(-\Delta)^s$, $s\in (0,1)$, on an arbitrary open set $\Omega$ satisfies $-\Delta u>\lambda_s(\Omega)^{1/s}u$ in $\Omega$. In one dimension, this yields strong concavity and settles a conjecture of Bañuelos, Kulczycki, and Méndez-Hernández. We establish a hierarchy of pointwise inequalities comparing different powers of the Laplacian. In balls, these inequalities give a quantitative Hessian estimate for $\log u$, and hence log-concavity in every dimension. By contrast, we show that log-concavity fails in general convex domains by constructing counterexamples in sufficiently thin ellipsoids. We also extend the superharmonicity principle to ground states of $\psi(-\Delta)$ for complete Bernstein functions $\psi$.

In this work, we establish the quantitative unique continuation on both $n$-dimensional simplex lattice $\Delta^{(n)}_{N}$ and $\mathbb{Z}^n$ for arbitrary $n\geq 3$.

In this paper, we study higher regularity of homogeneous functions of the gradient of solutions to the inhomogeneous $p$-Laplace equation $\operatorname{div}(|Du|^{p-2}Du)=f$. Although a solution need not be of class $C^2$ across its critical set, its gradient is locally Hölder continuous. Suppose that $Du\in C^{0,\alpha}_{\rm loc}$ with $\alpha\le 1/(p-1)$, and let $\Phi$ be smooth away from the origin and positively homogeneous of degree $m$. We prove that $\Phi(Du)\in C^k_{\rm loc}$ whenever $m>k/\alpha$. Moreover, all its derivatives of order at most $k$ vanish on the critical set. The proof uses the intrinsic scale $r\simeq |Du|^{1/\alpha}$, Schauder estimates for a normalized uniformly elliptic equation, and an extension lemma across the critical set. We also obtain corresponding results for autonomous anisotropic equations and for elliptic and parabolic $p$-Laplace systems, under the appropriate Hölder assumption on the gradient. Finally, the same argument gives $C^k$ regularity criteria for high powers of nonnegative solutions to the porous medium equation.

We construct an explicit pure-swirl solution of the forced three-dimensional Navier–Stokes equations in a circular cylinder. For every $T>0$ and $1\le p,q<\infty$ with $1/p+1/q>1$, the force belongs to $L^q(0,T;L^p(D))$ and is smooth for $t<T$. Moreover, the same construction always yields the additional energy-class property $f\in L^1(0,T;L^2(D))$. The solution is classical on $[0,T)$, extends strongly in $L^2(D)$ to the unique Leray–Hopf solution at time $T$, and satisfies the energy equality, while $\|v(t)\|_{L^\infty(D)}\to\infty$ as $t\uparrow T$. We determine the exact mixed-norm ranges of both the force and the velocity and obtain two-sided rates for the force, the velocity supremum norm and the enstrophy. The construction refines Zhang's profile by an annular cancellation that preserves smoothness at the symmetry axis. It is a direct, self-contained sharpening of the $k=1$ part of our previous weighted construction and supersedes its endpoint discussion. Since the convection term is absorbed by the pressure, the same example applies to the forced Stokes system.

In 1965, V. I. Arnold established a structure theorem guaranteeing the existence of a foliation by invariant surfaces for general three-dimensional steady Euler flows with non-constant pressure. In this paper, we investigate what foliation structures can arise in steady Euler flows. We consider a toroidal domain foliated by the level sets of a flux function $\Psi$, and prove that every $C^{1}$ steady Euler flow $(\boldsymbol{u},p)$ satisfying the assumptions $\iota_{\boldsymbol{u}}d\Psi=0$ and $p=p(\Psi)$ admits the tangential flow representation \[\boldsymbol{u}=c_1(\Psi)\boldsymbol{\xi}^{1}+c_2(\Psi)\boldsymbol{\xi}^{2},\] for some lifted solenoidal vector fields $\boldsymbol{\xi}^{1}$ and $\boldsymbol{\xi}^{2}$ associated with a natural basis of weighted harmonic one-forms on the toroidal leaves. Moreover, the flux function $\Psi$ satisfies a single scalar equation, referred to as the normal flux equation. These characterizations reveal the general foliation structure of steady Euler flows, with the Clebsch representation and the Grad–Shafranov equation recovered as the axisymmetric special case.

We characterize minimizers in Gamow's liquid drop problem.

We prove that for every even integer $N\geq 4$ or $N\in\{3,5,7\}$, every axially symmetric solution to the $Q$-curvature-type problem $$ \alpha P_N u + (N-1)!(1-\frac{e^{Nu}}{\int_{\mathbb{S}^N} e^{Nu}dw})=0 \ \ \ \ \ \mbox{on} \ \mathbb{S}^N $$ is constant, provided that $ \alpha\ge\frac{1}{2}$ and $\alpha \not =1$. The proof starts from a Gegenbauer expansion and its weighted $\ell ^2$ estimate on coefficients. Then one key estimate is the refined estimate of the seminorm $\lfloor G\rfloor^2$ using a specially designed integration-by-parts identity for general even integer $N\geq 4$. For $N\in\{3,5,7\}$, the seminorm estimates are established via a descent to a three–dimensional nonlocal difference quotient identity.

We prove the strict log-concavity of the positive first eigenfunction \(-u\) of the \(2\)-Hessian equation and the strict $1/2$-convexity of the solution for the corresponding torsion problem in smooth bounded uniformly convex domains in $\mathbb{R}^{n}$. As applications, we establish the associated Brunn–Minkowski inequalities. We also show that this transformed-convexity phenomenon fails for \(3\)-Hessian equations by constructing, in dimension four, a smooth uniformly convex domain whose admissible zero-boundary solution has a nonconvex sublevel set.

We study data-driven prediction of coarse-grained dynamics in multiscale PDE systems. Adopting a closure-free operator-learning viewpoint, we apply a linear coarse-graining map and learn a surrogate evolution operator for the resolved field directly from filtered high-fidelity trajectories. Motivated by the Mori-Zwanzig formalism, we propose a spatiotemporal neural operator mapping a resolved history slab on $\Omega\times[-T_{\mathrm{in}},0]$ to a resolved future slab on $\Omega\times[0,T_{\mathrm{out}}]$. Spatial mixing uses Fourier convolution, while temporal mixing uses a causal kernel operator with position-attention weights on time lags. This causal temporal operator encodes finite-memory effects in the resolved dynamics while preserving the directionality of the history-to-future map. To improve rollout robustness and suppress nonconservative artifacts, we embed a flux-form inductive bias by parameterizing the windowed update in explicit divergence form. We also provide a data-driven guideline for selecting the memory length $T_{\mathrm{in}}$ via the decorrelation time of a closure-injection diagnostic computed from filtered trajectories. We validate on the coarse-grained viscous Burgers' equation, the Kuramoto-Sivashinsky equation, and two-dimensional turbulent flows, obtaining stable autoregressive rollouts with improved long-horizon accuracy and statistical fidelity.

We prove the optimal convergence rate $O(\varepsilon)$ for periodic homogenization of convex Hamilton-Jacobi equations arising from infinite systems of indistinguishable particles on the torus, under the assumption that the initial data depend only on the mean configuration. This extends the finite-dimensional result [1], which is based on the large-time behavior of the Lagrangian action metric and a curve-surgery argument. Here, these tools cannot be applied directly because minimizing curves live in an infinite-dimensional Hilbert space, where local compactness and finite-dimensional topology are unavailable. We overcome this difficulty by cutting the finite-dimensional mean-time projection of a minimizing curve and gluing the lifted pieces in the Hilbert space using the compact quotient induced by periodicity and rearrangement invariance. We conclude with an example showing that this rate is sharp.

We develop a recursive algebraic framework for the compatibility matrices arising in moving-boundary problems for the heat equation. Such matrices naturally appear in the analysis of boundary trace operators, including Dirichlet-to-Neumann maps and first-passage problems for Brownian motion. The construction is based on two recursive hierarchies: transported differential operators in the interior and boundary operators generated by successive differentiation of the boundary condition. Their interaction produces a recursive family of compatibility matrices whose size grows linearly with the order of the transported operator. The main contribution of the paper is a decomposition of every compatibility matrix into a universal reference matrix and a sequence of transport perturbations. This representation isolates the boundary geometry from the recursive transport corrections and shows that only the transport rows change at each recursive level. Consequently, the determinant of the compatibility matrix may be interpreted as the determinant of a reference matrix together with successive low-rank perturbations. The recursive construction is illustrated through explicit examples, which suggest additional algebraic structure in the reference determinants. The resulting recursive representation provides a systematic organization of compatibility matrices and establishes an algebraic foundation for future investigations of recursive determinant formulas, boundary trace operators, and Dirichlet-to-Neumann maps for moving-boundary problems.

In this work we give a counterexample to maximal $\mathrm{L}^2$-regularity in Lions' problem for divergence-form differential operators. On a bounded interval, we construct a bounded, uniformly elliptic, real scalar diffusion coefficient that is $\frac{1}{2}$-Hölder continuous in time with values in spatial $\mathrm{L}^\infty$. It can be chosen arbitrarily close to the constant coefficient of the heat equation. For zero initial data and a forcing term that is continuous in time with square-integrable spatial values, the unique Lions variational solution has a time derivative that is not square integrable in space-time. Thus $\frac{1}{2}$-Hölder continuity alone does not imply maximal $\mathrm{L}^2$-regularity, even for arbitrarily small scalar perturbations of the heat equation. The construction is based on a lacunary family of oscillatory trigonometric modes localised on shrinking time intervals. The spatial profile and the oscillatory modes, together with their first spatial derivatives, vanish at both endpoints. This permits zero extension of the counterexample to the real line. Tensorisation and localisation by parabolic rescaling then yield real symmetric isotropic counterexamples on $\mathbb{R}^d$ and on every bounded domain $\Omega\subset\mathbb{R}^d$, for all $d\ge1$.

A Lotka-Volterra type system with porous diffusion, which can be used as an alternative model to the classical Lotka-Volterra system, is under study. Multiparameter families of exact solutions of the system in question are constructed and their properties are established. It is shown that the solutions obtained can satisfy the zero Neumann conditions, which are typical conditions for mathematical models describing real-world processes. It is proved that the system possesses two stable steady-state points provided its coefficients are correctly-specified. In particular, this occurs when the system models the prey-predator interaction. The exact solutions are used for solving boundary-value problems. The analytical results are compared with numerical solutions of the same boundary-value problems but perturbed initial profiles. It is demonstrated that the numerical solutions coincide with the relevant exact solutions with high exactness in the case of sufficiently small perturbations of the initial profiles.

We prove that for every integer $N\geq 3$ and $\alpha\geq \frac{1}{2}$, Beckner's inequality \begin{equation*} \frac{\alpha}{2}\int_{\mathbb{S}^N}u(P_{N}u) dw+(N-1)!\int_{\mathbb{S}^N}u dw-\frac{(N-1)!}{N}\ln\int_{\mathbb{S}^N}e^{Nu} dw\geq 0 \end{equation*} holds for any axially symmetric $u\in H^{\frac{N}{2}}(\mathbb{S}^N)$ whose center of mass is at the origin. The proof is mainly based on a weighted $\ell ^2$ estimate on Gegenbauer coefficients and a rigidity theorem for stable critical points. Hence, we answer the generalized Chang-Yang conjecture positively in the axially symmetric case for every integer $N\geq 3$.

In this paper, we establish a system-wide $\ell^1$-norm Harnack-type inequality for positive solutions of weakly coupled nonlocal diffusion systems on $\mathbb{R}$, without assuming cooperativity or irreducibility of the coupling matrices. As a primary application, we integrate this analytical tool with a rescaling argument to establish the existence of the minimal wave speed for traveling waves for a class of non-cooperative nonlocal diffusion systems with network structures. Furthermore, we derive the precise asymptotic behavior of wave profiles at negative infinity. This is achieved by combining Harnack-type inequalities and Ikehara's theorem with Riesz projections to analyze singularities of vector-valued Laplace transforms.

This paper develops a path-space theory of entropy production for a class of stochastic evolution equations on infinite-dimensional Hilbert spaces. Since such spaces have no canonical Lebesgue reference measure, the usual finite-dimensional density formulas do not extend directly. We instead work relative to the invariant Gaussian measure of a reversible Ornstein–Uhlenbeck reference process. Combining an infinite-dimensional Girsanov transform, time reversal of the reference process, and the stationary Fokker–Planck equation relative to the Gaussian measure, we derive an explicit entropy-production formula in terms of an irreversibility field. On the natural test class, this field represents the difference between the forward and reversed nonlinear drifts. Under the standing assumptions, vanishing entropy production is equivalent to vanishing stationary probability current, self-adjointness of the generator in the invariant Hilbert space, detailed balance, and invariance of the stationary path law under time reversal. The reversible case is therefore characterized by a Gaussian-reference gradient structure for the nonlinear drift.

In this note, we prove that there exist two families of bi-rotational asymptotically conical self shrinkers with boundary called `trumpets', and two families of bi-rotational self shrinkers with boundary which close up called `disks', so that these families together with two generalized self shrinking cylinders, and a minimal cone foliate the whole space except some compact set containing the origin.