Papers

We develop a variational theory of structured deformations for energies whose surface densities satisfy general growth conditions. This requires a formulation in the generalised space ${\rm GBV}_\star$, introduced by Dal Maso and Toader, which is the natural setting for surface energies that are linear near the origin and bounded at infinity. In this framework, we prove three main results: an approximation theorem for structured deformations, an integral representation theorem for abstract lower semicontinuous functionals, and an explicit representation formula for relaxed energies. The proofs rely on new density results for functions of bounded variation and on Poincaré-type inequalities tailored to ${\rm GBV}_\star$. Our results extend the applicability of structured deformations to cohesive models in fracture mechanics.

This paper addresses the backstepping boundary stabilization of coupled multidimensional first-order hyperbolic systems. We consider systems whose transport velocity fields are collinear, meaning that each velocity field is a scalar multiple of a common base velocity field. Building upon a recent framework developed for scalar multidimensional first-order hyperbolic equations, we introduce a change of variables, based on characteristic curves defined entirely in the spatial domain, that converts the original multidimensional system into a continuum of coupled one-dimensional first-order hyperbolic systems. By designing a backstepping controller for each system in the continuum representation, and assuming that the transit times of the characteristic curves are uniformly bounded, we achieve finite-time stabilization of the multidimensional system.

We study the Cauchy problem for inhomogeneous evolution equations with time-dependent, potentially degenerate, and unbounded coefficients. A key feature of our work is allowing the principal coefficients to undergo arbitrary blow-up at both the initial and terminal times.

We consider an inverse boundary value problem for the time-independent Schrödinger equation in dimension three. We prove that the local Dirichlet-to-Neumann map defined near a boundary point uniquely determines the potential in a neighborhood of the boundary point in the interior. In particular, we show that the uniqueness question can be reduced to the injectivity of a weighted X-ray transform, which links inverse boundary value problems to integral geometry.

We investigate a singular limit of a system of Patlak-Keller-Segel (PKS) equations modeling the evolution of multiple interacting species. Our primary motivation is the Differential Adhesion Hypothesis (DAH), introduced by Malcolm Steinberg in 1962, which posits that cell populations self-organize by minimizing adhesion energy, in a manner analogous to fluids minimizing surface tension. Our starting point is a continuum model describing the evolution of the density distributions of $N$ distinct species, representing different cell types. These species interact through attractive nonlocal forces governed by interaction kernels of similar form but different strengths (the $N\times N$ matrix of interaction coefficients encodes the key properties of the system). These attractive forces are balanced by a nonlinear pressure, depending on the total density and strong enough to prevent concentration. In the limit of short-range interactions, we establish sufficient conditions on the interaction matrix for cell sorting to take place (i.e., the spontaneous separation of the different species). We then prove a general $\Gamma$-convergence result for the associated energy functional, showing convergence to an interfacial energy where the surface tension coefficients are determined by a geodesic problem. A detailed analysis of this problem allows us to identify regimes in which engulfment occurs (more adhesive species cluster together and are surrounded by less adhesive ones) which is a key feature of the DAH. In the second part of the paper, we analyze the asymptotic behavior of solutions to the PKS system in the combined long-time and short-range interaction limit. Under a standard energy convergence assumption, we prove the convergence to a multiphase Hele–Shaw problem with surface tension (and contact angle conditions at triple junctions).

We establish higher order boundary Harnack estimates for solutions to kinetic Fokker-Planck equations with absorbing incoming boundaries. Unlike classical elliptic and parabolic equations with Dirichlet data, we show that the quotient of two solutions for kinetic equations is not $C^{\infty}$ up to the boundary. Instead, we develop a general theory showing that, near the grazing set, the quotient of two solutions is $C^{3/2}$ if the domain and data are sufficiently smooth, and $C^{1,1}$ in the absence of source terms. These exponents are optimal.

Our aim in this paper is to study principal eigenvalues, maximum principles and estimates for Lane-Endem related systems in nondivergence form.

We investigate the high Mach number limit for the scaled compressible Navier–Stokes system in the critical Besov framework. In the scaled momentum equation, the pressure force is represented by the term \(\varepsilon^2\nabla a^\varepsilon\), where $\varepsilon$ is the inverse Mach number; as \(\varepsilon\to0\), the formal limiting system is the compressible pressureless Navier–Stokes system. The analysis is complicated by the absence of density dissipation in the limiting model and by the highest-order coupling created by the viscous terms. For \(d\geq2\), we prove the global well-posedness of the scaled system for small initial data and obtain estimates that are uniform with respect to $\varepsilon$. A crucial ingredient is a parameter-dependent lower-order estimate for \(\varepsilon a^\varepsilon\), which compensates for the purely transport nature of the density equation and allows the uniform bounds to be closed. Based on these estimates, we justify the high Mach number limit and recover a global strong solution to the pressureless Navier–Stokes system. For \(d\geq3\), we further derive quantitative error estimates between the scaled solutions and the pressureless limiting solution. More precisely, on each fixed finite time interval, if the initial discrepancy is of order \(\mathcal{O}(\varepsilon)\), then the corresponding lower-order critical Besov error satisfies the same rate, which yields a quantitative justification of the pressureless limit.

We describe the linearized dynamics near the Lamb-Chaplygin dipole, a classical traveling solution of the two-dimensional Euler equations. Exploiting the Hamiltonian structure of the system together with its symmetries, we identify all possible sources of linear instability. For general perturbations in $L^1\cap L^p$, $p>2$, growth can occur through two explicit mechanisms triggered by: (i) a nonzero circulation on the core of the dipole, and (ii) a nontrivial component along the generalized eigenvectors associated with the eigenvalue $0$. In particular, we completely classify the spectrum and the Jordan chains of the operator associated with the linear dynamics. Both mechanisms hint for a nonlinear dynamics that may drift along the symmetry-generated family of traveling dipoles without moving away from it.

We determine the complete Benjamin-Feir spectrum near the origin for small-amplitude hydroelastic Stokes waves of the two-dimensional finite-depth irrotational Euler equations with surface tension and elastic bending. For the non-resonant Stokes branch and away from an intrinsic characteristic-collision surface $\mathfrak D$, we resolve all four Bloch eigenvalues bifurcating from the origin in the long-wave Floquet regime. Exploiting the Hamiltonian and reversible structure of the problem, we reduce the linearized Bloch operator to the four-dimensional spectral subspace bifurcating from the generalized kernel at the origin and conjugate the resulting matrix to the direct sum of a Benjamin-Feir block and a long-wave block. The long-wave pair remains purely imaginary, whereas the Benjamin-Feir pair is governed by an explicit closed-form instability index $\operatorname{Ind}(\mathtt{h},\kappa,b)$: a positive index produces a local figure-eight spectral curve with nonzero real part, while a negative index implies that all four small eigenvalues remain purely imaginary. Together with the Wilton-type resonance loci and the characteristic-collision surface $\mathfrak D$, this index yields a three-parameter spectral-stability diagram in the depth $\mathtt{h}$, surface tension $\kappa$, and bending rigidity $b$. The diagram recovers the classical pure-gravity critical-depth limit and, on the zero-bending boundary, the gravity–capillary stability diagram. It also reveals a genuinely hydroelastic phenomenon: all Wilton-type resonances disappear whenever $b\geq 1/14$ or $\kappa\geq 1/2$. This provides the first complete rigorous characterization of the local Benjamin-Feir spectrum for a hydroelastic free-boundary problem.

In this paper, we resolve the question of whether the Neumann problem for the parabolic PDE $-\partial_tu + \mathrm{div}(A\nabla u)=0$ on a Lipschitz cylinder $\mathcal O\times\mathbb R$ is solvable for some $p\in (1,\infty)$ under the assumption that the matrix $A$ is elliptic with bounded and measurable coefficients that satisfy a natural Carleson condition (a parabolic analog of the so-called DKP-condition). We prove that for any $1<p<\infty$ the Neumann problem is solvable under the assumption that both the Carleson norm of coefficients and the Lipschitz constant of the domain are sufficiently small (with dependence on $p$). The question of what happens in the "large Carleson norm/large Lipschitz constant" regime remains open, and even for elliptic PDEs this question has only been resolved in two dimensions. This paper complements results from our recent manuscript (by the same authors) in which the parabolic regularity problem has been fully resolved in both the small and large Carleson norm regime. Previously, the Dirichlet problem had been resolved under the same conditions by various authors.

We prove a sharp quantitative stability result for the affine \(L^p\)-Sobolev inequality, for \(p\ge2\), introduced by Lutwak–Yang–Zhang (J. Differential Geom., 62 (2002), 17–38). Moreover, the stability exponent is shown to be optimal, and equal to \(p\).

We study an initial-boundary value problem for a Zakharov type system in three space dimensions in a general domain. Under a smallness assumption on the initial data, we construct a unique global strong solution. The solution is obtained by a direct approach based on higher-order energy estimates and the construction of a Cauchy sequence in suitable Banach spaces, without employing compactness methods. Furthermore, we obtain estimates on the growth of higher-order Sobolev norms of solutions.

We develop an inhomogeneous form of Edwards' Jensen-measure duality for Perron envelopes constrained by Monge–Ampère lower bounds. The admissible subsolution families are convex but not cones; nevertheless, the dual measures remain the homogeneous Jensen measures, and the right-hand side enters through a scalar Jensen deficit \[ B_{\mathcal{A}}(x,\mu) = \inf_{u\in\mathcal{A}} \left(\int_{\partial\Omega}u\,d\mu-u(x)\right). \] Under natural structural hypotheses we prove a boundary dual formula \[ \sup\{u(x):u\in\mathcal{A},\ u\leq\varphi\text{ on }E\} = \inf_{\mu\in J_x^\partial} \left( \int_{\partial\Omega}\varphi\,d\mu - B_{\mathcal{A}}(x,\mu) \right). \] We apply the theorem to real Alexandrov subsolutions and to complex Bedford–Taylor plurisubharmonic subsolutions with continuous density. In one real dimension the deficit is the Green-potential correction; in higher dimensions it has intrinsic stress and current interpretations. On B-regular domains, a bounded Bedford–Taylor approximation theorem identifies bounded and continuous competitors and yields a duality proof of continuity for the corresponding Dirichlet solution. Finally, for smooth strictly elliptic solutions, optimal Jensen measures are the harmonic measures of the linearized Monge–Ampère operators, equivalently the boundary derivatives of the nonlinear solution map.

We study the Cauchy problem for a mixed-sign quadratic Dirac equation on a noncompact $N$-star metric graph $G$, \[ \mathrm{i}\partial_t \psi = D\psi - \mathcal N(\psi), \qquad \psi(0)=\psi_0, \] where $\psi=(\psi_1,\psi_2)^{\mathsf T}:\mathbb{R}\times G\to\mathbb{C}^2$ and $D$ denotes the self-adjoint Dirac-Kirchhoff operator on $G$. The nonlinearity acts edgewise and is given by a bilinear interaction between the positive and negative spectral parts, \[ \mathcal N(\psi)=\mathcal B\bigl(\Pi_+\psi,\Pi_-\psi\bigr), \] where $\Pi_\pm$ are the spectral projections of $D$ and $\mathcal B$ is a fixed bilinear map on $\mathbb{C}^2$ applied componentwise on each edge. This is a model quadratic interaction tailored to the mixed-sign Bourgain-space mechanism, rather than a general nonlinear Dirac equation on graphs. Using Bourgain-type spaces associated with the spectral resolution of $D$ and a mixed-sign bilinear estimate on $N$-star graphs, we prove local well-posedness in the operator Sobolev space $H_D^s(G)$ for \(s>-\frac18\). We also establish a blow-up alternative in $H_D^s(G)$ for the maximal forward lifespan.

We investigate the asymptotic behavior of the complex roots of Rayleigh-Lamb dispersion relations arising in elastic waveguides. We consider Neumann, Dirichlet, and fluid boundary conditions and derive explicit asymptotic expansions of the associated wavenumbers as their modulus tends to infinity. In the Neumann case, we provide a rigorous justification of asymptotic formulas that have long been used in the literature without proof, together with higher-order terms and explicit constants. Similar results are obtained for Dirichlet and fluid boundary conditions. The analysis relies on a reformulation of the dispersion relations as zeros of holomorphic functions and on asymptotic properties of the Lambert function. We also show how these asymptotic expansions can be used to establish modal decompositions and well-posedness results for elastic waveguide problems. Numerical experiments confirm the accuracy of the proposed formulas.

In this paper we derive a framework for topological expansions of shape functionals with respect to multiple holes based on the expansion of one hole. Our strategy is similar to deriving a Taylor expansion of order two of functions in $\mathbf R^d$ by applying $d$-times a Taylor expansion in $\mathbf R$. For shape functionals defined on subsets of $\mathbf R^2$ we relate the topological expansion with multiple holes to a Taylor expansion with respect to the ball volumes where usual derivatives are replaced by topological state derivatives. The first topological state derivative was introduced in a previous paper and relates the first topological derivative of cost functionals. The second topological state derivative at two distinct points is introduced, similar to a second directional derivatives of functions defined on Banach spaces, as the topological state derivative of the topological state derivative. The second topological state derivative at the same point is defined as the "far-away" component of the compound layer expansion of the state variable. In dimension three the obtained topological expansion still involves first and second topological state derivatives, but it does not follow the same pattern of a Taylor expansion in the ball volumes. As we will show this dimension dependent behaviour is rather of geometrical nature and already appears in cost functionals without PDE constraints.

We show that for a generic real or complex-valued compactly supported potential, the corresponding Schroedinger operator achieves maximal resonance density, in the sense that its integrated resonance counting function achieves the optimal asymptotic upper bound. For odd dimensions this follows from results of Dinh-Vu once we adapt an argument of Christiansen Hislop. The proof for even dimensions constitutes the bulk of the paper, and we prove several new results on resonances which have analogues in the odd dimensional case. This includes a sharp upper bound on the integrated resonance counting function for any compactly support potential, a proof that the characteristic function of a ball has resonance counting function which achieves the optimal upper bound, and an even-dimensional analogue of the result of Dinh-Vu on asymptotics of the resonance counting functions for complements of pluripolar subsets of analytic families of potentials. We use the characterization of resonances as zeros of certain Fredholm determinant functions related to the scattering matrix, allowing us to apply techniques and results from the theories of one and several complex variables. Our proof that the characteristic function of a ball has counting function achieving the optimal upper bound uses the uniform asymptotics of Bessel functions and follows ideas of Zworski, Christiansen-Hislop, and Dinh-Vu.

In this article, we focus on a MHD model that takes into account the Coriolis force. In this case, the strength of the rotation is measured by the Rossby number $\epsilon$ (positive). Under the assumption that the rotation is strong (that is, when $\epsilon$ goes to zero) and the kinematic viscosity goes to zero as $\epsilon$^$\alpha$ ($\alpha$ positive), we first establish the global convergence of weak solutions, and then prove the global existence and convergence of strong solutions. In particular, the Strichartz estimates used to obtain explicit convergence rates depend on the above parameter $\alpha$ and lead us to identify an admissible interval for $\alpha$.

We classify the long-time decay rate in dispersive estimates for the Schrödinger and wave equations on non-trapping asymptotically conic manifolds and exact metric cones in terms of the intensity of geometric focusing. Letting $X_0$ be a metric cone, one of our main results demonstrates that each multiplicity of conjugate points within distance $\pi$ on $Y=\partial X_0$ leads to a $|t|^{1/2}$-loss in the long-time decay order and a half-order shift in the regularity index in the dispersive estimate for the Schrödinger equation. Unexpectedly, conjugate point pairs on $Y$ at distance $\pi$ do not cause loss when the Legendre submanifold carrying the wave propagation satisfies a natural admissible condition that we propose. In sum, we give a robust framework for proving dispersive estimates that is stable under geometric perturbations and also accommodates perturbations by potentials.

We solve a boundary value problem arising from Maxwell's equations in the quasi-static approximation, which governs the time evolution of the so-called weighting potential $V_{\mathrm{w}}(t,x)$ in resistive solid-state particle detectors. The model reduces to the third-order time-dependent PDE $$ \varepsilon\,\partial_t \Delta V_{\mathrm{w}}(t,x) + \mathrm{div}\,\big(\sigma(x)\nabla V_{\mathrm{w}}(t,x)\big)=0 \quad\text{in } [0,T]\times\Omega,$$ supplemented with mixed Neumann-Dirichlet boundary conditions, possibly degenerate. Our analysis is based on the decomposition of the weighting potential into a static and a dynamic component. The static part solves a uniformly elliptic mixed boundary value problem, while the dynamic part satisfies a degenerate parabolic Cauchy problem. We also establish interior regularity results.

We construct an explicit sign-changing Poisson kernel for a uniformly elliptic divergence form operator in the unit disk. The coefficient matrix is non-symmetric, and its skew-symmetric part has a jump discontinuity across a fixed diameter, with the size of the jump determined by $k\in\mathbb R$. For each $1<p<\infty$, the associated $L^p$ Dirichlet problem undergoes a sharp transition as $k$ crosses a $p$-dependent threshold. In the unique solvability regime, the solution operator preserves positivity and is represented by a nonnegative Poisson kernel. Beyond the threshold, the homogeneous problem admits a nontrivial solution, and for real-valued boundary data, every bounded linear selection from this nonunique solution family is represented by a sign-changing kernel. Thus, even in bounded domains, uniform ellipticity and the maximum principle in the energy class do not prevent Poisson kernels from changing sign. Using a Riemann-Hilbert formulation, we obtain explicit solution formulas and show that the factorization index governs both nonuniqueness and sign change. To our knowledge, this is the first explicit example of this counterintuitive phenomenon in a bounded domain.

We construct here global in time mild solutions to the forced fractional Navier-Stokes-Boussinesq system in some critical Fourier-Herz spaces which will be based on Lorentz norms in the time and the frequency variables. For the initial data and for the external force we will consider Fourier-Besov spaces that will also be based on Lorentz spaces. Moreover, we will show, by performing a separate study of the velocity field and the temperature, that it is possible to consider one of the largest functional space -in the Fourier-Besov-Lorentz framework-for the initial velocity field.

This paper addresses the problem of quantitative rapid stabilization via the linear quadratic (LQ) theory. Specifically, for non-self-adjoint parabolic equations, by virtue of the LQ theory, we derive the quantitative rapid stabilization by selecting a special cost functional and combining it with a quantitative observability inequality. Furthermore, this approach reveals the equivalence between the quantitative rapid stabilization and the quantitative observability inequality for linear systems. In addition, we apply this framework to the Navier–Stokes equations and establish the quantitative rapid stabilization around nontrivial steady states. Finally, we extend this methodology to finite-dimensional feedback laws in self-adjoint cases.

We introduce a new family of weighted Gaussian $L^2$-Poincaré-type inequalities with explicit sharp constants, optimizers, and corresponding sharp $L^2$-gradient stability estimates. This family substantially extends the classical Gaussian Poincaré inequality. Owing to the singular nature of the weights involved, standard approaches to classical Gaussian Poincaré inequalities do not apply. To overcome this difficulty, we develop a new method based on generalized Laguerre polynomial expansions, spherical harmonic decompositions, and a Kelvin-type transform. As an application, we completely characterize the stability of the $L^2$-Caffarelli–Kohn–Nirenberg (CKN) inequalities by establishing sharp stability estimates, together with the stability of the stability inequality results, throughout the entire parameter range. Previous results were available only in a few special cases. We further establish weighted $L^p$-Poincaré inequalities for all $p>1$, and derive stability estimates for the $L^p$-CKN inequalities for $p\geq 2$ throughout the full parameter regime in which sharp constants and optimizers are known. In contrast, earlier $L^p$ results were restricted to highly limited parameter ranges.

We prove strong ill-posedness of the 2d incompressible Euler Equation for velocity field in the critical Besov Spaces $B^{1}_{\infty, q}$ for $1<q<\infty$.

In this paper, we investigate the well-posedness of the three-dimensional stationary incompressible resistive-viscous Hall magnetohydrodynamic (Hall-MHD) system within the framework of scale-invariant Besov-Morrey spaces. A key component of our analysis is the derivation of product estimates in these spaces, which are instrumental in handling the system's nonlinear terms. As an application, we also establish a well-posedness result for the stationary Navier-Stokes equations in this setting, thereby broadening the admissible classes of external forces and solutions.

We consider the following quasilinear reaction-diffusion equation $$ \partial_tu=\Delta u^m+(1+|x|)^{\sigma}u^p, \quad (x,t)\in\mathbb{R}^N\times(0,\infty), $$ in dimension $N\geq3$ and in the range of exponents $1<p<m$ and $-\infty<\sigma<-2$. We prove that, for initial conditions $u_0$ satisfying $$ u_0\geq0, \quad u_0\not\equiv0, \quad \lim\limits_{|x|\to\infty}|x|^{-(\sigma+2)/(m-p)}u_0(x)=0, $$ the solution $u$ to the corresponding Cauchy problem remains uniformly bounded from above and below: $$ C_1\leq \|u(t)\|_{\infty}\leq C_2, \quad t\in(0,\infty), $$ for some positive constants $C_1$ and $C_2$. Under suitable conditions on $p$, we also establish the rate of expansion of the upper limit $R(t)$ of the positivity set for compactly supported data, that is, $$ At^{\beta}\leq R(t)\leq Bt^{\beta}, \quad \beta=-\frac{m-p}{\sigma(m-1)+2(p-1)}, $$ and a different time scale in outer sets, that is $$ D_1t^{-\alpha}\leq u(x,t)\leq D_2t^{-\alpha}, \quad \alpha=\frac{\sigma+2}{\sigma(m-1)+2(p-1)}, \quad {\rm if} \ |x|\geq Ct^{\beta}. $$ The boundedness is in striking contrast with the property of grow-up as $t\to\infty$ established in previous works by the authors for $\sigma>-2$, illustrating the character of threshold of the exponent $\sigma=-2$.

In this paper, we investigate the relationship between the long time behaviour of solutions to the Kahler-Ricci flow on an asymptotically conical gradient Kahler-Ricci expander and the asymptotic behaviour of their initial data at spatial infinity.

We make use of the free data formalism developed in \cite{CK25} to construct solutions of the Einstein vacuum constraint equations by integrating in the forward direction. This, together with a new gauge condition based on effective uniformization, allows us to construct general solutions with limited decay at spacelike infinity. In particular, we construct solutions with minimal and even borderline decay, as considered in \cite{Shen23}, \cite{Shen24} in connection with the stability of the Minkowski space. In a forthcoming paper, we make use of the techniques we develop here to identify and construct a general class of short-pulse Cauchy data that lead to the formation of trapped surfaces, extending the well-known result of \cite{LiYu}.