Papers

A sharp lower bound for the first Dirichlet eigenvalue of the $p$-laplacian is derived for sets with prescribed $p$-torsional rigidity. The result provides an extension of the classical spectral inequality due to Kohler-Jobin. The proof is based on a careful analysis of the generalized $p$-torsional rigidity and on a sharp mass comparison result.

The coupling of the evolution of a surface with evolution equations defined on that surface is of relevance in many applications and has been in the focus of interest in the analysis of parabolic PDEs in recent years. In applications the evolution of two-phase vesicles and biomembranes is governed by flows decreasing an energy which involves Canham-Helfrich-type curvature energies coupled to a Ginzburg-Landau energy. We derive a new Cahn-Hilliard-Canham-Helfrich system for the evolution of two-phase membranes. The resulting system is highly non-linear and we use the theory of quasi-linear parabolic evolution equations in weighted $L_p$-spaces to show the existence of a strong local-in-time solution and hence demonstrate that the derived system is well-posed.

We establish a scaling-critical continuation criterion for the three-dimensional incompressible magnetohydrodynamic equations in $\mathbb{R}^3$ with arbitrary positive viscosity and magnetic diffusivity. Let $\beta(t)$ be a unit vector that is piecewise $H^1$ in time and has only finitely many jumps. For $3\le p<\infty$, set $\gamma_p=2p/(2p-3)$. We prove that an $H^1$ strong solution can be continued beyond $T_*$ if \[ \int_0^{T_*}\left( \|u(t)\cdot\beta(t)\|_{\dot H^{3/2}}^2+ \|b(t)\cdot\beta(t)\|_{\dot H^{3/2}}^2+ \|\beta(t)\cdot\operatorname{curl}b(t)\|_{L^p}^{\gamma_p} \right)\,dt<\infty . \] Thus the observed component may vary in time, and the magnetic assumption is reduced to one moving magnetic component together with one Serrin-type current-density component. The proof is based on a moving-frame formulation of the vorticity-current system, an anisotropic product estimate adapted to the moving frame, and a horizontal Hodge decomposition that controls the current-Jacobian residual.

In this article, we examine the mean-field limit of the non-exchangeable Cucker–Dong model. This model corresponds to a biologically more realistic version of the classic Cucker-Smale model, which is used to describe the alignment phenomenon in large animal groups. In addition to alignment forces, the non-exchangeable Cucker–Dong model integrates attraction/repulsion forces and network-structured interactions. In order to enable convergence towards a flocking profile, the attraction/repulsion forces are weighted by a second-order coefficient called the alignment measure, which is smaller when individuals are more aligned overall. Deriving the mean-field limit of this model relies on a new stability result that is in agreement with with both the second-order nature of the alignment measure and the non-exchangeability induced by the graph-dependent interactions.

We study homogenization of a regularized viscoelastic Oldroyd-type model in a periodically perforated bounded domain. The system describes an incompressible non-Newtonian fluid coupled to an elastic extra stress tensor and includes both nonlinear viscosity and nonlinear stress diffusion effects. The governing model, introduced by Kreml, Pokorný, and Šalom (2015), covers Oldroyd-A- and Oldroyd-B-type constitutive laws. We establish qualitative and quantitative homogenization results in suitable scaling regimes and show convergence toward an effective Darcy law on the macroscopic domain. In particular, we prove that, under appropriate assumptions on the scaling parameters, the polymeric stress does not contribute to the effective limit equation. The analysis combines uniform estimates, oscillating test-function techniques, and a relative energy method, and additionally yields a weak-strong uniqueness principle for the viscoelastic system.

We study local regularity properties of viscosity solutions to fully nonlinear elliptic equations with variable gradient degeneracy and Hamiltonian-type lower-order terms, \[ |\nabla u|^{p(x)}F(\nabla^{2}u) + a(x)|\nabla u|^{q(x)} = f(x). \] Here, $F$ is uniformly elliptic, while the exponents $p$ and $q$ are allowed to vary in space. We prove interior Hölder estimates for the gradient, with an exponent determined by the maximal degeneracy rate and by the regularity available for the associated homogeneous uniformly elliptic equation. We also obtain pointwise improvements at points where the source term and the Hamiltonian coefficient vanish with prescribed Hölder rates. Finally, at extremal points, we establish a Schauder-type estimate showing that the solution separates from its extremal value with order strictly larger than two. The proofs combine compactness estimates for shifted equations, stability of viscosity solutions, and improvement-of-flatness iterations.

In the setting of a complete, smooth properly immersed mean curvature flow, we assume uniformly bounded $|H|$ and $|\nabla H|$ on $M^n\times[0,T)$ and some bounded initial geometry to get local spatial $L^p$ estimates for the second fundamental form with $p\in[4,\infty)$. For $p>n+2$, this leads us to a local space time $L^\infty$ bound for the second fundamental form which allows us to smoothly extend the flow $F:M^n\times [0,T) \rightarrow \mathbb{R}^{n+1}$ past the singular time $T<+\infty$ for a short time.

We prove the lack of selection in the zero-noise limit for solutions to SDEs driven by a divergence-free, Hölder continuous vector field with exponent $\alpha\in(0,1)$, arbitrarily close to $1$ but fixed. The result applies to a broad class of regularizing additive noises, including fractional Brownian motion and stable Lévy processes. The proof combines pathwise Lagrangian arguments, based on the analysis of the deterministic flows associated to mixing velocity fields, with probabilistic estimates coming from the stochastic sewing lemma. This allows to show that lack of selection happens simultaneously on a large set of initial data, whose complement has arbitrarily small Lebesgue measure.

We study an inverse source problem for the Monge–Ampere equation \[ \det D^2u=f(x) \] on a bounded smooth uniformly convex domain. In the smooth classical regime, we prove that the Dirichlet-to-Neumann map associated with convex solutions determines the positive source uniquely. The proof uses a family of large boundary values and reduces the inverse source problem to the injectivity of the Euclidean X-ray transform.

We study the semilinear wave equation $ V(x) \partial_t^2 u + d(t) M(x,\nabla_x) u=\tilde{V}(x) \tilde{d}(t) |u|^{p-1}u$ on $ \mathbb{R}^N \times \mathbb{R}$ and show the existence of solutions which are localized in space and in time, called rogue waves, by means of variational methods. We introduce an energy functional on a suitable Hilbert space, and provide sufficient conditions on the coefficients $V, \tilde{V}, d, \tilde{d}$, the elliptic operator $M$ and $p>1$ for the existence of a critical point. Our approach is based on a detailed analysis of the wave type operator and in particular its spectral properties. Further regularity considerations show that critical points are weak solutions to our equation. Moreover, we provide examples of the coefficients and the elliptic operator which satisfy our assumptions.

We construct probabilistic strong solutions to the cubic Schrödinger equation on the three-dimensional ball with radial initial data, which is a significant improvement of a result by Bourgain–Bulut. These solutions lie in the supercritical regime with respect to the probabilistic scaling introduced by Deng–Nahmod–Yue. We achieve this result through gauge transformations that do not modify the equation, combined with a refined modulation analysis using random averaging operators.

This paper explores stability dichotomies of thermoelastic Timoshenko-Boltzmann systems with hereditary memory, comparing Fourier's parabolic (no thermal relaxation) and Cattaneo's hyperbolic (with thermal relaxation) heat conduction via semigroup theory and spectral analysis. The Fourier-semigroup achieves exponential stability through the $\delta$-condition plus a specific parameter condition, while sole $\delta$-condition ensures $1/2$-order polynomial stability. The Cattaneo-semigroup retains this polynomial stability but requires a thermal relaxation-included modified parameter condition for exponential stability. As thermal relaxation time approaches zero, Cattaneo's modified parameter converges to Fourier's, formalizing their asymptotic connection. Results show thermal relaxation alters exponential criteria but preserves polynomial decay, supporting stability analysis under time-dependent thermal loading in structural mechanics.

We give a variational formulation of the continuous adjoint Kutta condition for two-dimensional subcritical potential flow, with emphasis on the Kutta condition and the role of the wake. We show that the adjoint Kutta condition can be imposed by a penalty term evaluated at the trailing edge, with the corresponding Lagrange multiplier determined by stationarity of the Lagrangian with respect to circulation, and that a wake treatment is not required. Some of the implications of these results for adjoint consistency are also briefly discussed.

In this paper we utilize a convex integration scheme to construct non-trivial weak solutions to the $k$-generalized KdV equation which lie in $$ \bigcap_{\epsilon > 0} C_t^0 L_x^{k-\epsilon}([0,1] \times \mathbb{T}) $$ and, when $k \ge 3$, it may also be chosen in \[ \bigcap_{\epsilon >0} C_t^0 H_x^{\frac{1}{2} - \frac{1}{k} - \epsilon}([0,1] \times \mathbb{T}) \] attaining identically $0$ initial data. Since our solutions do not lie in $C_t^0 L_x^k$, this requires introducing a new notion of weak solution, which is in fact stronger than the classical notion of a weak solution when the nonlinearity is integrable. This result shows that a necessary condition for unconditional uniqueness for $k$-gKdV is that the nonlinearity lies in $C_t^0L^1_x$. In the case of KdV this is in fact also sufficient.

We study the dynamics of planar sets under area-preserving crystalline curvature flow. We prove under mild assumptions on $\phi$ that the flat flow solution from regular initial data coincides with a classical ODE evolution, extending the results of \cite{Almgren1995} to the area-preserving setting. We also show that for arbitrary initial data, the flat flow converges exponentially in time to a disjoint union of Wulff shapes, and under a non-bubbling assumption, the flow eventually becomes regular. Both of these results are novel for area-preserving crystalline flow of general sets, i.e. without assuming geometric properties such as convexity or star-shapedness. A key ingredient of independent interest is that planar almost-minimizers are Lipschitz $\phi$-regular, which we prove by exploiting a sharp minimality estimate for distinguished line segments, as opposed to the excess decay argument given in \cite{Ambrosio2002regularity}. The novelty of our approach lies in the application of $\phi$-minimal barriers for energy competition arguments, both for the geometric rigidity of the discretized flat flow and for the regularity of almost-minimizers.

We show that the small-data scattering map uniquely determines the nonlinearity for a class of nonlinear Schrödinger equations with radial, Hartree-type nonlinearities. Our assumptions on the convolution kernel require only a mild decay condition at infinity and permit a locally integrable singularity at the origin.

We discuss self improving properties of some local and nonlocal, elliptic and parabolic, equations on bounded domains. We employ a functional analytic approach wherein the solution space sits in a suitable interpolation scale. Utilizing a classical analytic perturbation result, we extrapolate the invertibility of the main operator from the base space to nearby spaces within the interpolation family.

We address the Euler immersion problem, a fluid-structure interaction problem in which an elastic body is immersed in an incompressible inviscid fluid governed by the Euler equations. We show that the system exhibits a loss of one derivative and formulate the problem in analytic function spaces. We then prove local well-posedness in analytic spaces under velocity-matching boundary conditions. Finally, by means of an example, we show that existence fails in analytic spaces when both velocity and stress-matching boundary conditions are prescribed.

In this paper, we derive a new two-dimensional rod-Kadomtsev-Petviashvili (rod-KP) equation from the incompressible and irrotational three-dimensional Euler equation under the shallow water scaling. We establish the local well-posedness of the Cauchy problem in a suitable Sobolev space and derive the blow-up criterion for strong solutions via the energy method. Then, by exploring two appropriate conservation laws of the rod-KP equation, we are able to construct its global strong solution when the physical dimensionless parameter $\sigma=0$, and on the other hand produce the finite-time blow up solutions under some certain conditions when $\sigma \neq 0$. Furthermore, we present a uniqueness continuation property for the solutions. Finally, we investigate the existence of traveling-wave solutions in order to highlight the influence of weak transverse effects on wave stability, and we also exhibit the symmetry of solitary waves in the propagation direction.

We prove modified scattering for small data solutions to the Vlasov-Poisson system in a functional framework where the initial data, scattering states, and asymptotic convergence are measured in the same topology. In addition, we show that the corresponding wave operators define homeomorphisms between the spaces of initial and scattering data, while enjoying a local Lipschitz continuity property in weaker norms. As a consequence, in the repulsive case, large spherically symmetric solutions are asymptotically stable. The proof relies in particular on the introduction of a suitable system of dynamic coordinates adapted to the asymptotic nonlinear flow.

We study the defocusing $k$-dispersion generalized Benjamin-Ono equation. For every even integer $k\geq 4$, we prove that solutions with initial data in the energy space $H^{\frac{\alpha}{2}}$ are global in time and scatter. The proof combines the concentration-compactness-rigidity method of Kenig and Merle with techniques based on the Caffarelli-Silvestre extension and Tao's monotonicity formula adapted to the fractional dispersion setting.

We study an inverse source problem for fully nonlinear elliptic equations of the form \[ F(D^2u)=f \quad \text{in } \Omega. \] The question is whether the source term can be recovered from the Dirichlet-to-Neumann map. In two dimensions, the first linearization does not immediately give uniqueness: it leaves a natural conformal ambiguity in the linearized coefficients. For homogeneous nonlinearities $F$ with injective differential $DF$, we show that this ambiguity has a precise meaning at the level of the equation itself, namely that the source is determined up to an explicit scalar factor. The main point of the paper is to show how this remaining factor can be removed. We use the second linearization to extract information which is invisible at first order, and combine it with an algebraic nondegeneracy condition on the nonlinearity. Under this condition, the residual ambiguity is forced to be trivial, and the Dirichlet-to-Neumann map uniquely determines the source. The result applies, in particular, to homogeneous admissible Hessian equations of Monge–Ampère type and related examples.

We study an inverse problem for translation-invariant symmetric stable operators of the form \begin{equation*} L_a u(x)=\mathrm{P.V.}\int_{\mathbb R^n}(u(x)-u(y))\frac{a((x-y)/|x-y|)}{|x-y|^{n+2s}}\,dy, \quad 0<s<1, \end{equation*} where the unknown is the even angular density $a$ on $\mathbb Sn$. For a bounded open set $\Omega\subset\mathbb R^n$, with $\Omega_e=\mathbb R^n\setminus\overline\Omega$, we consider restricted exterior Dirichlet-to-Neumann maps $\Lambda_a^{W_1,W_2}$, where exterior data are supported in $W_1\Subset\Omega_e$ and the nonlocal Neumann data are observed on $W_2\Subset\Omega_e$. We prove three recovery results for the leading angular density. In the overlapping regime $W_1\cap W_2\ne\emptyset$, the exterior diagonal singularity determines every smooth elliptic angular density. In the separated regime $\overline W_1\cap\overline W_2=\emptyset$, where this singularity is absent, we prove uniqueness in the finite harmonic angular class by an exact factorization of the stable symbol. We also prove separated-data uniqueness for real-analytic angular densities when the source and observation sets lie in the unbounded exterior component, using analytic continuation of the off-diagonal Dirichlet-to-Neumann kernel and a far-field asymptotic argument.

This addendum extends the spectral lower bounds established in the original paper to the Neumann problem. Under the formal positivity assumption on the elliptic tuple, we prove that the same bounds hold for Neumann boundary conditions as for the Dirichlet case.

We study some local and global properties of solutions of $-\Delta u- m\abs{\nabla u}^q-e^{u}=0$ in a punctured domain $\Omega\setminus\{0\}$, or in an exterior domain of $R^N$, $N\geq 2$, where $m$ is a positive parameter and $q>1$. We study particularly the local behaviour of solutions with an isolated singularity or the asymptotic behaviour for solutions defined in an exterior domain, and also the existence of solutions with the behaviours previously described. These behaviours change drastically according $q$ is smaller or larger than $2$. Many results are obtained by introducing various dynamical systems associated to the equation.

In this article we develop a quadratic energy which approximates the Canham-Helfrich energy for a tube-like surface with clamped boundary and area constraint. The energy is suited to the study of small deformations of biological membranes where the deformations are induced by point forces or point constraints due to the cytoskeleton or a phase dependent spontaneous curvature. Since the deformations we consider are small, we may assume that the surface of interest is a graph over a fixed, undeformed surface. A Lagrangian and the associated Euler-Lagrange equations for the graph are derived. Well-posedness of the Euler-Lagrange equations in suitable spaces is shown. Finally, we provide some illustrative numerical examples.

By now, several approaches have been proposed to endow phase-field fracture models with the ability to describe crack nucleation under multiaxial stress states. These include techniques for splitting the free energy, direct modifications of the phase-field driving or resisting forces that sacrifice the variational structure of the problem, and the introduction of additional internal variables, such as plastic strains or other nonlinear strains. In this paper, we propose a fundamentally different strategy for incorporating arbitrary elastic domains into phase-field fracture models, formulated within the variational framework of generalized standard materials. The proposed approach relies on letting the dissipation potential depend on the current state of the material. In this way, the variational structure of the problem is preserved, while elastic degradation and the strength criterion remain two distinct and independently controllable aspects of the material response. Simple yet representative models are presented and thoroughly discussed to demonstrate the effectiveness of the proposed methodology. The resulting evolution of the elastic domain is investigated in both strain and stress spaces. Moreover, numerical simulations demonstrate a range of crack nucleation processes under multiaxial loading conditions for various analytical strength surfaces. This work paves the way for future developments and applications in several directions.

Energy decay rates for solutions of the damped wave equation on the torus are known to be influenced by the geometry of the damped set and the growth properties of the damping. In this paper we produce lower bounds on energy decay rates for a class of damping which are positive on a superellipse and grow polynomially like the distance to the boundary of the superellipse. The energy decay rates we obtain depend explicitly on the exponent used to define the superellipse and the polynomial power. We show these rates are sometimes optimal. The proof adapts quasimodes from $y$-invariant damping using a simplification of the usual normal form argument.

We consider a two-dimensional periodic reaction-diffusion system under natural conditions on the reaction function and with initial condition $\theta$. We show that on the global attractor $\mathcal A$ of the resulting dynamical system $(u_\theta(t):t>0)$, a reverse Poincaré inequality holds true, and that as a consequence the map $\theta \mapsto u_\theta(t)$ satisfies a $L^2$-Lipschitz stability estimate on $\mathcal A$ for any $t>0$ fixed. We then show that statistical recovery of an initial condition $\theta$ in the attractor $\mathcal A$, as well as prediction of the states $u_\theta$, is possible from discrete measurements of the system at `fast' near parametric convergence rates.

In a smoothly bounded domain $\Omega\subset R^n$, $n\le 5$, the manuscript considers the variant of the Keller-Segel system given by \[ \left\{ \begin{array}{l} u_t = D \Delta u - \nabla \cdot (u\nabla v), \\[1mm] v_t = d \Delta v + \nabla \cdot (u\nabla v) - v + u, \end{array} \right. \] which involves an additional contribution $\nabla \cdot (u\nabla v)$ to the chemoattractant evolution, in line with refined modeling literature reflecting an anisotropic correction to the isotropic signal production term $+u$ in the classical Keller-Segel model. It is shown that for arbitrary $D>0$ and $d>0$ and any nonnegative intial data from $W^{1,\infty}(\Omega)\times W^{1, \infty}(\Omega)$, an associated Neumann problem admits a global weak solution $(u,v)$ which, inter alia, satisfies \[ \sup_{t \in (0,\infty)\setminus N} \int_\Omega e^{u^\alpha(\cdot,t)} < \infty \] with some $\alpha>0$ and some null set $N\subset (0,\infty)$.