Papers

We study the Dirichlet problem for the $p$-Poisson equation in the metric measure space setting equipped with a doubling measure and supporting a $(p,p)$-Poincaré inequality. We prove the existence of the solutions by using a variational approach. We prove the stability and uniqueness of the solutions, when the space is also geodesic.

We consider mean curvature flow with free boundary through cylindrical or half-cylindrical singularities, namely singularities of the types $\mathbb{R}^k\times S^{n-k}$, $\mathbb{R}^k_+\times S^{n-k}$ or $\mathbb{R}^k\times S^{n-k}_+$. Using the foundational results for free boundary Brakke flows by Edelen and the first author, and the recent classification of ancient asymptotically cylindrical flows by Bamler-Lai, we prove that all these singularities have a mean-convex neighborhood. Moreover, generalizing work of Hershkovits-White to the free boundary setting we show that the free boundary level set flow is nonfattening provided all singularities have a mean-convex neighborhood. We conclude that free boundary flow through singularities is well-posed as long as all singularities are of cylindrical or half-cylindrical type.

Starting from a spectral problem posed in a Clifford algebra with $d$ generators and Euclidean signature, we study an integrable, coupled system of PDEs that can be viewed as a vector perturbation of the Camassa–Holm equation with residual orthogonal symmetry. In the two-component case $d=2$, we show that the travelling wave solutions correspond to a Liouville integrable Hamiltonian system with two degrees of freedom, making use of a reciprocal transformation linking the coupled PDEs to a symmetry of the Hirota–Satsuma system. We also present a symmetry classification of all integrable two-component perturbations of Camassa–Holm, and find that besides the $d=2$ system analyzed here, the coupled 2CH system studied by Olver and Rosenau (as well as by Chen, Liu and Zhang, and Falqui), and equations related to either of those systems by Miura transformations, we also obtain a new system that (to the best of our knowledge) has not been reported previously. For the case of an arbitrary number of components $d$, we additionally investigate the short-pulse (high-frequency) regime, in which the limiting dynamics are governed by a vector-valued Hunter-Saxton type system. Furthermore, we provide a detailed analysis of the corresponding measure-valued (weak) solutions associated with this system.

We give a short proof of the quantitative isoperimetric inequality. Our argument is based on a notion of quantitative calibrations which induce a natural distance controlling both the Fraenkel asymmetry and the tilt excess. The proof of our key result which can be viewed as a nonlinear, geometric version of Fuglede's result in $BV$ is direct and self-contained. In particular, we do not make any use of regularity theory for almost minimizers.

In the present manuscript, we study an inverse problem related to a semilinear dynamical Schrödinger equation with lower order terms, in a bounded domain of $\Rb^{1+n},n\geq 2$. Our focus is on determination of the time-dependent coefficients appearing in the aforementioned equation, from the boundary measurements of the solutions. More precisely, we establish the {pointwise reconstruction} formulae for determining the time-dependent coefficients of linear and nonlinear terms from the knowledge of Dirichlet-to-Neumann map. Since the concerned non-linear Schrödinger equation possesses a trivial solution, we linearize the equation around the trivial solution and use the asymptotic solutions (with concentrated amplitudes) of the linearized problem for reconstructing the aforementioned coefficients. To be more specific, we use first-order linearization to reconstruct vector and scalar potentials associated with the coefficients of linear terms and the higher-order linearization technique is used to reconstruct coefficients of nonlinearity. The nonlinear equation considered in this manuscript can be seen as a generalization of the Gross-Pitaevskii equation (GPE), which is employed to describe the dynamics of dilute Bose-Einstein condensates (BEC).

We establish uniqueness of blow-ups, with sharp quantitative convergence, for several classes of singular minimizing cones in the Alt-Phillips problem, in the range $\gamma\in(0,2)$. As a consequence, we obtain uniqueness at every free boundary point in dimensions $d=2,3,4$ for $\gamma\in(1,2)$, and in dimensions $d\geq 5$ for $\gamma\in\left(1,\frac32\right)$. The proof of uniqueness is based on three new logarithmic epiperimetric inequalities. The sharp distinction between polynomial and logarithmic convergence is governed by a finite-dimensional integrability condition (sub-integrability) for the spherical linearized problem. We prove this sharpness for the radial cone and its cylindrical extensions through an explicit integrability and bifurcation analysis, showing in particular that logarithmic convergence may be sharp even in dimension two. In contrast, we show that the one-dimensional cone is exceptional: although the integrability condition fails, the convergence is polynomial. Finally, we characterize the minimality of the radial cone in terms of $d$ and $\gamma$ by means of a one-dimensional calibration argument, exhibiting in dimension $d\geq6$ a nontrivial regime in which the radial cone is stable but not minimizing.

Subwavelength elastic resonators can concentrate wave energy at length scales far below the incident wavelength, but their behavior becomes especially delicate when two resonators almost touch. In this paper, we give a rigorous analysis of a two-dimensional dimer made of two high-contrast hard inclusions embedded in a soft elastic matrix. The analysis confronts two features that are absent from the corresponding three-dimensional theory: the logarithmic low-frequency singularity of the two-dimensional elastic fundamental solution and the possible non-invertibility of the static single-layer potential. We overcome these difficulties by proving the invertibility of the correct frequency-dependent leading-order operator and then using it to reduce the resonance problem to a finite-dimensional system. For generally convex resonators satisfying natural symmetry assumptions, we derive six subwavelength resonant frequencies and identify their dependence on the material contrast $\delta$ and the inter-inclusion distance $\varepsilon$. We further quantify the resonant field concentration in the narrow gap. In the regime $\varepsilon=\Ocal(\delta^\beta)$, $0<\beta<2$, the gradients of the eigenmodes display sharply classified blow-up behavior: some modes attain the stronger rate $\Ocal(1/\varepsilon)$ at the closest point of the gap, while others blow up at the rate $\Ocal(1/\sqrt{\varepsilon})$ away from the centerline; the remaining mode is governed by a boundary mismatch mechanism. These results uncover resonance-induced singularities that are markedly stronger and more structured than those in static or non-resonant elasticity, and they provide a framework for analyzing larger clusters of closely spaced elastic subwavelength resonators.

We classify the dynamics of radial $H^1$ solutions at the (radial) ground state threshold for the $3d$ cubic NLS in the presence of a repulsive inverse-square potential.

In this paper, we investigate the existence of positive supersolutions for the following mixed local-nonlocal Lane-Emden type equation: $$ -\Delta u+(-\Delta)^s u=u^q\quad\text{in }\mathbb R^n, $$ where $n\geq 3$, $s\in(0,1)$, and $q>1$. More precisely, we prove that the equation admits positive distributional supersolutions if and only if $q>\frac{n}{n-2s}$. In the process, we establish several novel properties of mixed local-nonlocal operators, including sharp asymptotic estimates for the fundamental solution, a maximum principle in the distributional sense, and an equivalent integral inequality for supersolutions.

We study a backward inverse problem for a coupled Cahn-Hilliard-reaction-diffusion system. Our main result is a Carleman estimate for a fourth-order/second-order parabolic system with cross-diffusion terms, which allows us to derive conditional stability estimates for the reconstruction of past states from a single final-time observation: Hölder stability at positive times and logarithmic stability for the initial datum. We then apply the Carleman estimate to a phase-field tumour growth model coupling the tumour volume fraction with a nutrient concentration. In this setting, we obtain backward uniqueness and quantitative stability for the recovery of early tumour states, improving earlier results based on logarithmic convexity, which only yielded uniqueness under an additional smallness assumption on the chemotaxis coefficient. We also discuss how these results support Lipschitz stability on finite-dimensional admissible sets, which is relevant for ensuring convergence of iterative discretisation algorithms.

In [Problem 3.2.23, CBMS Regional Conference Series in Mathematics 83, 1994], Kenig asked whether the vanishing Carleson condition is the necessary and sufficient for the logarithm of the Poisson kernel of a perturbation of the Laplacian on the unit ball in $\mathbb{R}^n$ belonging to the space VMO. The sufficiency was proved by Escauriaza [Israel J. Math. 1996] and extended by Milakis, Pipher, and Toro [Contemp. Math. 2014] to more general setting. In this article, using the technique of bi-Lipschitz mappings, we construct a counterexample to show that the vanishing Carleson condition is not necessary and hence give a negative answer to the aforementioned problem.

This paper establishes time decay estimates for the following two-dimensional beam (plate) equation with a decaying real-valued potential $V$: \begin{equation*} \partial_t^2 u + (\Delta^2 + V) u = 0, \qquad u(0,x)=f(x),\quad \partial_t u(0,x)=g(x). \end{equation*} When zero is a regular point or a first-kind resonance of $H=\Delta^2+V$, we first prove sharp $L^1\to L^\infty$ estimates for the solution operators: \begin{align*} \left\|\cos(t\sqrt{H})P_{\mathrm{ac}}(H)\right\|_{L^1\to L^\infty} + \left\|\frac{\sin(t\sqrt{H})}{t\sqrt{H}}P_{\mathrm{ac}}(H)\right\|_{L^1\to L^\infty} \lesssim \frac{1}{|t|}, \end{align*} and obtain an enhanced decay $(|t|\log|t|)^{-1}$ in logarithmically weighted spaces $L^1_\omega\to L^\infty_{-\omega}$ with $\omega(x)=\log(2+|x|)$. For second-kind resonances of $H$ (the bi-Laplacian $\Delta^2$ belongs to this class), a non-zero trace moment $\langle |x|^2V,\phi\rangle\neq0$ for some second-kind resonance function $\phi$ induces severe threshold singularities, worsening the $L^1\to L^\infty$ estimate to $|t|^{-1}(\log|t|)^2$. Finally, for third-kind resonances or a zero eigenvalue, we prove that the presence of $d$-wave resonance leads to the worst $L^1\to L^\infty$ decay rate $\sim(\log|t|)^{-1}$. Several improved estimates are also obtained without a $d$-wave resonance. In particular, in the pure eigenvalue case (i.e., neither $d$-wave nor $p$-wave resonance), both propagators recover the optimal unweighted $L^1\to L^\infty$ estimate $|t|^{-1}.$

In $\mathbb{R}^{3}$, we prove that $L^q\to L^p$ restriction estimates associated with smooth surfaces with negative Gaussian curvature hold for all $p>\frac{22}{7}$ and $q'<\frac{p}{2}$.

We establish quantitative stability of the De Philippis–Rindler rigidity on conical convex regions. By constructing $\mathcal{A}$-quasiconvex functions with a tailored concavity—which additionally yields a new, elementary proof of the original theorem for constant-rank operators—we prove a spreading inequality that strictly limits the angular concentration of $\mathcal{A}$-free measures. This geometric bound establishes that singular concentrations cannot cluster arbitrarily close to a direction outside the Tartar wave cone. As a consequence, we obtain a new $L^1$ compensated compactness result: $\mathcal{A}$-free sequences with uniformly bounded mass are forced to be equi-integrable, thereby preventing the formation of mass concentrations, provided their targets are asymptotically restricted to cones whose aperture is controlled by a power of the distance to the wave cone of $\mathcal A$.

Under cone containment and the central ray hypothesis, we prove a determinant majorization result for hyperbolic polynomials on Euclidean Jordan algebras. In the case of real symmetric $n \times n$ matrices, this recovers the main theorem of Harvey and Lawson [Duke Math. J. 174 (2025), no. 13, 2749–2763] and also yields a new $\sigma_2$ majorization result. Moreover, for every $3 \leq k \leq n-1$, we construct explicit hyperbolic polynomials satisfying cone containment and the central ray hypothesis but for which the analogous $\sigma_k$ majorization fails.

It is now well known that the classical notion of admissible weak solutions (also known as weak entropy solutions) does not restore uniqueness for the multi-dimensional compressible Euler equations. Indeed, convex integration has shown that admissible weak solutions are in general highly non-unique. This has motivated additional selection criteria intended to rule out the counterintuitive solutions generated by convex integration. In this paper we prove that the local least action criterion introduced by H.~Gimperlein, M.~Grinfeld, R.~J.~Knops and M.~Slemrod does not serve as a proper selection criterion, since it fails to select the solution that is intuitively expected to be physically relevant.

We prove an operator-level convergence theorem for the mixed paracontrolled blocks \[ T^{i;j,k}_{\Lambda}(w) =I_i\bigl(w<\Psi_{j,\Lambda}\bigr)\circ\Psi_{k,\Lambda} \] on \(\mathbb T^d\), under Fourier-diagonal Gaussian covariance. At finite Galerkin cutoff, the Wick decomposition is \[ T^\tau_\Lambda=D^\tau_\Lambda+B^\tau_\Lambda, \qquad \tau=(i;j,k), \] where \(D^\tau_\Lambda\) is a deterministic Volterra multiplier on the external diagonal \(n=q\), while \(B^\tau_\Lambda\) is a centered second homogeneous Gaussian chaos with incidence \(n=q+\ell+r\). The centered coefficient has two Gaussian frequency legs and one input-output pair. Write \(\Delta_T=\{(t,s):0\le s\le t\le T\}\). An iterated rectangular non-commutative Khintchine argument, followed by the four associated oriented flattenings, yields \[ \bigl\|B^\tau_{\Lambda,N,Q,M}\bigr\|_ {L^p\bigl(\Omega;C(\Delta_T;\mathcal L(\ell_q^2,\ell_n^2))\bigr)} \lesssim_{p,\varepsilon} N^{d/2-\Gamma_\tau+\varepsilon} \bigl(M^{d/2}+Q^{d/2}\bigr), \qquad \Gamma_\tau=\lambda_i+\alpha_j+\alpha_k. \] Under the resulting strict Sobolev–Besov summability conditions, the centered cutoffs converge in \(L^p(\Omega)\) and almost surely in operator norm. The covariance contraction is kept separate from the tensor estimate: retained raw branches, or prescribed finite combinations of branches, enter through a scalar Volterra criterion imposed after the relevant cancellation has been formed. We give a frequency-envelope variant and verify the diagonal criterion for distinct-speed wave and Klein–Gordon contractions.

The paper studies the Q-tensor model for nematic liquid crystals, a system that couples a Navier-Stokes equation with an evolution equation for the order parameter tensor Q. The first goal of the paper is to establish the local well-posedness of the system in $\mathbb R^N$ and $\mathbb R^N_+$ for $N=2,3$ in the $L^2$ framework, improving existing results in the literature, where the existence of local strong solutions was obtained only under smallness assumptions on the initial data. Fundamental is the application of the energy method, which shows a cancellation phenomena on the nonlinear terms, allowing the use of a contraction argument to prove existence and uniqueness of solutions. Finally, with the same approach we establish global well-posedness in the three-dimensional case for small initial data.

We study the one-dimensional cubic nonlinear Schrödinger system \[ u_i''+2\left(\sum_{k=1}^N u_k^2\right)u_i=-\mu_i u_i \quad \mbox{in } \ \mathbb R,\ \ i=1,2,\cdots,N, \] where $u=(u_1,\cdots,u_N)\in \big(H^1(\mathbb{R})\big)^N$, $\mu_1\leq\mu_2\leq\cdots\leq\mu_N<0$, and $N\geq 2$ is arbitrary. In this paper, we prove the following results for any $N\ge 2$: (i). All nontrivial solutions of the system can be completely classified; (ii). The linearized operator at any nontrivial solution of the system is non-degenerate; (iii). For all $i=1, 2,\cdots, N$, the exact $L^2$-mass identity of $u_i$ is derived in terms of $2\sqrt {|\mu_i|}$, which yields a complete characterization of normalized solutions satisfying $\int_{\mathbb{R}}u_i^2dx=1$. These settle some conjectures of [R. Frank, D. Gontier and M. Lewin, CMP, 2021] and [Y. Guo, Y. Luo and J. Wei, APDE, 2026], where the system was addressed specially for $N=2$ and $N=3$, respectively.

We develop an anti-concentration method for volume-constrained nonlocal shape optimization. Precisely, we apply it to prove the nonexistence of minimizers of an optimal design problem driven by a Hartree-type energy in the large mass regimes. The proof refines and extends the ideas from [Lu-Otto, CPAM 2014], where Lu and Otto dealt with nonexistence of minimizers for the Thomas-Fermi-Dirac-von Weizsäcker energy. We also establish the asymptotic behavior of the associated isoperimetric profile of the energy.

Quasilinear parabolic boundary value problems are considered. Sharp conditions on the data ensuring existence and uniqueness of solutions in Sobolev classes are exposed. They are smoothness and consistency conditions on the data and some inequalities ensuring the maximum principle for a solution. The proof relies on a priori bounds and the method of continuation in a parameter.

We study the heat equation in the half-space with nonhomogeneous Dirichlet boundary data. For the caloric extension $v$ of the boundary data $g$, we prove maximal regularity estimates in mixed Lebesgue norms $L^p_tL^q_x$ for any order derivative of $v$ in terms of mixed Besov and Lizorkin–Triebel type norms of $g$. We also establish the corresponding reverse inequalities, which are caloric trace estimates recovering the boundary regularity of $g$ from the mixed-norm regularity of $v$. As a model case, our results show that the natural \[\dotc W^{1,p}\big(\R;L^q(\R^d_+)\big)\cap L^p\big(\R;\dotc W^{2,q}(\R^d_+)\big)\] regularity norm of $v$ is controlled by the \[\dotc {F}^{1-\frac{1}{2q}}_{p,q}\big(\R;\,L^{q}(\R^{d-1})\big)\cap L^{p}\big(\R;\,\dotc{B}_{q,q}^{2-\frac{1}{q}}(\R^{d-1})\big)\] norm of $g$. The maximal regularity estimate holds for $1\leq p,q<\infty$, while the caloric trace estimate holds for $1<p<\infty$ and $1\leq q\leq\infty$. In particular, the endpoint cases $p=1$ or $q=1$ in the maximal regularity estimate are included and appear to be new. These endpoint estimates may be useful in the analysis of free-boundary Navier–Stokes problems with small initial data, whereas the caloric trace estimates may be relevant to the construction of Stokes or Navier–Stokes flows exhibiting strong boundary singularities.

The large time behavior of non-negative solutions to the absorption-diffusion equation $\partial$\textsubscr{t} u = $\Delta$ u\textsuperscript{m} - |x|\textsuperscript{$\sigma$ }u\textsuperscript{m} in (0,$\infty$) x \textsuperscript{N}, with m > 1 and $\sigma$> $\sigma$\textsubscr{0}\,:= N (m-1)/(m+1) is identified. It is shown that all solutions approach a unique stationary solution in self-similar variables, which also provides a universal upper bound (friendly giant ), strongly contrasting to the standard case $\sigma$ = 0. On the one hand, the convergence proof exploits the variational structure of the equation and a suitable Caffarelli-Kohn-Nirenberg inequality, along with the Bénilan-Crandall homogeneity regularizing effect. On the other hand, the detailed study of the stationary problem combines elliptic estimates, Moser iteration and techniques from ordinary differential equations.

In this contribution, we investigate 2-dimensional model problems of three coupled equations. We assume that the underlying Hamiltonian presents a symmetric intersection between three eigenvalues of Dirac's type with a mass function that vanish along a curve. We investigate the existence of edge modes for these models: such functions are solutions of the semiclassical associated evolution problem that are asymptotic to a coherent state with zero energy and break the correspondence principle by propagating inside the crossing set and not along a classical trajectory.

For $p>1$ and a $p$-energy on a volume doubling metric measure space, we prove the Poincaré inequality under the elliptic Harnack inequality, two-sided capacity bounds, and additional geometric and analytic assumptions. Combining this with our previous results, we obtain, under the same structural assumptions, the equivalence between the elliptic Harnack inequality together with two-sided capacity bounds and the conjunction of the Poincaré inequality and the cutoff Sobolev inequality.

We consider the inverse problem of determining an inclusion contained in an elastic body undergoing time-harmonic oscillations at fixed frequency by local Cauchy data. Both the body and the inclusion are made by different homogeneous linearly elastic isotropic materials, with piecewise constant mass densities. Under mild a priori regularity assumptions on the unknown inclusion and with no spectral hypothesis on the frequency, we establish a logarithmic-type stability estimate by local Cauchy data. We also provide a stability result in terms of the so local Dirichlet to Neumann map for sufficiently small frequency.

We study mirror flows as Banach-space counterparts of Hilbertian gradient flows and show that they play an essential role in $p$-Laplacian eigenvalue problems on metric measure spaces. Under a Rellich–Kondrachov type compactness assumption and for $p\ge2$, we establish a Ljusternik–Schnirelman type existence theorem without assuming $C^1$-regularity of the associated energy. More precisely, we prove that every element $\lambda$ of the Krasnoselskii spectrum is an eigenvalue of the $p$-Laplacian $\Delta_p$; that is, there exists a nontrivial solution $f$ to $\Delta_p f=-\lambda |f|^{p-2}f$.We also investigate the large time behavior of the corresponding mirror flow and prove its convergence to an eigenfunction when the eigenvalue is simple and isolated.

We prove the global existence of segregated weak solutions of a one-dimensional degenerate cross-diffusion system with independent drifts, which is endowed with a Wasserstein gradient flow structure. We argue by a Lagrangian formulation written in terms of the (pseudo-)inverse for the cumulative mass function of the sum of the species, which we solve by a Minimising Movement Scheme in the setting of $L^2 \cap BV_{\mathrm{loc}}$. This Lagrangian problem gives rise to a parabolic PDE similar to a $p$-Laplace equation, with typical range $p \in (-\infty,1)$. We employ monotonicity methods \emphà la} Minty–Browder to obtain strong convergence and pass to the limit $\tau \to 0$ in the time-step of the discrete scheme. Our contribution simultaneously treats all porous medium degeneracies, the log-entropy, and fast diffusions of index $\alpha \in (\frac{1}{3},1)$.

We study the mixed local-nonlocal operator $\mathcal{L}_{p,q} := -\Delta_p + (-\Delta)_q^s$ in the isocritical regime $p^* = q_s^*$, i.e. $1 - N/p = s - N/q$, under which both operators become critical for the same nonlinearity. We consider \[ -\Delta_p u + (-\Delta)_q^s u = |u|^{p^*-2}u \qquad \text{in } \mathbb{R}^N, \] with $N \geq 2$, $1 < p < N$, $0 < s < 1$, $1 < sq < N$. In this regime the energy space reduces to $\mathcal{D}_0^{1,p}(\mathbb{R}^N)$, and both best Sobolev constants enter the variational structure simultaneously. We prove: $(i)$ existence of a nonnegative radial ground state via Nehari manifold methods and a double-threshold concentration-compactness analysis; $(ii)$ a logarithmic energy estimate, weak comparison principle, and strong maximum principle for all admissible exponents; $(iii)$ a weak Harnack inequality; and $(iv)$ sharp two-sided decay $U(x) \asymp |x|^{-(N-p)/(p-1)}$ for positive radial solutions, matching the fundamental solution of the $p$-Laplacian.

We consider nonhomogeneous kinetic equations that involve a free transport operator and a diffusion of porous medium type acting on velocities. The main novelty is a gradient flow interpretation of dynamics driven by an interplay of conservative and dissipative effects. We rely on a notion of discrepancy adapted to a phase space of positions and velocities, built upon second-order characteristics obeying Newton's laws. The equation appears as the steepest descent of the free energy functional. We also prove that approximate solutions constructed with an implicit Euler scheme converge to a solution of the kinetic equation. Thus, we generalise to a family of nonlinear kinetic equations the celebrated JKO scheme in mass transport theory. Most of our results are new even in the case of the linear Vlasov-Fokker-Planck equation.