Papers

Let $R$ be a normalized root system in $\mathbb{R}^d$ with reflection group $G$, let $k$ be a $G$-invariant multiplicity function, and let $\mathcal{F}_k$ be the associated Dunkl transform. We write $k_\alpha=k(\alpha)>0$ and $\mathrm{d}\mu_k(x)=w(x)\,\mathrm{d}x$, where $w(x)=\prod_{\alpha\in R_+}|\langle x,\alpha\rangle|^{2k_\alpha}$. We study observability, Hölder-type interpolation, and spectral inequalities for the Dunkl heat equation on $\mathbb{R}^d$. We establish a Bernstein inequality for ordinary derivatives and a Logvinenko–Sereda theorem with a spectral constant of the form $e^{C(1+N)}$ for functions whose Dunkl transforms are supported in $\overline{B(0,N)}$. We characterize observable sets as the measurable sets that are thick with respect to $\mu_k$, and prove the equivalence of the observability, Hölder-type interpolation, and spectral inequalities.

This paper is the final installment in our series on the geometric theory of complex Monge-Ampere equations. We study singular Kahler metrics on compact normal Kahler spaces with klt singularities whose volume densities lie in $L^p$ for some $p>1$. We show that a lower bound for the Ricci current in the sense of pluripotential theory is equivalent to a synthetic Ricci lower bound in the sense of RCD theory. As a consequence, every Kahler class on a compact normal Kahler space with klt singularities admits an RCD structure, which makes differential geometric analysis available on singular complex spaces. As an application, we show that the fundamental group of a compact Kahler Calabi-Yau space with klt singularities is almost Abelian. We further establish compactness results for singular Kahler-Einstein spaces.

This paper is the continuation of our earlier work on Holder estimates for Kahler potentials on compact normal Kahler spaces. We establish a geometric counterpart for analytic Holder regularity of Kolodziej. If a singular metric $\omega_\phi$ has Holder continuous potentials with respect to a smooth background metric, then its distance function is Holder continuous with respect to a smooth background distance. The result holds on normal Kahler spaces and yields compactness of the metric completion as well as its identification with the underlying complex space if $\omega_\phi$ is a Kahler current. Under this positivity assumption, our estimates establish the Holder equivalence between the intrinsic canonical Kahler metrics and the extrinsic smooth metrics on Kahler-RCD spaces, particularly on smoothable Kahler-Einstein spaces.

Let $X$ be a compact Kahler manifold. The analytic stability theorem of Kolodziej for complex Monge-Ampere equation states that for any Kahler metrics $\omega$ and $\omega'$ in the same cohomology class, if their volume measures are bounded in $L^p(X)$ (for some $p>1$) and close in $L^1(X)$, then their Kahler potentials are close in $L^\infty(X)$. In this paper, we establish the geometric stability for complex Monge-Ampère equations that $L^1$-closeness of volume measures implies $L^\infty$-closeness for the induced distance functions by $\omega$ and $\omega'$. Consequently, we prove that any non-smooth Kahler current with volume measure bounded in $L^p$ (for some $p>1$) and Ricci current bounded below induces a unique metric space, which turns out to be a compact RCD space homeomorphic to $X$ itself.

We study a class of complex-valued KdV-type equations with cubic second-order derivative nonlinearities and arbitrary complex coefficients. For sufficiently small initial data in $L^2$, we prove local well-posedness in $H^s(\mathbb R)$ for $s\ge3/4$. The key ingredient is a family of dyadic resolution spaces $Z_k=X_k+Y_k$, designed to overcome the logarithmic divergence arising from high$*$low$*$low interactions in which both derivatives fall on the highest-frequency factor. For the completely integrable third-order Kaup–Newell flow, we establish global-in-time $H^s(\mathbb R)$ bounds for $0\le s<1$ and global well-posedness in $H^s(\mathbb R)$ for every $s\ge3/4$, with no smallness assumption on the initial data.

This paper is devoted to the non-diffusive computation of solutions to $N$-dimensional hyperbolic systems of conservation laws (HCL) and more specifically the accurate approximation of simple waves. We develop the LeafNet algorithm, which relies on: i) a reformulation of HCL as a coupled system involving smooth solutions, and ii) an accurate physics informed algorithms approximating smooth functions. An error estimate analysis and two-dimensional numerical experiments are proposed to illustrate the presented strategy.

We prove that a compact immersed disk of constant mean curvature in $\R^3$ whose boundary is mapped diffeomorphically onto a round circle is a planar disk or a spherical cap.

This paper is concerned with the Weyl problem, i.e., the existence of isometric immersions or embeddings of two-spheres into the three-dimensional Euclidean space or general ambient three-manifolds. We establish two results on the degenerate Weyl problem, namely when the Gaussian curvature $K_g$ is only nonnegative rather than strictly positive. First, for a smooth metric $g$ on the two-sphere, if $K_g$ is strictly positive except at finitely many points where the Hessian of $K_g$ is positive definite, then $g$ admits a global $C^{2,1}$-isometric embedding into $\mathbb{R}^3$. It appears to be the first result on the degenerate Weyl problem with purely intrinsic conditions on $g$. Second, for a general simply-connected ambient three-manifold $(\mathcal{M},{\overline{g}})$, if $K_g \geq K_0 \geq {\rm sec}_{\overline{g}}$ for some constant $K_0$, $(K_g-K_0)^{-1/2} \in L^p$ with $p \geq 2$, and a certain uniform pinching condition holds for approximate nondegenerate isometric immersions, then there exists a $W^{3,p}$-isometric immersion. Alongside we also resolve the nondegenerate Weyl problem (i.e., when $K_g>0$) into general simply-connected ambient three-manifolds for $g$, ${\overline{g}} \in C^{2,1}$.

We study an optimal control problem for the two-dimensional Navier-Stokes equations projected onto the tangent space of the $\mathbb{H}$-unit sphere manifold $\mathcal{M} := \{\boldsymbol{u} \in \mathbb{H} : \|\boldsymbol{u}\|_{\mathbb{H}}=1\}$. This geometric projection yields a constrained evolution system of finite codimension, ensuring that the state remains on $\mathcal{M}$ for all times. Using a Faedo-Galerkin approximation technique, we establish the global existence and uniqueness of strong solutions to the constrained Navier-Stokes equations with external controls $\boldsymbol{U} \in \mathrm{L}^2(0,T;\mathbb{H})$, together with uniform energy estimates. The associated optimal control problem, minimizing a quadratic cost functional subject to the manifold constraint, is shown to admit at least one optimal pair $(\boldsymbol{u}^*,\boldsymbol{U}^*)$. We derive the first-order necessary optimality conditions by formulating an optimal control system whose state-control pair $(\boldsymbol{u}^*,\boldsymbol{U}^*)$ evolves on the tangent bundle $T\mathcal{M}$ of the constraint manifold $\mathcal{M}$. In this setting, the control satisfies $\boldsymbol{U}^*(t)\in T_{\boldsymbol{u}^*(t)}\mathcal{M}$ for all $t\in[0,T]$, ensuring that the dynamics remain confined to the tangent bundle $T\mathcal{M}$. Furthermore, we establish the Fréchet differentiability of the control-to-state mapping $\boldsymbol{U}\mapsto \boldsymbol{u}_{\boldsymbol{U}}$ in $\mathrm{C}([0,T];\mathbb{H})\cap\mathrm{L}^2(0,T;\mathbb{V})$, with the corresponding linearized state characterized as the unique weak solution of a linearized constrained Navier-Stokes system preserving the tangency condition. These results provide a rigorous analytical foundation for the optimal control of fluid flows constrained to nonlinear manifolds.

We study the Kuramoto–Sivashinsky (KS) equation in one, two, and three spatial dimensions subject to a codimension-one constraint. The constraint preserves the \(L^2\)-norm of the solution and consequently provides a uniform a priori bound that plays a central role in the higher-order energy estimates. In one spatial dimension, we establish global well-posedness by means of a Fourier–Galerkin approximation and compactness arguments. In two and three dimensions, we first prove the existence and uniqueness of local strong solutions using analytic semigroup theory and the Banach fixed-point theorem. In two dimensions, the preserved \(L^2\)-norm, combined with the Agmon inequality and suitable higher-order energy estimates, allows us to control the nonlinear terms and obtain global-in-time a priori bounds. Consequently, we establish the existence and uniqueness of global strong solutions for arbitrary sufficiently regular initial data, without any smallness assumption. This provides a global well-posedness result for the constrained two-dimensional KS equation, in contrast with the corresponding unconstrained problem, for which global well-posedness of strong solutions remains open in general. In three dimensions, although the constraint yields a uniform \(L^2\)-bound, the available interpolation and Agmon estimates are insufficient to close the higher-order energy estimate. This identifies a genuine dimension-dependent obstruction to extending the global regularity argument to three dimensions.

We construct a smooth solution of the focusing cubic nonlinear Schrödinger equation on a square two-dimensional torus which is global in both time directions and concentrates at one point as time tends to positive infinity. Its mass is the mass of the Euclidean ground state and its energy is zero. If $\lambda(t)$ denotes its concentration scale and $L$ the period, then $L/\lambda(t)=2\log t+\frac{11}{2}\log\log t+O(1)$. The solution approaches a phased, periodized ground state with an error tending to zero in every fixed Sobolev space. In particular, its $H^s$ norm is asymptotic to an explicit constant times $(\log t)^s$ for every $s>0$. We also prove uniqueness, up to constant phase and time translation, within the quantitative single-profile asymptotic class used in the construction. The mechanism is the exponentially small interaction of the profile with its periodic images. Exact mass and energy constraints select the concentrating branch, while a positive energy on the moving constrained space controls the terminal-value problem at every fixed regularity order.

We prove the existence of an essential spectral gap for convex cocompact complex hyperbolic manifolds under the hypothesis that the limit set does not contain a complex circle. We apply an extension to complex hyperbolic manifolds due to Quan of the approach of Vasy to meromorphic continuation of the resolvent. This allows us to adapt the method of Dyatlov-Zahl to prove fine microlocal properties of resonant states. Following recent work of Athreya-Dyatlov-Miller and Dyatlov-Jezequel, we then take advantage of the different expansion rates in different directions for the geodesic flow, and apply the one dimensional fractal uncertainty principle of Bourgain-Dyatlov along the fast expanding direction. Combining these techniques provides a new resolvent bound from which the essential spectral gap follows.

This article concerns Garding Dirichlet operators, and also the theory of Garding polynomials. Garding Dirichlet operators are those which arise from homogeneous polynomials on the space S(n) of symmetric $n \times n$ polynomials, which are hyperbolic with respect to any positive definite matrix. (In particular, any polynomial on ${\mathbb R}^n$, which is hyperbolic with respect to elements in $({\mathbb R_+})^n$, gives rise to a Garding operator on S(n), in many ways.) Here a new duality theorem is developed with many special cases and examples. The dual $g^*$ of a Garding operator $g$ is used to prove a new basic inequality for $g$. An important Lemma of Guo-Phong-Tong is improved upon as a final application.

This paper considers the existence of a unique global-in-time strong solution to the heat equations in two unbounded domains $\Omega_A$, $\Omega_B$ and the interface $\Gamma (= \partial \Omega_A \cap \partial \Omega_B)$. We introduce and study some function spaces in the two unbounded domains and the interface. We apply our function spaces and maximal $L^p$-regularity for Hilbert space-valued functions to show the existence of a local-in-time strong solution to our heat equations. By using an energy equality of our heat system, we prove the existence of a unique global-in-time strong solution to the system with large initial data when the slope of the interface is gentle and our parameters are limited. The key ideas for showing the existence of our strong solutions are to transform our system into a system of equations in two half spaces $\mathbb{R}^3_+, \mathbb{R}^3_-$ and the whole space $\mathbb{R}^2$, and to make use of nice properties of the heat semigroups and kernels for $\mathbb{R}^3_+$, $\mathbb{R}^3_-$, and $\mathbb{R}^2$. In Appendix (I), we derive our heat equations in the two unbounded domains and the interface from an energetic point of view. In Appendix (II), we study representation formulas for differential operators on unbounded domains and surfaces.

This paper proves generic structural stability for Riemann solutions to $n \times n$ systems of hyperbolic conservation laws in one spatial dimension. Under assumptions of strict hyperbolicity, genuine nonlinearity, and a regular manifold hypothesis on the Rankine-Hugoniot map, we show that for almost every pair of left and right states, any $n$-wave Riemann solution consisting of Lax-admissible shocks and rarefactions is structurally stable under perturbations of the left state, the right state, and the flux function in the $C^2$ topology. The central new idea is sequential transversality, which chains the $n$ waves through intermediate states and transports their tangent contributions to a common reference point via pushforward maps, reducing the structural stability condition to the invertibility of an $n \times n$ transversality matrix. We apply the results to the $p$-system, polydisperse particle-laden thin films, and machine-learned flux approximations.

These are extended notes based on a series of lectures given by the author at the Institut de Mathematiques de Toulouse in June 2025. The goal is to present classical problems as well as the most recent developments in spectral geometry of the Laplace operator in Euclidean domains, subject to Dirichlet, Neumann and Robin boundary conditions. Among the topics covered, there are embedded eigenvalues in unbounded domains, spectral isoperimetric inequalities, nodal-line and hot-spots conjectures, inverse problems of hearing the shape of a drum and geometrically induced eigenvalues and Hardy-type inequalities in curved tubes.

Let $(\mathcal{H}\_t)_{t \geq 0}$ denote the standard Ornstein-Uhlenbeck semigroup over ambient space $\mathbb R^d$ equipped with Gaussian measure $d\gamma(x) = e^{-|x|^2} dx$. Further let $\mathcal{H}^*(f)(x) = \sup_{t > 0} \mathcal{H}_t(|f|)(x)$ denote the associated maximal operator. Since $(\mathcal{H}_t)_{t \geq 0}$ is a Markovian semigroup over $(\mathbb R^d,d\gamma(x))$, it is well known that $\mathcal{H}^*$ is bounded on $L^p(\mathbb R^d,d\gamma(x))$ for all $1 < p \leq \infty$. We show that $\mathcal{H}^*$ is also bounded on the weighted space $L^p(\mathbb{R}^d,\omega(x)e^{-\frac{p}{2}|x|^2}dx)$ for $1 < p <\infty$, for weights $\omega\,: \mathbb R^d \to (0,\infty)$ belonging to a certain Muckenhoupt type class $A^\alpha_p$, where $\alpha \in [0,1)$. Essentially, a weight $\omega$ belongs to $A^\alpha_p$ if and only if $\omega(Q) \omega^{-\frac{1}{p-1}}(Q)^{p-1} \leq C |Q|^p$ for all cubes $Q$ contained in any reference cube $N_\alpha(R_x) \subseteq \mathbb R^d$, where the well chosen family $(N_\alpha(R_x))_{x \in \mathbb R^d}$ covers $\mathbb R^d$, $N_\alpha(R_x)$ contains $x$ and the side length of $N_\alpha(R_x)$ is comparable to $1/\max(1,|x|)^\alpha$, with constants depending only on $d$. The class $A^\alpha_p$ contains the usual Muckenhoupt class. We also use the weight class $A_p^{loc}$ from \cite[Definition 2.2]{B} defined in a similar way as $A^\alpha_p$ but with dyadic subcubes contained in some $N(R)$ built over the Gaussian dyadic grid (see below), and we show that the local part $\sup_{t >0}\mathcal{H}_t(|f|\chi_{N(R_x)})(x)$ is bounded on $L^p(\mathbb{R}^d,\omega(x)e^{-\frac{p}{2}|x|^2}dx)$ if and only if $\omega \in A^{loc}_p$.

In this paper, we prove that the defocusing mass-critical Hartree equation is global well-posed and scatters for any initial data $u_0\in L^2$, and prove the analogous result in the focusing case, provided the mass of $u_0$ is strictly less than that of the ground state $Q$. This result removes the radial restriction from [C. Miao, G. Xu and L. Zhao, J. Math. Pures Appl. (9), 91(2009), 49-79.], thereby proving the scattering conjecture for the Hartree equation at the scaling-critical regularity.

We propose and analyze a fully discrete finite element scheme for the Beris-Edwards system of nematic liquid crystal dynamics, in which the incompressible Navier-Stokes equations are coupled to a gradient flow for the Landau-de Gennes Q-tensor. The scheme combines a linearly implicit formally second-order accurate backward differentiation formula in time with an incremental Chorin projection step for the incompressibility constraint, conforming finite elements in space, and the invariant energy quadratization approach with mass lumping for the nonlinear bulk potential. Each time step requires the solution of one linear system and one Poisson problem. We show that the scheme is uniquely solvable, that it preserves the symmetry and trace-free structure of the discrete Q-tensor and molecular field, and that it satisfies a discrete energy law without any restriction on the time step. Our main result is that, as the mesh size $h$ and the time step $\Delta t$ tend to zero subject to $h^{2} = o(\Delta t)$, the approximations converge along a subsequence to a weak solution of the Beris-Edwards system. The convergence proof addresses two difficulties: the projection method produces two velocity approximations, only one of which is uniformly bounded in $L^2(0,T;H^1_0(\Omega))$, and the coupling term $\mathcal{H}\nabla Q$ requires strong convergence of $\nabla Q$, which we obtain from the structure of the equation for $\mathcal{H}$ rather than from any discrete $H^2$-bound. Numerical experiments in two dimensions exhibit approximately second-order convergence in space and time, and reproduce the splitting of a $+1$ point defect into two $+1/2$ defects and the transport and deformation of a skyrmion induced by a constant pressure gradient.

Under suitable relative homotopy and boundary admissibility assumptions, we establish a min-max theory in arbitrary codimension that produces nonconstant branched immersed disks or spheres in closed Riemannian manifolds, with controlled Morse index, where the prescribed mean curvature type tensor and the nonorthogonal, nonconstant contact angle condition of disks along a supporting submanifold are determined by the same differential 2-form. In Euclidean space, under suitable topological and convex barrier assumptions, we obtain such disks with free boundary on a closed supporting hypersurface and a Morse index bound depending only on the ambient dimension.

We study finite-time blow-up for a semilinear shifted Tricomi equation with decreasing propagation speed and an oscillatory scale-invariant mass. We focus on the Strauss-type critical regime and prove that every weak solution with finite speed of propagation, arising from nonnegative nontrivial energy data satisfying a suitable localization condition, blows up in finite time. For sufficiently small initial data, we also obtain the corresponding critical exponential upper bound for the lifespan. The proof relies on a positive self-similar solution of the homogeneous adjoint equation represented by the Gauss hypergeometric function. Combined with a previously established lower bound for the nonlinear term and new estimates adapted to the critical case, this construction reduces the PDE problem to a nonlinear differential inequality. An ODE comparison argument then yields both finite-time blow-up and the lifespan estimate.

We complete, in the Riemannian setting, the classification of isolated singularities of positive solutions to the Yamabe equation for \(3\le n\le 29\), initiated in the flat case by Caffarelli, Gidas, and Spruck in 1989. Precisely, every isolated singularity is either removable or asymptotic to a Fowler solution. The sharpness of this range is established by counterexamples for \(n\ge30\) in our subsequent paper.

We confirm a conjecture posed by T. Iwaniec in 1982: the $L^p$ norm of the Beurling–Ahlfors transform is $p-1$ if $p\geq 2$, and $(p-1)^{-1}$ if $1<p\leq 2$.

In this paper, we investigate the singular Dirichlet problem \begin{equation}\label{abs1} \left\{ \begin{array}{rclr} -\Delta_1u &=&\dfrac{\lambda}{|x|}\hbox{Sgn}\,(u)+\dfrac{f}{u^\gamma}&\quad\mbox{in}\; \Omega,\\[1ex] u &=&0& \quad\mbox{on}\; \partial\Omega, \end{array} \right. \end{equation} where $\Delta_1u=\hbox{div}\left(\frac{Du}{|Du|}\right)$ denotes the $1$-Laplacian operator, the parameters are $\lambda\in {R}$ and $\gamma>0$, and $f$ is a non-negative function belonging to the Lorentz space $L^{N,\infty}(\Omega)$. Our main goal is to establish the existence of non-trivial solutions under the sharp restriction $\lambda<N-1$, regardless of the magnitude of the datum $f$ or the value of the singular exponent $\gamma>0$. We also show that these solutions are globally bounded and that, on the contrary, no solution exists as soon as $\lambda\geq N-1$ and $f$ is positive. These results are achieved by rigorously analyzing the asymptotic behavior, as $p\to 1^+$, of the solutions to the approximating $p$-Laplace problems \begin{equation} \left\{ \begin{array}{rclc} -\Delta_pu&=&\dfrac{\lambda}{|x|^p}|u|^{p-2}u+\dfrac{f}{u^\gamma}&\quad\mbox{ in }\; \Omega,\\[1ex] u&=&0&\quad\mbox{ on }\; \partial\Omega. \end{array} \right. \end{equation} Finally, we provide a family of explicit examples designed to illustrate the sharp optimality of our main assumptions.

We study the Dirichlet problem for an infinite series of $p$-Laplacians, $$-\sum_{p=2}^{\infty}a_{p}\Delta_{p}u=f \ \text{ in } \Omega, \qquad u=g \ \text{ on } \partial\Omega,$$ where $\{a_{p}\}$ is a sequence of nonnegative numbers whose power series has radius of convergence $\sigma\in(0,\infty]$. The operator is formally $-\operatorname{div}( {A}(|\nabla u|)\nabla u)$ with $ {A}$ singular at $|\nabla u|=\sigma$; model cases are the mean curvature operator in Minkowski space and the Born-Infeld operator of nonlinear electrostatics. The radius of convergence forces the gradient constraint $\|\nabla u\|_\infty\le\sigma$, so the natural variational problem is constrained. A unique minimizer of the associated energy always exists (for boundary data compatible with the constraint) and always solves a variational inequality, and we identify the saturation flux $\Lambda:=\sum_{p\ge2}a_p\sigma^{p-1}$ as the quantity governing solvability of the equation: if $\Lambda<\infty$ a weak solution exists only if $|\int_E f|\le\Lambda P(E)$ for every set of finite perimeter $E\Subset\Omega$, so for $f\equiv\lambda$ no solution exists once $\lambda>\Lambda h(\Omega)$, $h(\Omega)$ the Cheeger constant of $\Omega$. The threshold is sharp on balls, where the minimizer has a saturation region $\{|\nabla u|=\sigma\}$ of positive measure. When $\Lambda=\infty$, as for Born-Infeld type operators, no such obstruction is present, and the minimizer solves the equation whenever a Lipschitz bound below $\sigma$ is available; for $f\equiv0$ we obtain such a bound, uniform in the truncations of the series, under a bounded slope condition on the boundary datum. As applications we obtain an isoperimetric bound on the charge densities supported by a nonlinear electrostatics with saturating displacement, and a quantitative convergence rate for the weak field expansion of the Born-Infeld model.

For every fixed $m>1$, we construct nonzero global forward self-similar mild solutions of the three-dimensional spatially inhomogeneous Boltzmann equation for Maxwell molecules. We assume that the angular kernel is bounded and satisfies a weighted integrability condition at the grazing and head-on angles. The initial data are homogeneous under the kinetic scaling, singular at the origin in phase space, and have infinite mass. Their size is small in a weighted norm invariant under this scaling. We first prove a bilinear estimate for the gain operator in this norm, using free-transport coordinates and the orthogonality of the collision displacements. This estimate gives a global solution of the equation without the loss term, which bounds the lower and upper approximations in the Kaniel–Shinbrot iteration. We prove that their difference tendsto zero to arbitrarily high order at the initial time on a fixed annulus in phase space. The argument uses a finite number of iterations of the collision estimate and stops characteristics when they leave a spatial annulus. Self-similarity then gives finite mass and a finite first absolute velocity moment for the difference. Comparing its mass balance with its scaling law shows that the two limits agree. The resulting solution attains its initial trace locally in $L^\infty$ away from the origin in phase space and has infinite mass at every positive time.

The axisymmetric profile \(\sin\psi(\rho)=c_0\rho\log(\rho/\rho_B)\) introduced by Naito, Okuda, and Ouyang is an exact solution of the unconstrained Helfrich shape equation for every \(\rho>0\), but its mean curvature diverges logarithmically at the dimple center. A covariant local analysis shows that in Cartesian graph coordinates the immersion is nondegenerate and belongs to \(W^{2,p}\) for every finite \(p\); it is \(C^{1,\alpha}\) for every \(\alpha<1\), but neither \(C^{1,1}\) nor \(C^2\). The conformal factor has a finite nonzero limit, so the point has branch multiplicity one and is not a branch point in the sense used in weak-immersion compactness theory. The Helfrich energy is finite, whereas the complete Euler–Lagrange operator carries the distributional residue \(\E_{\CH}=4\pi c_0\delta_p\) under the convention specified below. Hence the profile is a classical solution on the punctured surface and a finite-energy weak immersion, but it is not a free weak critical point of the unforced Canham–Helfrich functional. Stationarity is restored after adding the corresponding point-force potential or under a pinned-point constraint. Area and volume multipliers cannot cancel the Dirac residue. The apparent sign discrepancy between the shape equation used in the original papers and the modern convention is explained by the opposite definition of mean curvature. These results delimit precisely what can and cannot be inferred from modern existence and regularity theorems for Canham–Helfrich minimizers.

We study a class of Dirichlet problems for coupled quasilinear elliptic systems driven by the $p$–Laplacian in a bounded domain, where the nonlinear terms split into a critical homogeneous part and a convex subcritical perturbation. Using variational methods, a nonlinear eigenvalue theory based on the Fadell–Rabinowitz $\mathbb Z_2$–cohomological index, and a refined linking construction, we prove the existence of a nontrivial solution for every parameter value under suitable dimensional assumptions. We also treat a nonhomogeneous version with small forcing terms by applying a recent abstract perturbation theorem, yielding the existence of two distinct nontrivial solutions.

We investigate a class of mixed-order quasilinear elliptic equations in \(\mathbb R^N\) driven by the \(q\)-biharmonic and \(p\)-Laplacian operators, \[ \Delta_q^2u-\Delta_pu = \lambda\frac{|u|^{r_\sigma-2}u}{|x|^\sigma} +|u|^{p^*-2}u +|u|^{q^{**}-2}u, \] where \(1<q<N/2\), \(1<p<q^*\), \(p\ne q\), \(q\ne p^*\), and the weighted exponent is determined by the natural scaling of the equation. We establish compact weighted embeddings compatible with this mixed scaling and develop the corresponding variational framework. As an application, we prove existence and multiplicity of nontrivial solutions under suitable assumptions on the parameters. We also establish a local regularity result for the more general equation $\Delta_q^2u-\Delta_pu=f(x,u),$ where \(f\) is a Carathéodory function with local critical growth. By combining nonlinear potential estimates, a regularity-lifting argument, and a finite bootstrap for an associated second-order system, we obtain higher local regularity of both \(u\) and the nonlinear flux \(|\Delta u|^{q-2}\Delta u\). This regularity result, which is also of independent interest, enables us to derive a Pohozaev-type identity for the equations under consideration.

We prove nonlinear Landau damping for the two-dimensional Vlasov–Poisson system in the unconfined Euclidean setting near the homogeneous Poisson equilibrium \[ \mu(v)=\frac{1}{2\pi}(1+|v|^2)^{-3/2}. \] For small, smooth, localized, and neutral perturbations we establish global existence, quantitative pointwise decay estimates for the electric field, and scattering to free transport. The result we prove here appears to be the first nonlinear Landau damping result for solution of the unconfined 2D Vlasov–Poisson system. The uniform Penrose stability bound degenerates at low spatial frequencies, giving rise to weakly damped plasma oscillations. We separate a faster-decaying regular field from oscillatory components with temporal factors $\cos(t)$ and $\sin(t)$. The proof combines a Volterra representation of the density, weighted estimates for the characteristic flow and nonlinear moments, and integration by parts in time. Neutrality supplies the spatial cancellation needed in the critical two-dimensional estimates.