Papers

Given two conformal asymptotically hyperbolic metrics which are either both simple or both negatively curved, we show that if their (marked) renormalized boundary distances coincide for some choices of conformal representatives in their conformal infinities, then the two metrics are equal.

Spectral optimal partition and segregation problems are deeply connected with harmonic maps, eigenfunctions, and the fine structure of nodal sets for linear elliptic equations. In this paper, we show that replacing the spectral energy by torsional rigidity leads to a genuinely different theory. The resulting optimal configurations are governed locally not by harmonic equations, but by torsion-type energies and unstable free boundary problems, thereby creating a natural bridge between optimal partition theory and the analysis of sublinear free boundary phenomena. We prove existence of optimal torsional partitions and segregated torsional configurations, together with optimal Lipschitz regularity of the associated nonlinear eigenfunctions. We establish a strong unique continuation principle, characterize the admissible vanishing orders and the corresponding blow-up profiles, derive sharp Hausdorff dimension estimates for the nodal set and its singular subset, and prove $C^{1,\alpha}$-regularity of the regular part of the free boundary. The proofs combine variational arguments with Almgren-type and Weiss-type monotonicity formulae adapted to the intrinsically sublinear torsional regime, blow-up analysis, and tools from geometric measure theory.

We establish global-in-time Strichartz estimates without loss of derivatives for wave equations with time-dependent Lipschitz coefficients, which satisfy an additional structural assumption. Our approach is based on a parametrix construction through the Phillips functional calculus. We furthermore obtain the well-posedness of such wave equations with Lipschitz coefficients in $H^1$.

Inferring thermal fields and thermophysical properties from limited measurements is a fundamental challenge in micro- and nanoscale heat conduction, where the classical Fourier law breaks down and the phonon Boltzmann transport equation (BTE) is needed to capture non-diffusive transport effects. In this work, we extend Monte Carlo physics-informed neural networks (MC-PINNs), originally developed for forward phonon BTE problems [J. Comput. Phys. 542, 114364, 2025], to inverse multiscale heat conduction problems. Two representative classes of inverse problems are considered: (i) reconstructing the full thermal field from sparse interior temperature measurements when boundary conditions are unknown, and (ii) simultaneously inferring the unknown relaxation time together with the thermal field. Problem-specific MC-PINN architectures and training strategies are designed for each class. The mesh-free Monte Carlo sampling strategy enables a unified treatment across diffusive, transitional, and ballistic transport regimes without requiring a priori knowledge of the relaxation time. The proposed method is evaluated on quasi-one-dimensional, quasi-two-dimensional, and three-dimensional benchmark problems covering a wide range of Knudsen numbers, as well as on a realistic 3D fin field-effect transistor (FinFET) structure. Results demonstrate that MC-PINNs consistently outperform purely data-driven deep neural networks, particularly in the sparse-data regime, and can accurately infer spatially uniform relaxation times. For spatially varying relaxation times, the inferred distributions capture the dominant thermal response, and numerical simulations using the recovered parameters reproduce the macroscopic fields with good accuracy. These findings establish MC-PINNs as an effective and physically consistent framework for inverse thermal analysis at micro- and nanoscales.

A cross-diffusion system with advection is considered on the whole line, describing the dynamics of two segregating population species. Starting from two initial densities supported on the half-lines $x\leq 0$ and $x\geq 0$, respectively, we construct two distinct solutions of the system: one pair of densities remains confined to their initial supports and stay completely segregated, while the second pair of densities begins to invade the opposite half-line after a finite time (mixing). To the best of our knowledge, this provides one of the first examples of non-uniqueness for this class of equations, together with an explicit demonstration of mixing phenomena. The construction produces infinitely many mixing solutions and applies throughout the full range of pressure exponents. For a certain range of exponents, we additionally obtain quantitative estimates on the mixing process.

We study a one-dimensional cross-diffusion system of two populations. Their densities are diffused with a common pressure that depends on the total density, but are transported by two independent external potentials. Starting from segregated initial data with a single interface, we show that the way the two densities meet is not given by the equation alone: it depends on the notion of solution one chooses. When the drifts push the two phases towards each other at the interface, the vanishing-viscosity solution creates an overlap, whereas a segregated weak solution also exists. The two solutions are distinct, so the Cauchy problem is not well posed in the class of weak solutions. We first prove the result for an explicit stationary segregated solution, and then extend the non-uniqueness to general segregated data.

In this paper, we study the weighted $L^2$-Caffarelli-Kohn-Nirenberg inequalities for curl-free vector fields and second order derivatives. Firstly, we prove a family of the sharp weighted second order $L^2$-Caffarelli-Kohn-Nirenberg inequalities that complements the results in [C. Cazacu, J. Flynn and N. Lam, Calc. Var. Partial Differential Equations 62 (2023), no. 4, Paper No. 118, 26 pp.] and [A. T. Duong and V. H. Nguyen, On the sharp second order Caffarelli-Kohn-Nirenberg inequality. Ann. Fenn. Math., 50(1):275–286, 2025]. Secondly, we establish a stability version of the sharp weighted $L^2$-Caffarelli-Kohn-Nirenberg inequalities for curl-free vector fields proved by Cazacu, Flynn and Lam. Finally, we prove a stability estimate for the sharp weighted second order $L^2$-Caffarelli-Kohn-Nirenberg inequalities established in this paper. Our approach is based on the spherical harmonic decomposition method, the one dimensional integral inequalities and their improvements.

In this article, we study the semi-linear elliptic problem: \begin{equation*} -\Delta_{\mathbb{B}^N} u \, - \, \lambda u = |u|^{p-1}u, \quad u \in H^{1}(\mathbb{B}^N), \end{equation*} where $N \geq 3$, $\lambda > \frac{N(N-2)}{4}$, and $1 < p \leq 2^*-1$. Here, $\mathbb{B}^N$ represents the Poincaré ball model of the hyperbolic space, and $H^{1}(\mathbb{B}^N)$ denotes the Sobolev space on $\mathbb{B}^N$. In the critical case $p = 2^*-1$, with $\lambda < \frac{(N-1)^2}{4}$ and $N \geq 4$, we establish the existence and multiplicity of nonradial sign-changing solutions. Moreover, our result answers an open question: “the existence of sign-changing solutions in the lower dimensions $4 \leq N \leq 6$". We also prove a partial non-existence result for a large class of symmetric solutions when $\lambda > \frac{(N-1)^2}{4}$, based on a quantitative orbit packing condition.

There are relatively few results on the convexity of solutions to complex equations. In this paper, We prove a strict real log-concavity theorem for the first eigenfunction of the complex $\sigma_{2}$-Hessian operator on smooth, bounded, real uniformly strictly convex domains in $\mathbb{C}^{n}$. As an application, we obtain a Brunn–Minkowski inequality for the first complex $\sigma_{2}$-Hessian eigenvalue. The proof combines a Bian–Guan constant-rank argument, a new inverse-convexity lemma for the compressed real Hessian, and Salani's viscosity admissible-test-function method.

We study the local existence and blow-up behavior of solutions, with possible growth at space infinity, for the diffusive Hamilton-Jacobi equation $u_t-\Delta u=|\nabla u|^p$ ($p>1$), in a half-space with Dirichlet boundary conditions. Under optimal, polynomial growth assumptions on the initial data, characterized by the critical exponent $p/(p-1)$, we first prove local existence and uniqueness of a maximal classical solution and establish a blow-up alternative. This is complemented by a nonexistence result showing that $p/(p-1)$ is the sharp threshold for admissible growth at infinity. Next, for initial data with subcritical growth, we show global-in-time existence for $1<p\le2$ and that finite-time blow-up can occur only through gradient blow-up for $p>2$, and we derive sharp upper and lower estimates on the gradient, including a precise type~II boundary blow-up rate and a dichotomy in its behavior near the singular time. Finally, in the case $p>2$, for any type~II blow-up solution, we prove convergence after rescaling) to an explicit one-dimensional profile and providing a refined description of the asymptotic singularity formation.

In this paper, we consider a parabolic problem with constraint written as a differential inclusion, driven by a multiplicative colored noise and involving a p-Laplace operator (for $p \geq 2$), nonlinear random source terms and subject to Neumann boundary conditions on a bounded Lipschitz domain of $R^d$ with $d \geq 1$. This contribution aims at proving existence and uniqueness of a solution for such a multivalued problem. On one hand, the existence result is proved by the analysis of a semi-implicit time discretization scheme constructed on a smoother version of our problem, itself obtained by a regularization "à la Moreau-Yosida" of the subdifferential term. The key point of our approach consists in finding a clever relation between the time step denoted $\tau$ and the Moreau-Yosida regularization parameter denoted $\epsilon$ in view to pass simultaneously to the limit with respect to $\tau$ and $\epsilon$. On the other hand, the uniqueness of the solution is proved by standard arguments.

We prove the global-in-time existence of weak solutions to the isothermal Maxwell–Stefan system on the whole space $\mathbb R^3$. The main difficulty is that, unlike in bounded domains, the concentrations generally have infinite mass and the standard mixing entropy is not finite. We therefore work with the relative entropy with respect to a strictly positive constant equilibrium state. The proof proceeds by solving the problem on balls $B_R$ with no-flux boundary conditions, using the bounded-domain entropy theory, and deriving estimates independent of $R$. These estimates yield uniform control of the relative entropy, the gradients $\nabla\sqrt{c_i}$, and the fluxes $J_i$. Passing to the limit $R\to\infty$ is achieved by local compactness and a diagonal argument. The resulting weak solution satisfies the Maxwell–Stefan system in the sense of distributions and obeys a global relative entropy inequality.

Defeaturing is routinely employed in numerical simulations of medium- and high-voltage equipment, not only to reduce computational costs but also to make meshing possible. But the errors that this practice introduces in the predicted breakdown/inception voltages are currently neglected. This work presents a goal-oriented a posteriori error estimation framework that aims at assessing such defeaturing errors in inception voltage computations. The methodology combines a first-order approximation of the inception voltage error with dual-weighted residual estimators, providing an error bound that quantifies the impact of defeaturing on the simulation results. The approach builds upon the streamer integral model for inception voltage prediction and uses the recently proposed certified goal-oriented analysis-aware defeaturing estimator of~[1] for elliptic PDEs. The first-order approximation of the inception voltage is a linear functional J of the background electric field whose definition involves a line integral along the critical field line, and is therefore only of low regularity. To meet the abstract regularity assumption of the goal-oriented theory, we introduce a Gaussian mollification of J. The methodology is illustrated on a pin-plate benchmark with a small protrusion of varying size and shape, using a single shared adaptive mesh and an adaptive coupling of the mollification width to the local mesh size in order to ensure that the defeaturing error dominates over the discretization and the regularization errors.

We prove a three-dimensional positive-cone continuation criterion for the stress-diffusion-free Oldroyd-B system on the periodic torus. Writing the positive conformation tensor as A = exp(B), we show that finite-time breakdown of a strong H^s solution, s > 5/2, can occur only through loss of the logarithmic spectral envelope of A or divergence of the endpoint vorticity clock given by the time integral of the B^0_{infty,1} norm of curl u. The proof combines compact positive-cone envelopes, endpoint Biot-Savart estimates, and high-order logarithmic conformation estimates, without using stress diffusion. We also derive a positive-cone Reynolds admissibility criterion with an exact residual-work cost. The least L^2 conformation residual needed to pay positive pressure-free residual work is determined by the entropy-dual lever G = I - A^{-1}, and this cost degenerates quantitatively near the equilibrium A = I. Together, the two criteria identify the same positive-cone obstruction in the strong and relaxed regimes: before breakdown one must control the endpoint flow clock on a compact logarithmic cone, while after passage to a relaxed description positive residual work must be paid for by an exact entropy-dual conformation defect.

As part of Lutwak's broadening of the Brunn-Minkowski theory, and extending the notion of affine quermassintegrals and dual curvature measure discussed by Milman, Yehudayoff and Huang, Lutwak, Yang and Zhang, centro-sectional measures with real parameter q have been recently introduced by Cai, Leng, Wu, Xi. In this paper, we introduce the Lp cross sectional Minkowski problem analogously to the Lp dual Minkowski problem formulated by Lutwak, Yang and Zhang. We solve the Lp centro-sectional Minkowski problem for p>1 and q>0, discuss the regularity and uniqueness of the solution, and prove Lp Brunn-Minkowski-type inequalities when $p$ is relatively large.

Let $\mathbb{B}$ be the unit ball in $\mathbb{R}^N$, $W_0^{1,N} \left( \mathbb{B} \right)$ is a standard Sobolev space. Zhang proved the extremal function of the Moser-Trudinger inequality as follows, \begin{align*} \int_{ \mathbb{B}} h_\epsilon(x) e^{ \alpha_N \left( 1 + \epsilon \right) |u_{\epsilon}|^{ \frac{N}{N-1} } } dx, \quad u_{\epsilon} \in W_0^{1,N} ( \mathbb{B} ) \cap \mathcal{S}, \end{align*} where $\alpha_N = \omega_N^{ \frac{1}{N-1} }$, $\omega_N $ is the area of the unit sphere in $\mathbb{R}^N$(see \citep{26}) . In this paper, we consider the compactness of the sequence $\{ u_{\epsilon} \}_{\epsilon} $ and prove that it has a subsequence converging to a function in $C^1 \left(\overline{ \mathbb{B}} \right)$.

We audit a finite-scale program for the local regularity problem of the three-dimensional incompressible Navier–Stokes equations. The program develops critical ledgers, coarse-grained defect decompositions, pressure–flux work identities, quotient cleanings, and bad-scale counting mechanisms. These results form an obstruction calculus: they locate how Caffarelli–Kohn–Nirenberg badness may be transported, hidden, or reproduced across scales, but they do not by themselves provide a coercive estimate excluding a surviving obstruction. We prove a resolution lemma separating full CKN badness into coarse badness and subfilter residual, and show that no unconditional single-scale domination by a signed combined-work detector is available. The audit therefore identifies the next necessary target: a filtered stretching–diffusion estimate, including subgrid forcing, leakage, pressure tails, and direction-incoherence defects, capable of converting the existing decomposition theory into a genuine regularity or obstruction-exclusion mechanism.

This paper is concerned with propagation phenomena of spatially periodic combustion reaction-diffusion equations in exterior domains. It is known that there is a pulsating front connecting 0 and 1 with positive speed in $\mathbb{R}^N$ for any direction $e\in\mathbb{S}^{N-1}$. We first prove that there exists an entire solution originating from a pulsating front in the exterior domain. Then, we prove that the entire solution propagates completely. Additionally, by constructing appropriate super- and sub-solutions, we establish that the entire solution is a transition front connecting 0 and 1, and that it is trapped between two translates of the pulsating front as $t\rightarrow +\infty$. Finally, under a suitable assumption, we show that the entire solution converges to the same pulsating front as $t\rightarrow +\infty$, as well as the uniqueness of such entire solutions.

We prove a finite-scale coarse-grained decomposition for the three-dimensional incompressible Navier-Stokes equations near the Caffarelli-Kohn-Nirenberg local regularity framework. The first part is a local resolution lemma for the scale-critical quantity: for every spatial filter length ell > 0, Psi(r) <= 4 Psi^ell(r) + 4 Omega^ell(r), where Psi^ell(r) is the corresponding coarse-grained velocity-pressure quantity and Omega^ell(r) is the explicitly defined subfilter residual. Thus a CKN-bad scale is either visible at the resolved level or is carried by unresolved velocity-pressure oscillation. The second part is an exact fixed-chain depletion theorem for the combined pressure-flux work distribution G^ell = Pi^ell + div(P^ell U^ell), Pi^ell = -R^ell : grad U^ell, which is the signed work density appearing in the localized resolved-energy balance. For finite-dimensional active test families with common endpoint traces, we obtain a constructive active-work extraction and a weighted telescoping inequality: forward combined work and resolved dissipation are paid by the initial localized kinetic energy, explicit localization leakage, and negative combined work/backscatter.

We prove a residual-work compatibility theory for positive-cone viscoelastic Reynolds states. In Oldroyd–B, the entropy cancellation eliminates incompressible transport and upper-convected stretching against polymeric stress work. After quotienting pressure tensors and spatial means, positive pressure-free residual work can be paid only by the conformation residual through the entropy-dual lever (G=I-A^{-1}). This yields the closed residual-work cone (P+(\alpha/2)\int_{\mathbb T^d}G:S,dx\le 0), exact windowed tests, and the least-cost Hilbert-space repair; constrained closures pay through the projected lever. Strong residual-data limits preserve this cone, while weak–weak limits require the product-defect measure carried by (G:S). The augmented topology is sharp, as shown by localized residual packets. For FENE-P, the lever (G_b(C)=((b-d)/(b-\operatorname{tr}C))I-C^{-1}) makes the finite-extensibility boundary a genuine residual-work boundary.

Size-selective harvesting is often justified by the intuition that larger individuals have higher value and should therefore be harvested above a critical size. We study a controlled size-structured transport model in which a scalar environmental variable, generated by the population itself, modifies the vital rates \(g(E,l)\) and \(\mu(E,l)\). The first result is a corrected stationary closure theory: crowding-suppressed growth increases residence density at the inflow, so the closure derivative is not determined by pointwise profile monotonicity. Instead, $\Phi'(E)=\mathsf A(E)-\mathsf C(E)$, an integrated balance between residence-time amplification and cumulative survival loss. The second result is an exact stationary adjoint reduction. The nonlocal switching correction has rank one, $S=S_{\rm red}-\frac{A}{1-B}\psi$, and its zero-discount feedback gain satisfies the identity $B(0)=\Phi'(E^*)$. Thus the same scalar governs stationary closure sensitivity and threshold fragility. We also establish finite-horizon well-posedness and compactified optimal-control existence in a spatial-\(BV\) policy class. Numerical certification in a density-dependent von Bertalanffy model shows when minimum-size harvesting persists and when vital-rate feedback creates multiple-switch harvest windows.

We prove endpoint continuation criteria for stress-diffusion-free incompressible viscoelastic flows by working modulo pressure. In two space dimensions, the pressure-free part of any smooth spectral isotropic stress reduces to a single active deviatoric channel (q_1(a,|Y|^2)Y), where (C=aI+Y) and (\operatorname{tr}Y=0). This scalar quotient structure allows a weighted active-deviatoric energy to cancel the top-order coupling between polymer stretching and the divergence of the active stress. On compact conformation windows the resulting high-order estimate depends only on an endpoint velocity clock and a logarithmic conformation norm. For Oldroyd–B this gives continuation of strong two-dimensional solutions under (\nabla u\in L^1_tB^0_{\infty,1}), while for FENE-P it gives continuation under (\nabla u\in L^2_tB^0_{\infty,1}). In both models the compact conformation window and logarithmic bound are derived from the velocity clock and the model barriers, rather than imposed as independent hypotheses. The criteria are formulated in integer Sobolev strong-solution classes and do not assert Leray-type weak-solution or critical-space local well-posedness results. We also identify a static operator obstruction showing the functional necessity of the logarithmic threshold for the pressure-free stress map. In three dimensions the quotient contains an additional residual channel (q_2(Y^2)^\circ), so the exact scalar closure is intrinsically two-dimensional. On prescribed compact windows this residual can be absorbed by viscosity; for Oldroyd–B and FENE-P it vanishes because (q_2\equiv0).

We develop a rigorous theory of anisotropic mixed fractional Landau inequalities for rotating compressible fluid flows at high Mach numbers, incorporating Coriolis and centrifugal forces. We introduce rotating fractional Sobolev spaces $\mathcal{W}^{\nu,p}_{\alpha,\Omega}(\mathbb{R}^k)$, which encode directional scaling, fractional dissipation and rotational effects. We prove sharp fractional Landau inequalities with explicit dependence on the Mach number, the rotation rate $|\Omega|$ and the anisotropy vector $\alpha$. Key tools are a rotating Littlewood–Paley decomposition, anisotropic maximal estimates with rotational corrections, and commutator estimates for the Coriolis term. As applications, we establish stability bounds for neural operators approximating rotating compressible flows and derive optimal approximation rates of order $N^{-\nu/d_{\alpha,\Omega}}$, where $d_{\alpha,\Omega}=\sum_i\alpha_i^{-1}+\kappa|\Omega|^{2/\nu}$ is the effective anisotropic–rotational dimension. Our results provide a mathematical foundation for analysing high–Mach rotating flows and for designing physically informed neural architectures.

Let $W^T$ map Dirichlet data $f$ to time $T$ state $u^f(T, \cdot)$, we study the $K$-normal operator $C^T_K:= (W^T)^*KW^T$ where $(W^T)^*$ is the standard $L^2(dx)$ adjoint. We prove that it is a locally finite sum of Fourier Integral Operators (FIOs) away from the glancing directions with canonical relation involving collision data at time $T$ between pairs of velocities. However, if the operator is $M$-self-adjoint, then the canonical relation for $C^T_M$ loses the collision data. Thus, when system satisfies certain coupling requirement, we demonstrate off-polarization leakage detection and introduce a new collision rigidity problem. Combine these two parts, we prove an inverse problem of velocity recovery from $C^T_I$ for multi-velocity wave models and variable coefficient isotropic elasticity system, where $I$ is the identity matrix.

A Zoll magnetic system on an oriented closed surface $M$ is a Riemannian metric $g$ together with a function $\lambda\colon M\to \mathbb{R}$, such that every unit speed solution of the ODE $\ddot \gamma(t)=\lambda(\gamma(t))\gamma(t)^\perp$ is periodic and the minimal period depends continuously on $\gamma$. The trivial example is given by $g$ with constant curvature $K$ and $\lambda\equiv {\rm const.}$ such that $\lambda^2+K>0$. This article exhibits non-trivial Zoll magnetic systems for every genus-for genus $\ge 2$ these are the first such examples. The approach is twistor theoretic: To a general magnetic system $(g,\lambda)$ one associates its transport twistor space $Z(g,\lambda)$, which is the unit disk bundle $DM$, equipped with a degenerate complex structure that encodes the magnetic flow. For the trivial Zoll magnetic systems explicit holomorphic blow-down maps $\beta\colon Z(g,\lambda)\to W$ into certain ruled surfaces $W\to M$ are constructed, mapping $\partial Z(g,\lambda)$ onto a Lagrangian $P\subset W$. For small Lagrangian perturbations $P'\approx P$ the procedure can be reversed and this results in a large class of (non-trivial) nearby Zoll magnetic systems.

We investigate a two dimensional flux limited Keller Segel FLKS system using classical Lie symmetry analysis. The complete Lie point symmetry algebra is determined and shown to consist of temporal and spatial translations together with planar rotations. An optimal system of one-dimensional subalgebras is constructed, yielding stationary, translationally invariant, radially symmetric, travelling-wave, and rotatingwave reductions. The corresponding reduced equations are derived systematically. The results demonstrate that the flux limiting mechanism eliminates the scaling symmetries of the classical Keller Segel model and significantly restricts the class of admissible invariant solutions. Exact spatially homogeneous equilibrium solutions are determined, and integral representations for stationary radial and travelling-wave profiles are derived.

In this work we consider an extension of the Berestycki–Monneau–Scheinkman (BMS) model for speculative financial bubbles, in which the investor disagreement process is allowed to have jumps. While the original BMS framework assumes that disagreement evolves along continuous paths, our model accounts for sudden shifts in market sentiment through an independent Lévy jump process. Using optimal stopping theory and the Itô–Lévy formula, we show that the speculative bubble premium satisfies a non-local partial integro-differential equation (PIDE) with a moving obstacle. We develop a viscosity solution theory for this non-local obstacle problem. We prove a comparison principle by a doubling-of-variables argument adapted to the non-local jump integral, and we prove existence and uniqueness of the bubble price by Perron's method, constructing explicit continuous sub- and supersolutions. We then introduce a monotone Implicit–Explicit finite difference scheme for the bubble premium. Following the Barles–Souganidis framework, we show that the discrete operator preserves the M-matrix property and that the scheme converges locally uniformly to the unique viscosity solution under a state-dependent Courant–Friedrichs–Lewy (CFL) condition. At the end of the paper we implement the scheme via a PSOR–Picard algorithm and present numerical tests for four finite- and infinite-activity Lévy models.

We survey recent variational methods for strongly indefinite Schrödinger equations with sign-changing nonlinearities. The main object is an energy functional of the form \[ J(u)=\frac12\|u^+\|^2-\frac12\|u^-\|^2 -\int_{\mathbb{R}^N}F(u)\,dx+\lambda\int_{\mathbb{R}^N}G(u)\,dx, \] where the splitting $X=X^+\oplus X^-$ is induced by a spectral gap of the linear Schrödinger operator, and where the nonlinear part \[ I(u)=\int_{\mathbb{R}^N}F(u)\,dx-\lambda\int_{\mathbb{R}^N}G(u)\,dx \] is allowed to change sign. We discuss the generalized linking theorem developed for such functionals, and the abstract multiplicity theory for critical orbits in dislocation spaces. In the final part, we prove a new ground state result for the competing pure-power case \[ f(u)=|u|^{p-2}u,\qquad g(u)=|u|^{q-2}u,\qquad 2<q<p<2^*. \] More precisely, for $\lambda>0$ sufficiently small, the corresponding strongly indefinite functional possesses a nontrivial critical point of least energy among all nontrivial critical points.

We introduce a new Calderón-type problem for fractional powers of Schrödinger operators, with local boundary conditions. The associated Dirichlet-to-Neumann operator maps Dirichlet data to Neumann data on the boundary. We show that this operator determines the Taylor series of the potential at the boundary. In particular, analytic potentials are uniquely determined. These are the first results for a fractional Calderón problem with sources and measurements on the boundary. Our proof builds on recent advances for pseudodifferential boundary value problems to compute the complete symbol of the Dirichlet-to-Neumann operator.

In this article, we investigate the existence and nonexistence of weak solutions to higher-order doubly critical elliptic problems with weights, driven by a polyharmonic double phase operator. More precisely, we deal with the following problem \begin{equation} \begin{cases} \mathcal{L}^m_{p,q}(u) = f(x,u) ~&\text{in } \Omega,\\[6pt] u=\nabla u=\cdots\nabla^{m-1} u=0 &\text{on }{\partial\Omega}, \end{cases} \end{equation} where $\Omega \subset \mathbb{R}^N$ with $N \geq 2$ is a smooth bounded domain with Lipschitz boundary $\partial\Omega$, $m \in \mathbb{N}$, $1 < p < q < \frac{N}{m}$ with $(N-1)q\leq Np$, the nonlinear term $f\colon\Omega\times\mathbb{R}\to \mathbb{R}$ is a Carathéodory function, which has doubly critical growth, and $\mathcal{L}^m_{p,q}$ represents a polyharmonic double phase operator. By establishing new compactness results within a suitable Musielak–Orlicz–Sobolev framework and applying variational methods, we prove the existence of nontrivial weak solutions. In addition, we derive nonexistence results under appropriate assumptions by establishing a Pohozaev-type identity for higher–order derivatives. Our approach extends classical techniques to capture the intricate features of the double-phase operator for higher–order derivatives, and addresses the difficulties arising from critical nonlinearities, in particular extending the results of [F. Colasuonno, K. Perera, J. Differ. Equ., 422 (2025), 426–488] in a polyharmonic double phase setup overcoming the non-closedness of truncations in higher-order Sobolev spaces.