Papers

The emergence of cooperation in structured populations is fundamental to the success of human societies. Physical and online networks can drive behavioural change by altering who people interact with, thereby modifying social pressures. In this paper, we study imitation dynamics in a population of self-interested agents playing a continuous strategy Prisoner's Dilemma on a dynamically evolving weighted network. In the fast-network regime, we incorporate the edge weights into the strategy evolution before deriving and analysing the large population mean-field limit. Without noise, we establish well-posedness and show the solution collapses to a single Dirac mass. For initially separated clusters, we identify a payoff threshold and sufficient conditions for the overall level of cooperation to increase. We then introduce stochastic strategy updates, and obtain a non-local Fokker-Planck equation in the mean-field limit. We rigorously prove existence and uniqueness of stationary distributions, and show linear stability under sufficient noise. Numerics illustrate that noise can transform the deterministic consensus into stable cooperative stationary behaviour. These findings show that the fast adaptive interactions and stochastic exploration can jointly support the emergence of stable cooperation at a population level.

In this article, we study some classes of solutions for a stationary Navier-Stokes equation where we consider a rough drift given by a singular integral operator which does not belong to the classical Calderón-Zygmund family of singular integral operators. Given a small external force, we will construct solutions to this system in the framework of weighted Morrey-Sobolev spaces. The use of Morrey-based Sobolev spaces provides a more general setting than the usual Lebesgue-based Sobolev spaces, and the presence of Muckenhoupt weights will allow us to present some existence and uniqueness results from several points of view.

This paper investigates the well-posedness issue for a reduced two-dimensional model of micropolar fluids. This reduced model presents a coupling between an Euler-type equation for the velocity field of the fluid and a transport-diffusion equation for the microrotation field (which is a scalar field, in this setting). We establish the local existence and uniqueness of strong solutions in the scale of Besov space $B^s_{p,1}$ having regularity index $s\geq1+2/p$. Furthermore, in the subcritical case when $s>1+2/p$, we prove that these solutions exist globally in time. The global persistence of regularity in the critical setting $s=1+2/p$ remains open.

We prove that, for a generic $C^m$-domain $\Omega \subset \mathbb{R}^d$ (with $m\geq3$, $d \geq 2$) in the sense of Baire category for the Micheletti topology, every Dirichlet-Laplacian eigenfunction has nonzero average. This gives a standalone spectral-geometric answer to a question raised by Steinerberger and Venkatraman. The proof is based on a direct shape-perturbation argument. A key ingredient, which appears to be of independent interest, is a Baire-category theorem for shapes: domains with real-analytic boundary form a meager subset of the space of $C^m$ domains. It is this input that allows us to bypass, rather than resolve pointwise, the overdetermined elliptic configurations in which the first shape derivative of the mean degenerates. We then derive two consequences. First, from a geometric viewpoint, a generic domain has trivial Euclidean isometry group. Second, in control theory, a generic domain makes the Dirichlet heat equation approximately controllable and rapidly stabilizable, that is, stabilizable at any prescribed exponential decay rate, by means of a single spatially homogeneous scalar control.

Global nonnegative martingale solutions are shown to exist for a stochastic Keller-Segel system with a nonlocal Fisher-KPP source and super-linear multiplicative noise. The result is obtained for nonnegative initial data with no smallness assumption, provided that the nonlocal source term is dominant. The main difficulty stems from the absence of a coercive structure and the super-linear nature of the noise. An additional cut-off with finite L^2 norm in the classical Galerkin method is added to establish a well-posed approximation problem. Moreover, due to the nonlocal Fisher-KPP structure, it is necessary to prove the positivity of the approximating solution in order to obtain uniform estimates. In the compactness arguments, the usual tightness argument in the framework of Hilbert spaces cannot be directly applied to the uniform estimates obtained in this paper. As a result, we develop a more general version of the compactness argument and tightness criterion, presented in the appendix, which will be applied throughout the paper. This allows for the global existence of nonnegative martingale solutions to be derived from Jakubowski's version of the Skorokhod Theorem, along with a thorough discussion of the convergence properties.

We formulate and prove a conditional finite-scale reduction theorem for the local one-component regularity problem for suitable weak solutions of the three-dimensional Navier–Stokes equations. Starting from a scale-invariant bound Phi(1) <= M and smallness of the critical vertical component C_3(1) = delta, the argument compares the solution with a strict two-and-a-half-dimensional shadow class. The comparison is made in the harmonic-pressure quotient, which is the natural local topology for pressure compactness. The Reynolds commutator produced by coarse graining is treated as a positive covariance stress and is absorbed by an unresolved-variance buffer; consequently this stress contributes additively, while the genuinely vertical residuals carry a positive power of delta and may pass through finite-stage exponential constants. The theorem is deliberately stated as a reduction theorem. Under the explicitly listed structural inputs–prepared pressure-covariance closure, weak horizontal-defect admissibility, sharp admissible-time trace tightness, singular-stratum tangent-cone inputs, strict limiting smoothing and decay, finite-window trace-cost/Newton solvability, and the vertical-duality active-residual estimate–we derive r_reg(0,0) >= c_{M,theta} |log C_3(1)|^{-sigma/3}. The paper does not constitute an unconditional resolution of the logarithmic one-component regularity problem. Its contribution is a theorem-driven reduction: strict-shadow selection failure is reduced to a finite-mode flat trace obstruction, and that obstruction is eliminated, conditionally, by vertical duality forced by the full three-dimensional vertical momentum equation.

We consider the homogenization of three dimensional viscous incompressible non-Newtonian flows satisfying certain general $r$-structure in perforated domains. We focus on the case of `small holes' by assuming the holes under consideration are of size $\varepsilon^{\alpha}$ with $\alpha >3$, where $\varepsilon$ is the perforation parameter used to measure the mutual distance between the holes. We show the limit equations remain unchanged in the homogenization limit under the constraint $ \frac{6(\alpha - 1)}{4\alpha-5}< r<3-\frac{3}{\alpha}$, which seems optimal in the sense of Sobolev capacity of holes as explained in Remark 1.3. Quantitative convergence rates are further derived for both the velocity field and the pressure. To the best of our knowledge, both the qualitative and quantitative homogenization results are firstly given for non-Newtonian flows in the `small holes' case.

In this paper, we investigate regularity properties for viscosity solutions to a general class of degenerate or singular fully nonlinear elliptic equations with Hamiltonian terms, \[ \left\{ \begin{alignedat}{2} \Phi(|Du|,x)F(D^{2}u,x)+H(|Du|,x) & =f(x) \quad && \text{in } \quad \Omega,\\ u(x) & = g(x) \quad && \text{on } \quad \partial\Omega. \end{alignedat} \right. \] Here, $F$ is uniformly elliptic, while $\Phi$ and $H$ satisfy suitable structural and growth conditions allowing for both degenerate and singular regimes. Our first result establishes sharp global $C^{1,\alpha}$ regularity for this general class of equations, thereby providing a unified regularity framework for both degenerate and singular regimes. We next develop an oscillation-based approach for fully nonlinear equations with unbalanced variable degeneracy and Hamiltonian terms. Under a Hölder-type decay assumption on $ f $, we derive sharp boundary $C^{1,\beta'}$ estimates with an explicit exponent, and establish a quantitative non-degeneracy estimate in the singular region. Finally, as applications of the general theory, under a suitable viscosity curvature condition on the level sets of the solution, we establish global $ C^{1,\frac{1}{3}} $ regularity for the infinity-Poisson and global $ C^{1,\frac{1}{p-1}} $ regularity for the $ p $-Poisson with $ p>2 $. These results may provide a new perspective on these two long-standing open problems.

This paper investigates the three-dimensional compressible Navier–Stokes system with a polytropic pressure law and its pressureless counterpart arising from the high Mach number limit. We focus on the different critical scaling structures of these two models. In the presence of the pressure term, the pressure gradient is balanced with the inertial and viscous effects, and thereby selects a fixed critical scaling for the pressure system. In contrast, once the pressure term is ignored, the pressureless system admits a more flexible one-parameter family of invariant scalings. For both systems, we establish the global well-posedness of strong solutions under low-regularity assumptions on the initial data, allowing vacuum and large oscillations. This improves the global result of Wen (Adv. Math. 482 (2025), Paper No. 110628), where higher regularity assumptions on the initial data are required. A central feature of our result is that the smallness conditions are exactly invariant under the intrinsic critical scalings of the corresponding systems. These scaling structures differ from the usual parabolic scaling used in the critical-space framework of Danchin (Invent. Math. 141 (2000), pp. 579–614), where the system is reformulated around a reference state. We also derive uniform a priori estimates and obtain exponential decay estimates for the global strong solutions. The results show that the pressure term not only changes the analytic estimates, but also plays a decisive role in selecting the critical scaling structure and in determining the dynamical behavior of compressible flows.

This paper investigates an inverse problem for the simultaneous reconstruction of two spatially varying reaction coefficients, the local proliferation rate and the competition (saturation) coefficient, together with the unknown initial condition, in a nonlinear, density-dependent reaction-diffusion model motivated by cell invasion and tumor growth dynamics. Using Carleman estimates, we establish a global uniqueness result together with a Lipschitz-type stability estimate for the reaction coefficients and a weaker, logarithmic stability estimate for the initial condition. For the numerical reconstructions, we develop a two-stage algorithm employing a time-shift strategy to decouple the coefficient and the initial condition. Numerical experiments are presented to illustrate the feasibility, accuracy, and robustness of the proposed inversion method.

We investigate the Schrödinger bridge problem on a finite state space with a strictly positive Markov reference kernel. Starting from the semi-dual convex formulation, we introduce a continuous-time evolution for the terminal Schrödinger potential and show that its equilibria coincide with the unique solution of the bridge problem. The proposed dynamics induces an evolution for the terminal marginal. This marginal equation is governed by a state-dependent nonlocal Onsager operator, identified with the Hessian of the semi-dual functional. We derive its associated Dirichlet form, establish coercivity estimates on the appropriate quotient space, and interpret the resulting equation as a nonlocal gradient-flow formulation of relative entropy. Under natural positivity assumptions, we prove global well-posedness of the SBOF, convergence to the Schrödinger bridge, convergence of the induced couplings and path measures, and exponential relaxation of the terminal marginal. The latter follows from a uniform Poincaré inequality on compact sublevel sets together with entropy–variance comparison estimates. We also discuss the connection with finite-state generative modeling through the Doob transform and illustrate the theory on finite-grid examples involving rare states.

We investigate a free boundary PDE model for vascularized tumor growth with two time-periodic coefficients representing nutrient supply $\phi(t)$ and nutrient demand $\psi(t)$. Under radial symmetry, the model reduces to a nonautonomous ODE for the tumor radius. We prove a vanishing-persistence dichotomy governed by the averaged coefficients $\bar{\phi}$ and $\bar{\psi}$: if the average nutrient supply does not exceed the average nutrient demand, namely $\bar{\phi}\le \bar{\psi}$, the tumor radius tends to zero; if the average supply exceeds the average demand, namely $\bar{\phi}>\bar{\psi}$, the tumor persists and converges to a unique positive periodic solution. We also study the linear stability of this periodic solution when nonradial perturbations are imposed. Under a stronger condition on the nutrient supply and demand, we show that the radially symmetric periodic solution is linearly stable when the tumor aggressiveness parameter is sufficiently small. Numerical simulations are provided to illustrate the analytical results.

Let $\Delta$ be the Neumann Laplacian on the interval $(0,\pi)$, and let $T>0$. An integration by parts formula is proven for the spectral fractional Laplacian, $(-\Delta)^\alpha$, for $\alpha \in (0,1)$. As an application, we prove well-posedness results for the structurally damped beam equation $$u_{tt}+\Delta^2 u+\rho (-\Delta)^\alpha u_t=0, x\in (0,\pi),t>0$$ with various boundary conditions including $$ u_x(0,t)=u_{xxx}(0,t)=0;\ u_x(\pi,t)=f(t),\ u_{xxx}(\pi,t)=0, $$ and $f\in L^2(0,T)$ and appropriate initial conditions. Viewing $f$ as a control, we prove null-controllability. Analagous results are proven for higher order controls, and for the Dirichlet Laplacian.

In this paper, we apply the I-method to establish global well-posedness for the fractional nonlinear Schrödinger equation with initial data u_0 \in H^s(R^d) for s <\alpha/2, i.e., below the energy threshold. Moreover, for radial initial data, we combine a modiffed Morawetz estimate-recovered via Balakrishnan's formula-with the I-method to obtain improved results. In the same spirit, we employ the "upside-down" I-method to derive polynomial-in-time growth bounds for the higher-order Sobolev norm. The main difffculty stems from the fact that Strichartz estimates for the fractional Schrödinger equation has a loss of derivatives, and the problem is always L^2-supercritical, thereby requiring more delicate analysis.

We present a new proof that an integer rectifiable current with finite mass, and whose boundary has also finite mass, is integral. We deduce the result from De Giorgi's structure theorem for integer-valued $BV$ functions and a cylindrical projection argument. As a consequence, we also give a new proof of the compactness of integral currents that is ultimately based on the $BV$ theory.

We develop a global wellposedness theory for weak solutions of the pressureless Euler Alignment system with measure-valued density $\rho$, bounded and unidirectional velocity $\mathbf{u} = (u,0, \ldots, 0)$, and a communication protocol $\phi$ that may be bounded or weakly singular. This appears to be the first such theory that admits shocks outside one space dimension. We recast the system as a family of nonlocally coupled scalar balance laws—one for each horizontal slice of $\mathbb{R}^d$—and establish existence, uniqueness, and stability of entropy solutions of the reformulated system, before translating back to the level of $\rho$ and $u$. This grants us access to certain key one-dimensional tools in the direction of the flow. The analysis in the directions transverse to the flow, however, still presents substantial challenges: The horizontal slices are coupled and therefore cannot be treated independently, and two solutions need not distribute their mass over horizontal slices in the same way, so that there is no canonical way to compare them slicewise. Accordingly, our analysis is centered around the nonlocal alignment force, and our framework leverages optimal couplings between the projections onto $\mathbb{R}^{d-1}$ of the density profiles under consideration. We construct our solutions as limits of atomic density and momentum profiles whose atoms follow sticky particle Cucker–Smale dynamics, and under additional regularity assumptions, we obtain quantitative rates of convergence. Finally, our investigation of the long-time behavior of unidirectional solutions highlights the role of lateral communication in driving the system toward a limiting density profile: We prove that flocking occurs, at a rate independent of the number of agents, under assumptions on $\phi$ that bound its size from below only outside a cylindrical neighborhood of the axis parallel to the flow.

In this article we consider a system of coupled partial differential equations that mirrors the interconnection between the evolution of an electromagnetic field – described by the Maxwell's system – and that of the temperature distribution in a bounded region in three-dimension. A discussion of the mathematical model is provided. We establish the well-posedness of the initial-boundary value problems associated with the thermoelectromagnetic system, in an appropriate functional-analytic framework. Then, our investigation and main results pertain to the long-time behaviour of the solutions. It is shown that either exponential stability or convergence to stationary solutions hold true, according as the conductivity is positive definite or semidefinite, respectively; in the latter case, strong stability is attained in specific topological or analytical settings. This complements and expands earlier results obtained for the uncoupled Maxwell's system.

In this paper, we study the Cauchy problem for a class of semilinear evolution equations with scale-invariant time-dependent dissipation \begin{equation*} \begin{cases} u_{tt} + L_{w^2}u + \dfrac{\mu}{1+t}u_t = \Delta^{\theta} f(u), & t>0,\ x\in\mathbb{R}^n,\\ u(0,x) = 0,\qquad u_t(0,x) = u_1(x), & x\in\mathbb{R}^n, \end{cases} \end{equation*} where $\mu>0$, $f(u)=|u|^\alpha$ with $\alpha>1$, $\theta\in\{0,1\}$, and the operator $L_{w^2}$ is defined on the Fourier transform by multiplication by $w(\xi)^2$. We prove the global (in time) existence of small data solutions for $\alpha>\alpha_{\mathrm{crit}}$, where the critical exponent $\alpha_{\mathrm{crit}}$ depends on the choice of the operator $L_{w^2}$, the parameter $\mu$, and the nonlinear term. In particular, we consider two model cases. For Boussinesq-type operators with $w(\xi)=\sqrt{|\xi|^2+|\xi|^4}$, combined with the derivative-type nonlinearity $\Delta |u|^\alpha$, we obtain a Strauss-type critical exponent. On the other hand, for plate-type operators with $w(\xi)=|\xi|^\sigma$, $\sigma\geq2$, and power-type nonlinearity $|u|^\alpha$, the critical exponent is of Fujita type.

We exhibit a family of smooth incompressible velocity fields on the two-dimensional unit sphere such that the time evolution of any mean-free initial data passively advected by any of them is mixed exponentially fast. In the presence of molecular diffusivity, we show that the solution to the associated advection-diffusion equation experiences enhanced dissipation with optimal decay rates. Each member of this family is an alternating combination of two Rossby-Haurwitz flows with random amplitudes and constitutes a spherical analogue to the sine shear-alternating example of Pierrehumbert.

We study isolated singularities of positive solutions to a fractional Hartree equation with Riesz interaction, \[ (-\Delta)^s u = \left( \int_{\mathbb{R}^N\setminus\{0\}} \frac{u^p(y)}{|x-y|^\mu}\,dy \right)u^q \quad \text{in } \mathbb{R}^N\setminus\{0\}. \] The puncture changes the passage from the differential equation to its integral form: a fractional fundamental solution term may occur at the singular point. Under an upper bound at the scale of the fundamental solution and a weighted integrability condition on the source, we derive a Riesz decomposition with a nonnegative singular term. Under the same assumptions, if in addition the solution blows up at the puncture, \(p,q\ge1\), and both Kelvin weights are nonnegative, an off-center moving spheres argument yields radial symmetry and strict radial decrease. We also classify positive radial homogeneous singular solutions under the corresponding convergence conditions. If a nonzero positive radial homogeneous scaling limit exists and solves the limiting equation, this classification uniquely determines its coefficient.

We consider maps between two Riemannian manifolds and study a functional given in terms of the $L^\infty$-norm of the derivative. This functional is not differentiable, but we can define critical points with the help of a subdifferential. The resulting notion includes, for example, minimisers in a given homotopy class. We derive a geometric condition equivalent to criticality in this sense. The condition is formulated in terms of a vector-valued $1$-current on the domain manifold, which encapsulates some of the key properties of the critical point. Moreover, this $1$-current is itself a critical point of a generalised mass functional.

This article studies the high-frequency behavior of eigenstates of perturbations of the Laplace-Beltrami operator on the two-sphere $\mathbb{S}^{2}$ by a measure supported on an equator. We are interested in understanding to what extent this behavior can be described in terms of the geodesic flow of the sphere. This is done by analyzing quantum limits and semiclassical measures of sequences of high-frequency eigenfunctions, which describe how their $L^2$-masses concentrate in phase space. When the Laplacian on $\mathbb{S}^{2}$ is perturbed by a bounded potential, it is known that the family of all possible semiclassical measures is contained in the set of positive measures on the unit cosphere bundle $S^*\mathbb{S}^{2}$ that are invariant under geodesic flow (with equality in the unperturbed case). In this article, we show that the presence of a singular delta potential on a closed geodesic results in the existence of sequences of eigenfunctions whose semiclassical measure is not invariant under geodesic flow. In particular, one can find a sequence of eigenfunctions whose energy asymptotically concentrates on the hemisphere bounded by the equator on which the potential is concentrated.

We study a one-sided degenerate Cahn-Hilliard equation with anisotropic traction flux, arising as a reduced continuum description of mechanically biased cell interactions in digit-forming organoids. The equation combines a one-sided degenerate mobility with a density-weighted anisotropic higher-order transport term. This traction term is not generated by the variational derivative of the Cahn-Hilliard energy and therefore produces sign-indefinite contributions in the energy balance. For nonnegative initial data, we prove the global-in-time existence of nonnegative weak solutions. The proof combines an energy estimate for the diffusive flux with a mobility-matched entropy method adapted to the vacuum degeneracy. A key point is that the entropy variable cancels the mobility, turning the anisotropic traction contribution into a coercive first-order term in the entropy inequality, while the energy estimate supplies a weighted control of the diffusive flux.

We investigate the long-time asymptotics of a full soliton gas for the Camassa–Holm equation. The analysis starts from a pure-soliton Riemann–Hilbert (RH) problem with \(2N\) poles and two distinct types of residue conditions. We prove that, as \(N\to\infty\), this discrete RH problem converges to a limiting soliton gas RH problem whose jump matrix contains two nonzero reflection coefficients. In this sense, the limiting problem gives a full soliton gas model for the Camassa–Holm equation, in contrast to the previously studied half soliton gas models, whose jump matrices involve only one nonzero reflection coefficient. The limiting RH problem is analyzed by the Deift–Zhou nonlinear steepest descent method. The presence of two nonzero reflection coefficients requires two different types of triangular factorizations of the jump matrix and leads to a more delicate \(g\)-function mechanism. The main difficulty lies in the construction of suitable \(g\)-functions adapted to the Camassa–Holm phase, together with the precise control of their behavior near the distinguished point \(k=i/2\) and at infinity. Depending on the location of the spectral endpoints \(\eta_1\) and \(\eta_2\), different \(g\)-function mechanisms arise. In this paper, we focus on Case I and derive the long-time asymptotic formulas in three elliptic-wave regions of the self-similar plane. In each region, the leading term is given by a finite-gap elliptic function, while in the central region the first correction is of order \(\mathcal O(t^{-1/2})\) and involves parabolic cylinder functions.

We prove a phase space criterion for dynamical Amrein-Berthier uncertainty principles. The abstract result says that, for a Fourier integral operator $A\in FIO(\chi)$ associated with a tame canonical transformation $\chi$, the localized operator $\mathbf{1}_E A\mathbf{1}_F$ is compact on $L^2(\mathbb {R}^d)$ whenever $\chi$ satisfies a vertical non refocusing condition: high frequency covectors issued from a spatially localized region cannot return to a vertical direction over the observation region. In the linear symplectic case this condition is equivalent to the familiar nondegeneracy $\det B\neq0$ of the upper right block of the symplectic matrix. We apply this compactness theorem to Schrödinger propagators for Yajima–type Hamiltonians, including quadratic electric and linear magnetic growth, and obtain two–time Amrein–Berthier inequalities for compact localization sets at all nonrefocusing times. The result extends the compactness mechanism behind the dynamical Amrein-Berthier principle to a genuinely microlocal setting.

We study the approximation of certain renormalized conserved quantities for the NLS hierarchy with nonzero boundary conditions, in the long-wave regime, by the energies of the KdV hierarchy. We extend this to all $n \in \mathbb{N}$ by proving an approximation result for the transmission coefficient of the Lax operator of the NLS hierarchy, which is a Dirac operator in the nonrelativistic regime, by the transmission coefficient of a Schrödinger operator, which is the Lax operator of the KdV hierarchy. This yields a formal approximation result between the hierarchies, which we quantify using energy methods and previously established well-posedness results.

We prove dynamical Amrein-Berthier uncertainty principles for fractional Schrödinger flows. For the free Hamiltonian $H=(-\Delta)^\alpha$ on $L^2(\mathbb{R}^n)$, with $\alpha>\frac{1}{2}$, we show that two–time localization on finite measure sets $E,F$ forces the quantitative estimate \begin{equation*} \|u(t)\|_{L^{2}}\lesssim_{E,F,T,n,\alpha} \|u(0)\|_{L^{2}(E^{c})} + \|u(T)\|_{L^{2}(F^{c})}, \qquad T\neq0,\ t\in \mathbb{R} \end{equation*} for $u(t)=e^{-itH}u(0)$ at every time. The threshold $\alpha>\frac{1}{2}$ is tied to the stationary phase structure of the fractional kernel. If $\alpha\ge1$ the sets can be arbitrary finite measure sets; if $\frac{1}{2}<\alpha<1$ we impose the finiteness of a natural interaction energy \begin{equation*} \textstyle \mathcal{I}_{\gamma}(E,F) = \int_{\mathbb{R}^n \times \mathbb{R}^n} \mathbf{1}_{F}(x)|x-y|^{2\gamma}\mathbf{1}_{E}(y)\,dx\,dy<\infty, \qquad \gamma = \frac{n(1-\alpha)}{2 \alpha-1} \end{equation*} of the pair $(E,F)$, essentially equivalent to a sufficiently fast joint decay of the measure of the sets at infinity. In particular, compact support at two distinct times is impossible for a nonzero solution. We also prove corresponding results for one dimensional fractional Hamiltonians $(-\partial_x^2+V)^\alpha$ under weighted scattering assumptions, and for higher order Hamiltonians $(-\Delta)^m+V$ for suitable classes of decaying potentials $V$.

In this paper, we mainly consider positive solution to the $D^{1,p}(\R^{N})$-critical quasi-linear Schrödinger system with $p$-Laplacian: \begin{equation*}\begin{cases} -\Delta_p u = u^{\alpha}v^{\beta} \, \ \ \ \ \ \text{in}\,\ \ \R^N, \\ -\Delta_p v = u^{\beta}v^{\alpha} \,\ \ \ \ \ \text{in}\,\ \ \R^N, \end{cases}\end{equation*} where $1<p<N$, $N\geq2$, $0\leq \alpha \leq \beta,$ and $u,v\in D^{1,p}(\R^N)$. We establish regularity and the sharp estimates on asymptotic behaviors for any positive solution $(u,v)$. Then, we prove that all positive solutions are radially symmetric and strictly decreasing about some point. Furthermore, we obtain the uniqueness and complete classification of positive solutions. Our results extend the uniqueness results in \cite{LM,QS} for $p=2$ to general cases $1<p<N$.

We prove a local higher integrability result for the gradient of Hölder continuous weak solutions to the parabolic double phase equation \[ \partial_t u - \operatorname{div} \left(|Du|^{p-2}Du + a(z)|Du|^{q-2}Du\right) = 0 \qquad \text{in } \Omega_T. \] We work under a relaxed gap condition on the exponents $p$ and $q$. The coefficient $a$ is assumed to belong to the class $\mathcal{Z}^{\kappa}(\Omega_T)$ for some $\kappa \in (0,\infty)$. The functions in this class satisfy a one-sided pointwise bound that controls how fast $a$ can grow away from its zero set, and the class contains the Hölder continuous functions. We also impose a mild almost increasing condition on $a$, which motivates the introduction of a new mollification, which we call the slanted Steklov average. For $u \in C^{0,\gamma,\gamma/q}_{\mathrm{loc}}(\Omega_T)$ with $\gamma \in [0,1)$, our main result holds under the gap bound \begin{equation}\tag{G}\label{eq:G} 2 \le p \le q \le p + \frac{q\kappa}{q - 2\gamma}. \end{equation} The new gap condition \eqref{eq:G} is purely parabolic in nature and is stricter than the optimal gap relation associated with the Lavrentiev phenomenon for the elliptic double phase functional.

We consider polyconvex integrands that are conformally invariant and frame indifferent. In two dimensions, we prove that the corresponding stationary points are smooth outside a discrete set; this result is new even for minimizers. We further show that every orientation-preserving stationary point is $C^1$. Since such solutions are closely related to Teichmüller-type variational problems, our result also confirms, in the case of integrands with linear growth in the distortion, a conjecture of Astala, Iwaniec, Martin, and Onninen from 2005.