Papers

This paper considers boundary stabilization problems for a first-order hyperbolic partial integro-differential equation (PIDE) subject to concurrent input and state delays. The coexistence of these two types of delays complicates control design, especially under the case of large input delay that requires to predict more state information. A backstepping-based boundary controller is developed to achieve stabilization and delay compensation. The design relies on two affine Volterra transformations involving both Fredholm- and Volterra-type integral terms, which results in a four-PIDE kernel equations. To establish their well-posedness, the kernel domain is partitioned along characteristic lines into a finite number of triangular subregions, and the kernel solution is constructed successively in the preceding subregion to the next one. The finite-time stability of the resulting closed-loop system is established. Numerical simulations are provided to demonstrate the effectiveness of the proposed controller.

For integers $N\ge0$ and $n\ge2$, let \[ \Delta_N^{(n)} =\left\{\alpha\in\mathbb Z_{\ge 0}^n: \alpha_1+\cdots+\alpha_n=N\right\}. \] We study discrete unique continuation for functions on this lattice simplex. For an integer $R\ge1$, let $g:\Delta_{nR}^{(n)}\to\mathbb R$ satisfy \[ \sum_{i=1}^n g(\beta+e_i)=0, \qquad \beta\in\Delta_{nR-1}^{(n)}, \] where $e_i$ is the $i$th standard basis vector. We prove that nonvanishing at the center implies \[ |\operatorname{supp} g|\ge c_n R^{\lceil n/2\rceil}. \] Here $\operatorname{supp} g$ is the set of points where $g$ is nonzero, and $c_n>0$ depends only on $n$. The exponent $\lceil n/2\rceil$ is optimal. The proof represents the values of $g$ as polynomial coefficients and uses a Pascal uncertainty principle, which bounds from below the total number of nonzero coefficients of a one-variable polynomial and its unit translate in terms of their degree.

We construct an autonomous Lipschitz fast dynamo on the flat three-torus whose particle flow has zero topological entropy and whose ideal induction equation has no exponential growth. For every sufficiently small positive magnetic diffusivity, the induction operator has an eigenvalue with real part bounded below by a positive constant independent of the diffusivity. A corresponding non-zero real-valued magnetic field satisfies an exact exponential growth law in $\mathrm{L}^2$. For the same velocity field, every ideal solution grows at most linearly in $\mathrm{L}^2$. Moreover, the particle flow and its inverse have Lipschitz constants growing at most linearly in time. The velocity field is real-valued, divergence-free, differentiable everywhere and smooth away from one circle, but is not $\mathrm{C}^1$.

We study the dependence on time of the spatial regularity of solutions to the periodic Benjamin–Ono equation with real, mean-zero initial data of bounded variation. For every datum with a nonzero jump, we prove that continuity and essential boundedness at an irrational time $t=2\pi\alpha$ are equivalent to the arithmetic condition $\sum_j\log(q_{j+1})/\sqrt{q_j}<\infty$, where $q_j$ are the continued-fraction denominators of $\alpha$. We also determine the exact Hölder regularity and the Besov scale for spatial exponents $2<p\leq\infty$. We derive a high-frequency energy formula and prove the same sharp local criteria on every spatial interval at every irrational time for an open dense set of BV data. At rational times, piecewise smooth data give finite sums of logarithmic and jump kernels modulo a continuous remainder. For arbitrary BV data with a jump, the nonlinear and linear solutions have different logarithmic Poisson coefficients at every sufficiently large reduced denominator, uniformly in the numerator. The proofs combine the smoothing of Tao's gauge transform with estimates for quadratic Fourier series. A finite averaging argument isolates an atom of the initial gauge measure while retaining the real nonlinear reconstruction.

We investigate a broad class of mixed local and nonlocal degenerate $p$-Laplace equations with general right-hand sides. The degeneracy is governed by Muckenhoupt $A_p$-weights, yielding a highly nonuniform elliptic framework in which both the local and nonlocal operators may degenerate simultaneously. We establish a comprehensive local regularity theory, including local boundedness and lower semicontinuity of weak subsolutions, weak Harnack inequalities for weak supersolutions, Harnack inequalities, and local Hölder continuity of weak solutions. Our approach combines weighted analytic techniques with the De Giorgi–Nash–Moser iteration method, adapted to the mixed local–nonlocal setting. To the best of our knowledge, this is the first systematic regularity theory for mixed local and nonlocal equations with Muckenhoupt weights. In particular, our results are new even for homogeneous linear equations ($p=2$) under the natural assumption $w\in A_2$, and therefore substantially extend the existing regularity theory for mixed local–nonlocal equations to a degenerate weighted framework with general right-hand sides.

We study the long-time behavior of the one-dimensional defocusing cubic nonlinear Schrödinger equation with initial data in $L^2(\mathbb R)$. We first prove that the standard fixed-profile modified-scattering ansatz fails at the unweighted $L^2$ endpoint for the one-dimensional defocusing cubic nonlinear Schrödinger equation. More precisely, there exists a real-valued datum $$u_*\in L^1(\mathbb R)\cap\bigcap_{m\in\mathbb N}H^m(\mathbb R), \qquad xu_*\notin L^2(\mathbb R).,$$ for which the corrected Fourier profile has no strong $L^2$ limit. Secondly, for every $L^2$ datum, we prove generalized modified scattering: in the physical ray coordinates $x=2tz$, the rescaled solution admits a strong $L^2$ asymptotic representation whose amplitude is the square root of the conserved transmission density and whose unit complex phase is determined by the initial datum.

Countable subsets of solutions of higher-order nonlinear ODEs and elliptic PDEs with indefinite non-coercive operators from the reaction-diffusion, thin film and dynamical system (DS) theories are obtained via a gluing/matching argument. In particular we study some classic and quasilinear degenerate ODEs in $\mathbb{R}$, with boundary conditions at infinity $F(\infty)=0$, with non-odd nonlinearities such as $$ \begin{matrix} F^{(4)} =-F+F^2,\, \,\, F^{(4)}=-F+F^2{\rm e}^{F-1}, \, (|F''|F'')''=-F + F^2, \\ F^{(4)} =-|F|F+F^2, \,\,\,F^{(4)} =-F^3+F^4, \,\,\, F^{(6)}=F-F^2, \end{matrix} $$ etc, as well as equations with odd and non-smooth nonlinearities like $$F^{(4)}=-F+F^3, \,F^{(4)}=-F-(|F|F-F)'', \,\, F^{(4)}=- \frac F{\sqrt{|F|}}-(F^3-F)'',\,\mbox{etc.}$$ Some of these ODEs are Hamiltonian and were studied in detail in the DS theory. On the basis of nonlinear operators, elliptic PDEs and variational theory, related polyharmonic elliptic equations in $\mathbb{R}^N$, $F(\infty)=0$, such as $$\Delta^2 F=-F+F^2, \quad \Delta^2 F=-F -\Delta (F^2-F), \quad \Delta^3 F=F-F^2, \quad \mbox{etc.};$$ are also shown to admit countable families of solutions. For such equations with non-odd functionals associated Lusternik–Schnirel'man (L–S) genus/category variational theory guaranteeing existence of a sequence of critical points does not apply. These ODEs and elliptic PDEs (e.g., in the radial setting) are shown to admit at least two basic countable families ${\mathcal F}_{1,2}$ of positively dominant solutions connected with two periodic orbits $\Gamma_{\rm max/min}$. Patterns obtained by gluing together via exponentially decaying tails of arbitrary finite samples from $\Gamma$'s form a countable subset of homoclinics in $\mathbb{R}^4$ of an arbitrary complexity.

In this paper, we establish quantitative Denjoy–Koksma type estimates for higher-dimensional quasi-periodic torus rotations. For Diophantine frequency vectors, we establish quantitative estimates on the discrepancy between Birkhoff averages and spatial averages for observables with various Besov-type regularities. The Besov framework is naturally adapted to the small-divisor structure of the problem, allowing a sharp analysis of resonant frequencies and yielding optimal rates throughout the entire parameter range. These estimates yield, to the best of our knowledge, the sharpest currently available convergence rates for Hölder continuous observables. As applications, we obtain substantially improved quantitative homogenization results for Hamilton–Jacobi equations in spatially quasi-periodic settings, as well as nearly optimal statistical regularity estimates for invariant measures under perturbations.

For piecewise constant fan subsolutions to the isentropic Euler equations, the entropy and action rates of any associated convex integration solution, relative to a common reference solution, depend only on the fan and are computed explicitly. Moreover, these explicit expressions may be decomposed into a kinetic energy mismatch and an internal energy mismatch, from which we characterize agreement of Dafermos' entropy rate criterion with action rate criteria. The known disagreement for the Chiodaroli–Kreml two-shock data is recovered. For the planar contact reference, we prove that both criteria strictly prefer every convex integration solution associated with a strictly dissipative admissible fan to the classical contact discontinuity. Certified computations locate the Krupa–Székelyhidi solutions in this region, where the criteria agree, while Horimoto's analytical construction provides such fans for every strictly increasing pressure law.

In this paper, we consider the Cauchy problem for the Vlasov-Poisson-Fokker-Planck/Navier-Stokes-Poisson (VPFP/NSP) system, which couples the VPFP system with the compressible NSP system through a friction force dependent on the relative velocity and a self-consistent Poisson equation. Motivated by the spectrum analysis for the Vlasov-Poisson-Boltzmann (VPB) system, we introduce a suitable norm to capture the effect of the forcing induced by the Poisson equation and give a detailed spectrum analysis of the linearized system around a global equilibrium. Our results show that the two coupling mechanisms lead to an essentially different spectrum structure of the coupled system from those of the individual VPFP and NSP systems. More precisely, the low-frequency spectrum contains a pair of acoustic branches with the propagation speed $\sqrt{\frac{\gamma+1}{2}}~(\gamma\ge1)$ and two diffusive branches, thereby restoring the usual acoustic wave propagation of classical compressible fluids. Moreover, we establish the global existence of the solution to the nonlinear system and obtain the optimal time decay rate $(1+t)^{-\frac34}$, with a faster rate $(1+t)^{-\frac54}$ for the electric field and relative velocity. The present analysis also provides a useful framework for studying related kinetic-fluid models coupled through friction and self-consistent fields.

We establish two pointwise endpoint limits for nonlocal operators: one based on normalized integrals and a distributional limit based on weighted weak-type norms. As applications, the limits yield pointwise Bourgain–Brezis–Mironescu, Maz'ya–Shaposhnikova, Brezis–Seeger–Van Schaftingen–Yung and Gu–Yung formulas, together with higher-order and mean-oscillation variants. We then investigate endpoint limits for fractional powers generated by semigroups and for approximation processes, and derive sharp strong and weak endpoint estimates for operators in harmonic analysis. Finally, while Domínguez and Milman obtained weak-type estimates on product spaces for $p>1$ and left the endpoint $p=1$ open (see [Adv.~Math.~411 (2022), Paper~No.~108774, p.~22]), we prove a sharp pointwise limit in the parameter variable and obtain two–sided iterated weak-type estimates at $p=1$.

This paper is concerned with the Cauchy problem for the 3D incompressible magnetohydrodynamic (MHD) equations in supercritical Sobolev spaces. It is well known that the system is locally well-posed in subcritical Sobolev spaces, whereas the supercritical regime remains largely open. In this work, we establish norm inflation for the incompressible MHD equations, both with and without Laplacian dissipation, in supercritical Sobolev spaces, thereby revealing strong ill-posedness of the system at this regularity level. A distinctive feature of our approach is the introduction of a novel geometric construction, termed the “Magnetic-solo ansatz”, through which, for the ideal MHD system, norm inflation occurs exclusively in the magnetic field $b$ in $H^s$ with $0<s<\frac{5}{2}$, while the $H^s$-norm of the velocity field $u$ remains uniformly bounded. This asymmetric behavior shows that supercritical ill-posedness can be driven exclusively by the magnetic field, highlighting its essential role in the breakdown of well-posedness. Our findings fill a significant gap in the supercritical regularity theory for incompressible MHD and shed light on the distinct mechanisms governing the fluid and magnetic dynamics.

We consider a diffusion-relaxation system and investigate the conditions on parameters leading to equilibration versus localization. When the diffusion is dominant, solutions converge toward homogeneous equilibria. By contrast, when the effective diffusion is weak, localization emerges. Such behaviors have been studied for various models through formal asymptotic arguments and linearized stability analysis, but rigorous understanding of the associated nonlinear phenomena remains limited, particularly in higher dimensions. In the equilibration regime, we establish convergence toward constant equilibria by exploiting an energy dissipation structure and invariant-region estimates. In the localization regime, we study self-similar solutions and transform the problem of their existence into an autonomous dynamical system. The existence of self-similar profiles associated to localizing solutions is reduced to the construction of a heteroclinic orbit for an autonomous dynamical system. Their existence is obtained through an application of geometric singular perturbation theory. Our analysis provides a rigorous characterization of the transition from equilibration to localization.

In this manuscript, we determine the critical exponent for the three-dimensional semilinear wave equation with strong damping and thereby resolve an open problem posed in 2014. The threshold for the power nonlinearity $|u|^p$ is \begin{align*} p=p_{\mathrm{crit}}=\frac{7}{3}. \end{align*} Sufficiently small data generate global in-time solutions for $p>\frac{7}{3}$, whereas there exist arbitrarily small smooth compactly supported data whose solutions blow up in finite time for $1<p\leqslant\frac{7}{3}$. Positivity of the full velocity fundamental solution, combined with a dimension-descent formula, yields a positive half-line kernel and replaces the finite propagation property unavailable for strongly damped waves. This leads to a nonlinear lower-bound parabolic iteration without radial symmetry or pointwise sign assumptions on the initial data. At the critical power, a refined slicing argument on moving shells converts the borderline logarithmic gain into the growth required for blow-up.

We study linear parabolic equations of the second order in non-divergence form in a general set which is non-cylindrical with respect to the spatial and time variables. The restriction of the set on any bounded time interval is bounded, but the spatial diameter of each fixed-time cross section can be unbounded as time tends to infinity. For homogeneous problems, we obtain exponential, power and other intermediate decaying rates for the solutions in different scenarios. For inhomogeneous problems, we obtain all-time and asymptotic, as time tend to infinity, estimates for the solutions in terms of the data on the parabolic boundary and forcing functions. The analysis requires subtle properties of general space-time sets and an iteration scheme to bootstrap the Growth Lemma. The time steps for such an iteration need not be constant and are adapted to the growth of the diameter.

The narrow escape problem is a prototypical example for studying entropic metastability, motivated by the analysis of biological and chemical systems. The problem concerns the determination of the exit time and position of a Brownian particle trapped in a domain with a reflecting boundary pierced by narrow holes. Our goal is to investigate this problem in a general domain in any dimension (greater than or equal to two), using the quasi-stationary distribution approach to metastability. In particular, we derive first order asymptotic expansions with error bounds of the mean exit time and the law of the exit position in the limit where the hole sizes tend to zero. Our analytical predictions are illustrated by numerical simulations, using dedicated Monte Carlo techniques.

For $p>1$, we consider the boundary value problem for fully nonlinear degenerate elliptic equations $-\lambda_i(D^2u)+|Du|^p+\gamma u=f(x)$ in bounded domains with Dirichlet or boundary blow-up conditions; here $\lambda_i(D^2u)$ denotes the $i$-th eigenvalue of the Hessian. We study existence and nonexistence of solutions together with the asymptotic behaviour of the solutions when $\gamma$ goes to zero. A priori Lipschitz estimates play an important role. The interplay between the operator's degeneracy and the superlinear growth of the Hamiltonian gives rise to phenomena that are very different depending on which of the two terms dominates, e.g. the ergodic dichotomy takes place only when $i=N$, while new phenomena arise for $i<N$ in which case, under mild conditions, solutions that blow up even in just one point do not exist, and conditions on the size of $f$ must be imposed for the existence of solutions to the Dirichlet problem with homogeneous boundary condition.

We establish a unified local theory for the persistence of Sobolev regularity of vortex patch boundaries in a family of two-dimensional active scalar equations with radial convolution kernels $K(|x-y|)$. The class includes the 2D Euler equation, the generalized SQG equation in the locally integrable range $0<\beta<1$, and the quasi-geostrophic shallow water equation. Under natural assumptions on $K$ (smoothness, integrability near the origin, monotonicity, and polynomial growth), we prove that if the initial boundary belongs to $H^3(\mathbb T)$ and satisfies the arc-chord condition, then the contour dynamics equation admits a unique local solution in $C([0,T];H^3(\mathbb T))$. Under a stronger integrability condition on the kernel, we also obtain local existence of $H^2$ solutions. The proof combines Sobolev energy estimates for the contour equation with quantitative control of the arc-chord quantity.

For solutions to the Minkowski problem of the even dual curvature measures $\widetilde{C}_q$ in $\mathbb{R}^n$, $n\ge 2$, we prove the non-uniqueness for $q>n$ and $n\ge 2$, and the uniqueness for $0<q<n$ and $n=2$. These results are governed by our established logarithmic Brunn-Minkowski inequality for dual quermassintegrals $\widetilde V_q$.

We study the Euler–Riesz system on the torus $\mathbb T^d$, $d=2,3$: the compressible Euler equations with barotropic pressure $p(\varrho)=a\varrho^\gamma$ ($\gamma>1$, $a>0$), coupled to a repulsive nonlocal force ($\approx \varrho \nabla_x K \ast \varrho$) with Riesz kernel $K(x)\propto|x|^{\beta-d}$ of order $\beta\in(0,2)$. Since $K$ is the kernel of the inverse fractional Laplacian $(-\Delta)^{-\beta/2}$, we recast the force through the Caffarelli–Silvestre extension as the trace of a local stress tensor, replacing the nonlocal interaction by a local identity in one extra variable. For a repulsive kernel the total energy is coercive, and we use this to introduce a notion of global-in-time dissipative solution for arbitrarily large finite-energy data. Our main result is weak (measure-valued)–strong uniqueness, for every order $\beta\in(0,2)$ and every $\gamma>1$ independently: on any interval on which a strong solution exists, every dissipative solution with the same initial data coincides with it and all defects vanish. The proof rests on a suitable adaptation of relative energy.

We establish exponential mixing for the randomly forced and weakly damped KdV equation in $L^2(\mathbb{T})$. The noise is bounded, localized, and degenerate in high frequencies. Our proof relies on a general probabilistic framework in [11,33], nonlinear smoothing for KdV and its linearization via normal form transformation, and stabilization of the system by localized force. This paper continues a series of works connecting asymptotic compactness, control theory, and ergodicity and mixing for randomly forced dispersive PDEs.

In this paper, we show existence and uniqueness of solutions to the viscous Burgers' equation in $\mathbb{R}^d$, when the initial condition $u_0$ is in Hermite-Sobolev space of index $p$, for suitable non-negative integers $p$. Our solutions are local in time. We also have a regularity result, viz. if $u_0$ belongs to Schwartz space, then so does the solution.

In this paper, we establish necessary and sufficient conditions for continuous embeddings between weighted local Morrey-type spaces and weighted complementary local Morrey-type spaces. Rather than relying on standard multidimensional approaches, our methodology is based on a one-dimensional reduction. Specifically, we reduce the multidimensional embedding problems to corresponding weighted inequalities involving one-dimensional Cesàro and Copson function spaces. Applying this reduction alongside recent results in the characterization of the embeddings between Cesàro and Copson spaces, we provide complete characterizations for the embeddings \[ \LM_{p_1,q_1}(v_1, w_1) \hookrightarrow \LM_{p_2,q_2}(v_2, w_2)\] and \[\dual\LM_{p_1,q_1}(v_1, w_1) \hookrightarrow \LM_{p_2,q_2}(v_2, w_2).\] Our results cover all possible finite cases of the parameters $0 < p_1, p_2, q_1, q_2 < \infty$.

For every $n\geq 3$, we construct a nonconstant real-valued function $U\in C^\infty(\overline{\mathbb R^n_+})$, harmonic in the upper half-space, whose complete boundary jet vanishes on a compact nowhere dense subset of $\partial\mathbb R^n_+$ of positive $(n-1)$-dimensional measure. The essential construction takes place in two dimensions and yields the case $n=3$; higher-dimensional examples follow by cylindrical lifting. In every dimension, the exceptional set may occupy an arbitrarily large proportion of a fixed boundary cube. This resolves, in the negative, the smooth case of the boundary unique-continuation problem left open by Bourgain and Wolff in 1990 \cite[p.~260]{BourgainWolff1990}. In dimension three, a Möbius–Kelvin transfer gives the corresponding counterexample in the unit ball. It disproves Nadirashvili's smooth unit-ball conjecture on boundary singular sets \cite[Conjecture~4, p.~232]{Nadirashvili1997} and, a fortiori, disproves the gradient-only formulation subsequently recorded by Logunov and Malinnikova \cite[Section~7.4]{LogunovMalinnikova2020} and by Lin \cite[Conjecture~3, pp.~15–16]{Lin2020Current}. The proof uncovers a hidden flexibility principle for nonlocal elliptic equations: microscopic modifications can exert macroscopic control over exterior data. A quantitative correction mechanism for the half-Laplacian, iterated across scales, produces flat nonlocal Cauchy data on a set of positive measure. Thus nonlocality has a striking dual character: the same long-range interaction that drives unique-continuation rigidity can also furnish the flexibility through which that rigidity fails in the smooth category.

We investigate a Cucker-Smale type flocking model with a biological memory term and conclude that memory makes flocking harder. The model introduces $h(t)$ to describe memory, with dynamics $\frac{dh}{dt}=x(t)+v(t)-\lambda h(t)$ reflecting path, velocity dependence and exponential decay. In microscopic level, the flocking phenomenon becomes conditional, requires communication kernel $\psi(t)$ has polynomial lower bound and memory deviation $H(t)$ decays exponentially. In mesoscopic level, flocking rates exhibit exponential or algebraic decay and the range of communication strength $\beta$ become smaller.

We obtain closed explicit formulas for the spectral determinants of the smooth hyperbolic Bolza surface and the Klein quartic. In each case, a multiplicative relation expresses the determinant of the surface in terms of determinants of singular quotient orbifolds of genera zero and one. The elliptic factors are evaluated by applying the singular Polyakov anomaly formula to explicit Belyi maps on CM elliptic curves of discriminants -8 and -7, while the genus-zero factors are evaluated by explicit determinant formulas for constant-curvature spheres with conical singularities. The same multiplicative relations hold fibrewise on the corresponding equisymmetric deformation strata and yield determinant and first-variation identities. We also prove that every compact quasiplatonic hyperbolic surface is a critical point of the spectral determinant on its Teichmüller space; in particular, this applies to the Bolza surface and the Klein quartic.

We consider the Riemann problem for a 3x3 system of conservation laws with generalized Chaplygin pressure $p(\rho,v)=-\frac{A(v)}{\rho^\alpha}$, $0<\alpha\leq 1$, where the pressure depends on an additional transported variable. We analyze the system's wave structure and classify the Riemann solutions. Whenever classical solutions consisting of shocks, rarefaction waves, and contact discontinuities fail to exist, singular solutions arise. We verify that these satisfy the conservation laws in the distributional sense within the classical Dirac delta framework, and compare them with Nedeljkov's shadow-wave construction, giving two complementary descriptions of the same singular solution. We further study admissibility via the Dafermos maximum entropy dissipation principle, with several examples showing how it selects the physically relevant solution. Lax-Friedrichs simulations illustrate the Riemann wave patterns and provide a comparison with the analytical results. To construct viscous profiles for the isolated overcompressive $\delta$-shock, we assume $\alpha\in\mathbb{Q}\cap(0,1]$ and apply the Dafermos regularization together with a spherical blow-up. Working in three directional charts, we construct the reduced singular concatenation consisting of the left outer orbit, the middle orbit on the blown-up boundary, and the right outer orbit. We then prove that, for sufficiently small positive viscosity, this singular concatenation perturbs to a heteroclinic orbit. Consequently, the isolated overcompressive $\delta$-shock is realized as the zero-viscosity limit of a family of self-similar Dafermos viscous profiles.

The planar Pompeiu problem, originating in 1929, and the associated Schiffer conjecture are long-standing rigidity questions linking rigid-motion integral transforms and Fourier zero sets to overdetermined Neumann eigenvalue problems. We construct a bounded simply connected noncircular domain $\Omega\subset\mathbb{R}^2$ with real-analytic Jordan boundary and a nonconstant function $u$ such that $(\Delta+k^2)u=0$ in $\Omega$, $u=1,\partial_\nu u=0$ on $\partial\Omega $ for some $k\in(31.967007261,31.967007293)$. Thus $u$ is a Neumann eigenfunction which is constant on the boundary, and $\Omega$ is a counterexample to Schiffer's conjecture. Green's identity also gives $\widehat{\mathbf 1_\Omega}(k\omega)=0$ $(\omega\in\mathbb S^1)$, so $\Omega$ fails the Pompeiu property and is also a counterexample to the planar Pompeiu conjecture for bounded simply connected Lipschitz domains. We obtain the domain as $\Omega=\phi(\mathbb{D})$, where $\phi$ is a ten-fold symmetric conformal map close to an explicitly listed polynomial of degree $301$. On the unit disc, the analytic problem becomes a cubic operator equation on real coefficient spaces, $F(g,p)=g+|p|^2(1+Kg)=0$, where $K$, expressed in a disk-polynomial basis, is an explicit inverse of the Laplacian on the range compatible with zero Dirichlet and Neumann traces, and $p=k\phi'$. Positivity of the disk-polynomial linearisation coefficients, sharp bounds for $K$, and monotone control of the infinite tails establish an a posteriori contraction near the listed polynomial in a weighted coefficient algebra, and hence an exact zero of $F$.

We discuss entire solutions $u \in C^2(\mathbb{R}^2)$ of the equation ${\rm div}(\nabla f(\nabla u)) = 0$ with strictly convex density $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ of subquadratic growth and prove that $u$ is an affine function provided that at least one partial derivative is bounded from one side. Further results concern the behaviour of non-affine entire solutions.

This paper studies the fully nonlinear Alt-Phillips free boundary problem \begin{equation} F(D^2u) = u^\gamma \chi_{\{u>0\}}, \end{equation} for $\gamma \in (-1/3,1)$ without any convexity assumption on $F$. We establish that flat free boundaries are smooth. This result is new even for the fully nonlinear obstacle problem when $\gamma = 0$. Our approach is based on a partial hodograph transform, which converts the free boundary into a fixed boundary and produces a fully nonlinear degenerate equation. For this equation we prove a Harnack inequality, yielding an improvement of flatness, and a Schauder estimate for the associated linearized equation. Both estimates appear to be new and are of independent interest.