Papers

By adapting the techniques used in the IPM blowup result of Córdoba-Martínez-Zoroa with spatially smooth force, we prove finite-time blow-up for the IPM equation on $\mathbb T^2$ with a uniformly spacetime smooth force. In particular, we show there exist a smooth odd initial density, a smooth odd force $F\in C^\infty([0,1]\times\mathbb T^2)$, and a classical solution $\rho$ on $[0,1)$ whose density gradient and spatial velocity gradient diverge in $L^\infty$ as $t\uparrow 1$. Nevertheless, $\rho(t)$ converges in $C^\eta$ for every $0\leq\eta<1$.

We study finite-time blowup for the parabolic–elliptic Keller–Segel system on $\mathbb{R}^d$ in dimensions $d\geq11$, where the problem is mass supercritical. For every integer $l\geq2$, we construct smooth radially symmetric solutions whose radial mass variable concentrates the normalized stationary state $Q$ at a quantized scale. Each blowup regime can be realized by solutions with nonnegative population density throughout their classical lifespan. More precisely, near the blowup time $T$, \[ u(t,r)=\frac{1}{\lambda^2(t)}\left[Q\left(\frac{r}{\lambda(t)}\right) +\epsilon\left(t,\frac{r}{\lambda(t)}\right)\right], \qquad \lambda(t)=c(T-t)^{\frac{l}{\gamma(d)}}(1+o(1)), \] where $c>0$ and $\gamma(d)=\frac12\bigl(d-2-\sqrt{(d-2)(d-10)}\bigr)$. The remainder $\epsilon$ converges to zero in local $L^\infty$ norms and in a range of high-order homogeneous Sobolev norms. Since $2l>\gamma(d)$, the concentration scale is strictly smaller than the parabolic scale $\sqrt{T-t}$, and the resulting blowup is of type II. The $l$-th regime has exactly $l-1$ unstable radial modulation directions and is stable within a codimension-$(l-1)$ class of suitably regular radial initial data. The proof combines a generalized-kernel expansion driven by the algebraic tail of $Q$, modulation analysis, coercive weighted high-order energy estimates, and a finite-dimensional topological argument. This yields a quantized hierarchy of stationary-state concentration rates for the high-dimensional Keller–Segel flow.

We prove global strong well-posedness for the normalized isothermal, isobaric ternary Maxwell-Stefan diffusion system on bounded $C^{2+\alpha_0}$ domains, for arbitrary positive constant binary frictions and compatible nonnegative initial data, without a smallness condition. For distinct frictions, trace–determinant variables produce two separated spectral roots satisfying scalar parabolic equations with antisymmetric quadratic coupling. Euler-Poisson-Darboux one-sided truncations, stabilized by the mixing entropy, yield boundary De Giorgi oscillation decay and positive-time Hölder regularity. The mixing-entropy Hessian then provides a local symmetrizer; conormal Schauder estimates give uniform $C^{1+a}$ control, and frozen maximal $L_p$-regularity yields a uniform terminal trace bound excluding finite-time breakdown. Repeated frictions reduce to triangular scalar parabolic systems. Every component with positive conserved mass is strictly positive for positive time.

By addressing the problem of describing the differences between the saturation monoids of the Prandtl and Leray-$\alpha$ boundary layers, and algebraically distinguishing the action of the Helmholtz operator, we can show that, when the modification of the vector field is halted at a minimum scale determined by the differential filter, the paradox of constructing a solution that reaches infinite velocity in a finite time is avoided. Furthermore, it allows us to obtain the coefficients of the Taylor expansion near the wall for boundary layers where the Helmholtz operator has been applied.

We study the semilinear heat equation ${}^{S}D_0^{\alpha(t)}u=\Delta u+u^p$ on $\RR^n$, where $p>1$ and the Scarpi order changes exponentially from $\alpha_1$ to $\alpha_2$, with $0<\alpha_1,\alpha_2<1$. We establish local and maximal mild well-posedness for abstract Scarpi–Volterra equations. For the whole-space heat problem, positive resolvent families yield nonnegative solutions, comparison, a mass identity, and an $L^\infty$ blow-up alternative. To study finite-time growth, we use a Gaussian version of Kaplan's weighted-moment method. It reduces the PDE to a scalar nonlinear Volterra inequality and requires no pointwise lower estimate for the non-self-similar Scarpi heat kernel. Consequently, every nontrivial solution blows up in finite time when $1<p<1+2/n$, and sufficiently large data blow up for every $p>1$. For $u_0=A\varphi$ with fixed nonzero $0\le\varphi\in L^1\cap L^\infty$, the maximal lifespan satisfies \[ T_A\asymp A^{-(p-1)/\alpha_1} \qquad(A\to\infty), \] while a subcritical small-amplitude upper bound is governed by the long-time order $\alpha_2$. The lifespan bounds are expressed through the inverse of the integrated memory. In the numerical section, we complement these estimates by comparing the Scarpi dynamics with both Caputo endpoint models. Continuous Laplace inversion shows that the logarithmic slope of the integrated memory varies nonmonotonically across the transition. At large amplitudes, the growth thresholds approach those of the Caputo $\alpha_1$ model. A nonlinear space–time rescaling probes the long-time regime and reveals nonmonotone threshold-time ratios relative to Caputo $\alpha_2$. For fixed-width initial profiles, increasing the transition rate delays the prescribed growth threshold at larger tested amplitudes but advances it at smaller ones.

We establish a general mechanism that turns qualitative smoothness into uniform, quantitative regularity estimates for fully nonlinear elliptic equations. The central conclusion is that, for compact classes preserved by the natural rescalings of the equation, qualitative and quantitative regularity have the same critical threshold. At second order, mere twice differentiability throughout the corresponding centered hull already forces a uniform $C^{2,\alpha}$-theory for some $\alpha>0$. A one-sided tangent version gives a new partial regularity criterion for a single operator: every point at which a solution is twice differentiable is regular, whereas the singular set has universal positive codimension. The proof combines compactness of the full renormalized class with a scale-adaptive improvement of flatness, converting information available separately at each profile into estimates uniform across the entire class. The argument requires no quantitative control of the assumed smoothness and opens a path in problems where smoothness is visible, but estimates remain out of reach.

We study quantum systems modeled by Liouville-Von Neumann equations, with a finite number of energy levels. The system is coupled to a wave equation, intended to reproduce environmental effects on the quantum system. The whole dynamic is energy conservative, but damping effects are expected to be embodied into the energy exchanges with the environment. We analyse how these effects can effectively damp quantum coherences. We exhibit key mechanisms and structural properties for such a damping to arise, with a neat distinction between pure and mixed cases

A classical consequence of Caffarelli's fully nonlinear regularity theory [L. A. Caffarelli, Ann. of Math. (2) 130 (1989), no. 1, 189-213] is that viscosity solutions of uniformly elliptic equations $F(D^2u)=f$, with $f\in L^p$, $p > n$, are locally $C^{1,\gamma}$ for every $\gamma < \min\{\alpha_H,\sigma_p\}$. Here $\alpha_H=\alpha_H(n,\lambda,\Lambda)\in(0,1)$ denotes the universal Hölder exponent for gradient regularity of $F(D^2h)=0$, while $\sigma_p=1-n/p$ is the scaling exponent of the source term. In the source-limited regime $\sigma_p < \alpha_H$, the singularity of $f$ is the decisive obstruction and the endpoint $\gamma=\sigma_p$ is attainable. In the homogeneous-limited regime $\alpha_H\le\sigma_p$, however, classical theory only yields $\gamma < \alpha_H$, leaving the limiting differentiability estimate unquantified. This is the endpoint gap addressed here. When $\alpha_H < \sigma_p$, we prove that solutions admit pointwise Taylor expansions satisfying $|u(x)-u(x_0)-Du(x_0)\cdot(x-x_0)|\lesssim |x-x_0|^{1+\alpha_H}\left(1+\log\frac{1}{|x-x_0|}\right)^m$. Thus the homogeneous differentiability scale is reached up to an explicit logarithmic defect. At the critical threshold $\alpha_H=\sigma_p$, finite logarithmic powers no longer close the iteration; nevertheless, a slower selection of scales yields $|u(x)-u(x_0)-Du(x_0)\cdot(x-x_0)|=O\left(|x-x_0|^{1+\alpha_H}\exp\left(A\sqrt{1+\log\frac{1}{|x-x_0|}}\right)\right)$. Both estimates improve the full family of classical sub-endpoint $C^{1,\gamma}$ bounds, $\gamma < \alpha_H$, by quantifying differentiability at the limiting homogeneous exponent. The proof introduces a new scale-selection mechanism for endpoint Campanato-type recurrences, suggesting a flexible tool whenever the limiting smoothness is dictated by the homogeneous theory itself. We also discuss the role and possible optimality of the resulting logarithmic defects.

This paper is concerned with multivariable stochastic Hamiltonian systems governed by an ordinary differential equation for the position and an Ito stochastic differential equation for the momentum. The momentum dynamics involve both conservative and nonconservative forcing including Langevin viscous damping and the Wiener process as a random force. The setting allows for rotational degrees of freedom, a non-quadratic potential energy function and position-dependent mass, diffusion and damping matrices. In addition to modelling nonlinear physical dynamics subject to random forcing, the system is motivated by a stochastic optimisation viewpoint, where the search for global minima of the potential energy (for example, using the gradient descent) is carried out through convergence to a Boltzmann equilibrium measure which favours smaller values of this function. Under certain regularity conditions, we show that the $\chi^2$-divergence or, equivalently, Renyi's second-order relative entropy with respect to the invariant measure has a nonpositive time derivative and thus provides a Lyapunov functional candidate for this convergence. However, this derivative vanishes at some moments of time, which is related to the behaviour of the probability distribution of the momentum conditioned on the position and has a link to a family of auxiliary quantum harmonic oscillator Hamiltonians parameterised by the position space of the underlying classical system. This position-momentum conditioning is used in order to prove that such pre-equilibrium break times in the entropy dissipation are isolated, thus making the $\chi^2$-divergence a strictly decreasing Lyapunov functional for the Fokker-Planck-Kolmogorov equation which governs the joint position-momentum distribution in such systems.

We establish a quantitative characterization of the Mokobodzki condition for two adapted càdlàg barriers on an arbitrary filtered probability space. For every $p\geq1$, we introduce a mean co-variation functional $Var_p(L,U)$, which for $p=1$ and $L=U$ reduces to Rao's mean variation, and show that it is quantitatively equivalent to the minimal $\underline H^p$-norm among all semimartingales lying between the barriers. The result covers both the case $p>1$ and the endpoint $p=1$, where the natural scale is based on Doob's class $(D)$.

We consider a sharp-edge delta wing of small aspect ratio $A>0$, which is an example of a 3D airfoil whose aerodynamic properties cannot be modeled using the potential lift}only. An important role is played by a pair of attached vortices, which generate the vortex lift. We review in detail the leading-edge suction analogy, developed by Edward Polhamus (NASA Technical Note D-3767, 1966), which shows that the most accurate description of the vortex lift can be obtained by avoiding any direct analysis of vorticity of the flow around the wing. We also describe a 2D slender wing theory, which was introduced by Jones (NACA report no. 835, 1946) to approximate the potential lift. We show that, neglecting certain two borderline nonintegrability issues and an elliptic regularity issue, the slender wing theory is sufficient to yield accurate prediction of the potential lift as $A\to 0$. We also show that this 2D approximation is necessary to justify the modelling assumptions of the leading-edge suction analogy, so that the slender wing model can also be used to approximate the vortex lift as $A\to 0$. For completeness, we provide additional explanations of the necessary airfoil theory.

We establish precise upper and lower estimates for harmonic functions vanishing on the slit of a $C^1$ slit domain, with no assumption that the slit lies in a hyperplane. The classical $\sqrt{d}$ growth near the edge, $d$ being the distance to the slit, persists in this generality, up to an explicit factor \[\exp\Big( \pm C \int_\rho^r \omega(s)\, \frac{ds}{s} \Big)\] determined by the $C^1$-modulus of continuity $\omega$ of the slit, where $\rho < r$ are the two scales being compared. The upper estimates allow a right-hand side and non-zero boundary data. The correction factors remain bounded above and below by positive constants as $\rho\to0$ precisely when $\omega$ satisfies the Dini condition. Moduli of continuity beyond the Dini regime, as is the case for the logarithmic moduli arising at singular sets in relevant free boundary problems, were not covered by the previous $C^{1,\alpha}$ theory. Previously, the $\sqrt{d}$ growth was known for slits contained in a hyperplane, which additionally have a $C^{1,\alpha}$ edge (De Silva, Savin). For Lipschitz slits, there are boundary Harnack principles, but no growth rate is identified precisely. The main technical ingredient is a change of coordinates flattening a Lipschitz slit domain onto the model half-hyperplane slit, with quantitative estimates up to second order. The construction is geometric and does not use the equation, so we expect it to be useful for other boundary regularity problems.

The existence of a positive radial solution to the problem \begin{align*} -\Delta u + \lambda u = u^p + u^q, \quad u \in H^1(\mathbb{R}^N), ~~ N \ge 2, \end{align*} where $\lambda > 0$ and $1 < q < p < 2^* - 1$, has been known for a long time. For the pure-power nonlinearity, it is well known that this solution is unique. In contrast, the uniqueness problem for the double-power case is substantially more delicate and depends on the dimension and the exponents. It has been known that the uniqueness result is in general not true for the equation with double-power nonlinearities in dimension three and it is a long-standing open question to prove the uniqueness of positive radial solutions throughout the full subcritical range for $N \ge 4$ since the work of H. Berestycki, P.-L. Lions [2] (Arch. Ration. Mech. Anal. 1983). In this paper we provide the uniqueness results for all $\lambda > 0$ and all $1 < q < p < 2^* - 1$ when $N \ge 5$. In dimensions $N = 2,3,4$, with additional conditions on $p$ and $q$, we obtain the uniqueness results for all $\lambda > 0$. Our results are in sharp contrast to those of J. Dávila, M. del Pino and I. Guerra [6] (Proc. London Math. Soc. 2013), where some non-uniqueness results were obtained for $N=3$. At the same time, we give a positive answer and rigorous proof to the implication of their numerical simulation, clarify the open issues left by [6]. Finally, we completely resolve the uniqueness problem of the above equation with $L^2$-mass constraints in the mass-subcritical and mass-critical regimes for all $N \ge 2$.

Establishing global curvature estimates for $k$-convex solutions of prescribed curvature equations is a longstanding problem in fully nonlinear partial differential equations and geometric analysis. We develop a new approach that combines fractional-linear transformations with the logarithmic concavity of hyperbolic polynomials. The key step is a coercive estimate for the quadratic forms arising from the third-order terms in the maximum-principle argument. Together with a complementary concavity inequality, this estimate yields global curvature bounds for closed, strictly star-shaped $k$-convex hypersurfaces in $\mathbb{R}^{n+1}$ satisfying $\sigma_k(\kappa)=f(X,\nu)>0$, throughout the range $3\leq k<n<2k$.

In this paper we prove the existence of monotone traveling wave solutions for an age-structured population model with a monotone and bistable nonlinearity. We formulate the problem in terms of a discrete-time recurrence $u_{n+1} = Q[u_n]$, with operator $Q$ encoding the role of the speed into its definition. For each speed, we prove the existence of a traveling wave by the general theory of monotone evolution systems, and from appropriate estimates on the speed, we deduce the existence of a traveling wave of the original problem. Also, the sign of the speed is established under an invasion condition, and the asymptotic behavior of the profile is determined by some Tauberian results.

We develop a theory of Sobolev, Besov and Triebel–Lizorkin spaces associated with weighted subcoercive operators on any real connected Lie group.

We study three classical problems in the theory of linear second-order uniformly elliptic equations: (i) gradient estimates for the Dirichlet problem, (ii) estimates and asymptotics for the first eigenvalue of an elliptic operator, and (iii) the Landis conjecture on exponential decay. Our main results give versions of (i) in which the constant is sharply specified in terms of the norms of the coefficients of the operator and the size of the domain; and use these to strongly improve on known results for (ii) and (iii) by allowing both more general operators and weaker regularity assumptions on the coefficients, and by giving quantitative estimates. The proofs use a unified approach, relying on three ingredients: interior and boundary Harnack inequalities with sharp constants, duality arguments, and a $C^1$ estimate based on a rescaling procedure. The sharpness of the gradient and spectral estimates is demonstrated through various counterexamples.

Let $0<s<1$ and $A$ be a constant symmetric positive definite matrix. Given the quadratic form $$ Q_A(u)=\int_{\R^N} A\nabla^s u(x)\cdot \nabla^s u(x)\,dx, $$ where $\nabla^s$ denotes the Riesz fractional gradient, we establish necessary and sufficient conditions such that $Q_A$ defines a Dirichlet form. In strike contrast with the local case $s=1$, positive definiteness of a constant matrix is not sufficient for the Markov property when $0<s<1$. We also describe the associated symmetric Markov semigroup and its $L^p$ realizations, prove the existence of a smooth nonnegative heat kernel and establish a bounded $H^\infty$ functional calculus.

In the compressible primitive equations, gravity is not merely a source term: through the hydrostatic balance it forces, it organizes the system's energy geometry. The hydrostatic balance forces the sum of the barotropic enthalpy and the geopotential to be independent of the vertical coordinate an intrinsic rst hydrostatic integral, coinciding with the classical Montgomery potential of geophysical uid dynamics and testing the continuity equation against it gives the total mechanical energy identity, with no separate estimate on the vertical velocity. The core of this note is an augmented hydrostatic identity, obtained by testing the momentum equation against a BreschDesjardins-type augmented velocity: dierentiating the rst integral again produces two structural functions governing its cross terms, whose vanishing cases are characterized exactly here one vanishes identically if and only if the pressure is quadratic, the other if and only if it is isothermal, no law annihilating both. The resulting augmented identity also isolates a coupling involving the horizontal gradient of the vertical velocity that is not controlled by the energetic structure developed here, a limitation of this direct route to closing a BreschDesjardinstype entropy, not an impossibility theorem. For the isothermal law, the vanishing of the second function forces the vertical density prole to be a pure exponential, exactly the factorization underlying the Ersoy NgomSy change of variables the multiplicative expression of the Montgomery rst integral at that one law on which basis we give a detailed return-to-original-variables argument for the existence theorem of Wang, Dou and Jiu.

We give a Green-function proof of the stable Bernstein theorem in $\mathbb R^7$ for smooth, connected, complete, two-sided minimal hypersurface, thus resolving the last case in stable Bernstein problem.

We establish an intrinsic Harnack inequality with an optimal tail term for weak solutions of parabolic fractional $p$-Laplace equations with $p>2$ and $s\in(0,1)$ whose kernels are symmetric, measurable, and comparable to $|x-y|^{-n-sp}$. It constitutes the nonlocal analogue of the intrinsic Harnack inequality of DiBenedetto, Gianazza and Vespri. In contrast to the exponential change of variables and expansion-of-positivity used in the local case, our technique is nonlocal and relies on comparison arguments and barriers in a variational framework. Indeed, we obtain the stronger conclusion where the forward comparison starts immediately after the reference time, without an additional time gap, which is false in the local case. The latter leads to elliptic-type Harnack estimates for parabolic equations. Our result is new even in the linear case.

We study bounded entire solutions of the Allen-Cahn equation in dimension three with quadratic energy growth. We prove that such a solution of finite Morse index I has at most 2I ends. Furthermore, we show that Morse index one solutions have two ends and are axially symmetric, which generalizes an earlier result by Florit-Simon.

Quadrotor navigation in cluttered 3-D environments requires global guidance while local motion remains subject to collision and motion limits. Harmonic potentials provide dense guidance from a global boundary value problem, but coupling a volumetric harmonic field to constrained physical quadrotor motion remains an open experimental problem. We couple a precomputed volumetric harmonic field with a constrained predictive planner that queries the field at predicted positions instead of extracting a global reference path. In Structured 3-D tests, harmonic guidance yields larger minimum clearance and lower RMS jerk than matched Dijkstra guidance, at the cost of longer paths; the same pattern remains when both methods use the same passage. Long maze tests span routes far beyond one prediction horizon, and Crazyflie trials validate physical execution. To the best of our knowledge, this is the first physical quadrotor demonstration of volumetric harmonic field navigation. The results show that globally constructed harmonic guidance can directly support local constrained motion generation on a physical quadrotor.

We prove global Sobolev convergence of Howard policy iteration for finite-horizon Bellman equations with control-dependent diffusion. In one dimension, the result holds for bounded measurable coefficients under uniform ellipticity, without a large discount, a short horizon, a small diffusion perturbation, or regularity assumptions on the improving policies. We also obtain multidimensional extensions under structural conditions on the diffusion matrix. The key observation is that vanishing policy improvements force the Bellman residual to converge in measure; higher integrability then yields strong convergence and constructs the unique strong solution. We extend the argument to entropy-regularized models, including zero-temperature limits, and establish quadratic convergence for special one-dimensional models.

We prove that triangles minimize the Monge-Ampère eigenvalue among bounded planar convex domains of a given area, thus resolving the two-dimensional case of Le's simplex conjecture for the Monge-Ampère eigenvalue (The eigenvalue problem for the Monge-Ampère operator on general bounded convex domains, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 18 (2018)). The proof uses a neighbor-decreasing property of the Monge-Ampère eigenvalues of convex polygons along vertex elimination parallel chord movements.

We establish a weak Harnack inequality for nonnegative strong supersolutions and a Harnack inequality for nonnegative strong solutions of the nondivergence-form kinetic equation \[ \partial_tu+v\cdot\nabla_xu+a(t,x,v):\nabla_v^2u +b(t,x,v)\cdot\nabla_vu+c(t,x,v)u=0, \] where the coefficients $a$, $b$, and $c$ are merely Borel measurable, the matrix $a$ is uniformly elliptic, and $b$ and $c$ are bounded. As a consequence, we derive interior kinetic Hölder estimates for strong solutions. Our approach develops a probabilistic kinetic analogue of the Krylov–Safonov method. The main new ingredient is a quantitative version of Krylov's estimate for kinetic Itô processes with progressively measurable coefficients, which in turn yields the global existence of weak solutions to the associated kinetic SDE with bounded measurable coefficients. These results appear to be the first Harnack and Hölder regularity theory for kinetic equations with merely measurable coefficients.

This paper is concerned with the combined limits of large bulk viscosity and small angular viscosity for the two-dimensional compressible micropolar fluid equations. We first establish the global well-posedness of strong solutions to the Cauchy problem with large bulk viscosity, when the initial velocity is well-prepared in the sense that $\sqrt\nu \|{\rm div} u_0\|_{L^2}$ is uniformly bounded. The global a priori estimates obtained are uniform with respect to both the bulk viscosity $\nu$ and the angular viscosity $\varepsilon$. Based on these uniform estimates, we then justify the combined limits as $\nu\to\infty$ and $\varepsilon\to0$, and show that the solutions of the compressible micropolar system converge to those of the viscous and non-diffusive incompressible micropolar system. The convergence rates for the incompressible limit and the vanishing angular viscosity limit are also derived. In particular, the convergence rate for the incompressible limit is shown to be of order $\nu^{-1/4}$ in general, and can be improved to $\nu^{-1/2}$ under an additional decay condition on the initial density. Compared with previous works, the vanishing angular viscosity limit is established without the extra technical assumption $\lim_{\varepsilon\to0+}\zeta/\varepsilon^\alpha<\infty$ for some $1/2\leq \alpha<\infty$, which extends the relevant results from the incompressible case to the compressible setting. As a by-product, the global regularity of large solutions for the 2D compressible micropolar equations with/without angular viscosity is obtained, provided the bulk viscosity is large enough.

We establish the local boundedness of time derivatives of bounded viscosity solutions to the normalized parabolic $p$-Laplace equation for every $p\in(1,\infty)$. The proof combines a logarithmic Bernstein argument with quantitative stability to derive a recursive estimate between different regularizations. For $p=1$ and $n\geq2$, we construct a bounded viscosity solution which is not Hölder continuous in time with any positive exponent.

In this paper, we study the asymptotic behavior of least energy solutions to Lane–Emden problems on compact metric graphs as $p \to \infty$. For the problem with Dirichlet–Kirchhoff boundary conditions, we characterize the limiting variational problem in terms of the Dirichlet–Kirchhoff Green function and show that least energy positive solutions converge, up to a subsequence, to a normalized Green function centered at a maximizer of the diagonal Green function. We also show that their maximum points approach the set of such maximizers. For the problem with Kirchhoff–Neumann boundary conditions, we characterize the limiting variational problem on compact metric graphs without cycles and show that least energy solutions converge, up to a subsequence, to a limiting profile associated with a pair of points realizing the maximal distance on the graph. Moreover, the distance between their maximum and minimum points converges to the maximal distance on the graph.

We study the Faber–Krahn problem for the fractional–logarithmic Laplacian $(-\Delta)^{s+\ln}$, $0<s<1$, with Fourier symbol $|\xi|^{2s}\ln|\xi|^2$. We prove a small-volume and large-volume dichotomy. For sufficiently small volume, balls minimize the first Dirichlet eigenvalue, with rigidity up to translation.