Papers

We establish well posedness of the Dirichlet problem in weak local John domains $\Omega\subset\R^\dmn$ with Ahlfors regular boundaries of possibly non integer dimension $0<\bdmn<\dmn$, for linear second order elliptic equations with real coefficients, and with data in Besov spaces with a very broad range of acceptable parameters.

In this paper, we establish global $C^{1,1}$ solvability of the Dirichlet problem for the degenerate $2$-Hessian equation on bounded strictly mean convex $C^{3,1}$ domains $\Omega$, with general $C^{3,1}$ boundary values and nonnegative right-hand sides $f\in C^{1,1}(\overline{\Omega})$, resolving the $k=2$ case of a longstanding open problem. We uncover a global semiconvexity structure for general $2$-admissible solutions by proving a lower bound for $\sigma_{3}[D^{2}u]$ independent of $\inf_{\Omega}f$. For $3\leq k\leq n-1$, we also establish global semiconvexity for $k$-admissible solutions on the unit ball with $C^{3}$ boundary values and $f^{1/(k-1)}\in C^{1,1}(\overline{B}_{1})$, with a bound independent of $\inf_{B_{1}}f$. A counterexample shows that the $C^{3,1}$ boundary value assumption for global $C^{1,1}$ solvability is sharp in the Hölder scale.

We prove that the knowledge of local Cauchy data for a fractional Schrödinger equation on a smooth closed and connected Riemannian manifold of dimension at least two determines the manifold, its metric, and a smooth complex- valued potential, up to an isometry fixing the observation set. The observation set may be any nonempty open subset and no further geometric assumptions are required. The proof uses functional calculus identities to recover the local action of the resolvent of the Laplace-Beltrami operator.

We construct an odd smooth initial density for the unforced incompressible porous media (IPM) equation on $\mathbb T^2$ whose solution develops a finite-time blowup. The components $\partial_{x_2}\rho(t,0)$ and $\partial_{x_1}u_1(t,0)$ tend to $+\infty$ as the maximal smooth existence time is approached, while the density and velocity remain bounded in $L^2$. Starting from a stationary solution, we iterate the angular amplification of localized oscillations with rapidly growing frequencies, adapting the mechanism of Córdoba and Martínez-Zoroa [7] for forced IPM. Each perturbation is constructed from time zero and creates the geometry configuration for the next iteration stage. A one-sided upper bound for the pressure Hessian is applied to control backward preparation and initial perturbations, following an idea from the Euler construction of OpenAI [14].

We consider the defocusing quintic wave equation $\partial_t^2u-\Delta u+u^5=0$ for real-valued functions on $\mathbb{R}^{1+10}$, which is energy-supercritical. We construct a discretely self-similar solution in a backward light cone, of the form $$u(t,x)=(T-t)^{-1/2}W^*\big(\omega^*\log\frac1{T-t},\frac{|x|}{T-t}\big),$$ where the nonzero profile $W^*$ is $2\pi$-periodic in its first variable and real-analytic up to and across the light cone. Cutting off its initial data gives smooth, compactly supported, radial data whose solution coincides with the discretely self-similar solution in the backward light cone, is smooth on $[0,T]\times\mathbb{R}^{10}$ except at the vertex $(T,0)$ of the cone, and blows up there at the self-similar rate. The solutions we construct here appear to be the first real-valued blow-up solutions of defocusing energy-supercritical NLW. Existence of the real profile is proved by a computer-assisted Newton–Kantorovich argument in a Banach algebra of Fourier–Chebyshev coefficients, with rigorous error control. We find the approximate profile by numerically continuing rotating self-similar profiles of the complex-valued equation. This step is not essential for the blow-up proof, so it is left as non-rigorous numerics. The main analytic ingredients are an exact description of the mode operators for the linear part, which are hypergeometric and upper triangular in the Chebyshev basis, and a bound for their inverses that is uniform at high frequencies. All computer-assisted proofs, including the requisite codes and the approximate objects, are accessible on GitHub at \cite{code}.

We establish an $L^1$-theory for a class of elliptic operators with unbounded coefficients, extending our earlier semigroup generation results and those of Canale–Rhandi–Tacelli to the endpoint $p=1$. The $L^1$-realization generates a positive analytic $C_0$-semigroup that is compact for every positive time. We characterize the generator domain through weighted estimates and prove consistency with the corresponding $L^p$-semigroups for $1<p<\infty$.

This paper introduces a new extrapolation framework within the Calculus of Variations. We develop a robust quantitative theory of the $\Gamma$-convergence approach of De Giorgi by combining it with Gagliardo's work on relative completions in interpolation theory. A central new tool is the extrapolation of \emph{variational $K$-functionals}, which associates to an arbitrary family of functionals a dynamic penalty method that simultaneously characterizes their $\Gamma$-limits and yields explicit rates of convergence for minimizers. As a technical foundation we prove compactness extrapolation theorems also involving Gagliardo completions. To illustrate the mechanism, we apply the technique to variational models arising in image processing, notably the Rudin–Osher–Fatemi (ROF) and Chan–Esedoglu models. Our main result in this setting is a contribution to the inverse regularity problem associated to discrete approximations of the ROF model: we show that the required regularity of the observed image that guarantees a prescribed optimal rate of convergence is measured in terms of the decay of variational $K$-functionals. Surprisingly, our characterization is explicitly captured by using the recently introduced spaces of Brezis, Van Schaftingen, and Yung. Moreover, our methods unify, and considerably extend, classical pointwise convergence and $\Gamma$-versions of the Bourgain–Brezis–Mironescu and Maz'ya–Shaposhnikova formulae, as well as related results in a variety of settings that involve BMO-type functionals, heat kernels, Sobolev spaces on metric measure spaces, ... By bridging De Giorgi's variational convergence with Gagliardo's completion through variational $K$-functionals and extrapolation of compactness, this work establishes a quantitative approach to the Calculus of Variations.

In this work, we study transparent boundary conditions for the time-fractional heat equation on a metric star graph. This equation is the standard continuum model for subdiffusive heat and mass transport in disordered and porous branched media. Using the Laplace transform to solve the exterior branch problems, we demonstrate that the non-local vertex operator observed by the incoming bond is an Abel-type convolution kernel of order $\beta/2$, which generalizes the classical half-derivative kernel recovered at $\beta=1$. Our main result is that the diffusion-coefficient sum rule $\kappa_1=\kappa_2+\kappa_3$, known to eliminate thermal backflow for the classical heat equation, remains unchanged for every fractional order $\beta$: the transparency of the junction depends only on the network's diffusivities and not on the memory of the transport law. We rigorously prove this and uniqueness for the coupled vertex problem. We confirm our results numerically at the level of the underlying coupled partial differential equation system with an implicit L1 finite-difference scheme. We compare the star-graph solution to a reference solution on an unbranched line with matching diffusivity.

We study a notion of backward stochastic differential equation on the Wasserstein space, called a W-BSDE. This work extends to the common noise setting recent results of the first author. The equation is formulated in Markovian form along admissible flows of conditional laws generated by McKean-Vlasov dynamics with common noise. A solution is a pair of fields $(Y,Z)$, where $Y$ is a functional on the space of probability measures and $Z$ is the corresponding intrinsic Wasserstein gradient. Our main result shows that the well-posedness of the W-BSDE can be reduced to a characteristic equation for a field $p$. Whenever this equation admits a suitable solution, we construct a W-BSDE solution and identify $Z$ with both the characteristic field $p$ and the intrinsic gradient of $Y$. This gives an existence result for the W-BSDE and, at the same time, a probabilistic construction of a functional on the Wasserstein space together with its intrinsic gradient. We then develop several consequences of the theory. We obtain verification principles for McKean-Vlasov control problems and for mean field games with common noise, without relying on convexity or separability assumptions. We also show that the characteristic field admits a dynamic representation through a McKean-Vlasov forward-backward SDE. The first component $Y$ solves the associated semilinear PDE on the space of probability measures in the viscosity sense, including the second-order term induced by common noise. Under additional smoothness, $Y$ is a classical solution. Finally, we prove that W-BSDEs arise as intrinsic mean-field limits of classical finite-dimensional BSDEs written on empirical measures, with convergence of both the value component and the rescaled martingale integrands.

Let $(M^n,g_0)$ be a closed connected Einstein manifold with positive scalar curvature, where $n\ge6$. Using an Obata-type identity, we prove that every smooth metric conformal to $g_0$ with constant sixth-order $Q$-curvature is Einstein.

In this paper, we rigorously justify the low Mach number limit of the full compressible Navier-Stokes equations in a three-dimensional smooth bounded domain, subject to the no-slip boundary condition for the velocity and well-prepared initial data. In the presence of heat conduction and large temperature variations, we establish the uniform estimates for strong solutions on a time interval independent of the Mach number by constructing an energy functional in a Sobolev framework. The no-slip boundary condition cannot provide the vorticity boundary relation used in the slip-boundary setting. A key ingredient is to employ a divergence-free Stokes lifting of the boundary trace of the time derivative of modified velocity to define a test function vanishing on the boundary.

We study the surface Stokes-Cahn-Hilliard system governing tangential flow and phase separation in viscous fluid membranes. The equations are posed on a closed, stationary $C^2$ hypersurface $\Gamma\subset\mathbb{R}^{d+1}$, $d=2,3$. For constant viscosity and mobility and a class of regular double-well potentials, we prove global existence and uniqueness of weak solutions for initial phase fields in $H^1(\Gamma)$, with the velocity chosen orthogonal to the Killing fields. These solutions satisfy an energy equality and conserve mass. We establish Lipschitz continuous dependence in energy norms on every finite time interval, including for initial data with different means, and derive higher regularity results for $d=2$ under additional smoothness assumptions on the surface, potential, and initial data. We also develop a fully discrete unfitted finite element method that conserves the total phase and satisfies an exact discrete energy identity. The method combines a penalty on approximate Killing fields with projections in the transport and capillary terms to control these modes while preserving the energy balance. Numerical experiments examine the identification of Killing fields and illustrate phase separation on a torus and a liposome with experimentally informed material parameters.

A nonlinear observer is designed for a one-dimensional quasilinear diffusion equation with Neumann boundary conditions, using as an output the state at one end of the spatial domain. The observer consists of a copy of the system, with a nonlinear (in the output) correction term at the boundary. A Lyapunov-based analysis is performed to ensure global exponential stability of the zero equilibrium for the estimation error dynamics in the $L^1$ norm. The result holds for a general nonlinear diffusivity, under the assumption that it is lower-bounded by a positive constant, i.e., that the diffusion equation is uniformly parabolic.

We prove finite-time gradient blow-up for the unforced incompressible porous media equation on the two-dimensional torus, removing the forcing used in the IPM blow-up constructions of Alpöge, Buckmaster, and Coiculescu \cite{lit:ABC} and Córdoba and Martínez-Zoroa \cite{lit:CMIPM}. The key is an identity that expresses the logarithmic growth of a layer's gradient as the negative pressure curvature transverse to its transported phase. A nearly vertical parent layer rotates and amplifies a new phase, with a gain exponential in the reciprocal parent tilt. An asymmetric periodic profile gives a uniform upper bound for the pressure Hessian, preventing rapid decay of the gradient magnitude. Each layer is seeded at time zero; its phase means are selected by a terminal displacement problem, and a finite order expansion is corrected to an exact unforced solution. The scale induction then shows blow-up of two derivative components at the origin and convergence to a terminal density in every $C^\eta$, $\eta<1$.

In this paper we prove the existence, uniqueness and regularity of classical solutions \break of a semilinear parabolic equation with a mixed local and nonlocal diffusion operator \break $\mathcal{L} = -(-\Delta)^s + \Delta$ and Dirichlet boundary conditions. Here, $(-\Delta)^s$ is the integral fractional laplacian and $\Delta$ is the classic local laplacian. We then study the interior regularity of said solutions and conclude that they are Hölder continuous in both space and time, and they are $C^{2,\alpha}_{loc}$ in space for all positive times.

We study a general class of fully non-linear elliptic Hessian equations on compact Riemannian manifolds. In particular, we replace the standard assumption that the equation is a symmetric function of the eigenvalues of the Hessian made by Caffarelli, Nirenberg and Spruck by a weaker one. After deriving a priori estimates for solutions under suitable assumptions on the equation, we solve it with a continuity method.

A synthetic view is provided on scattered results which were never presented in this form in the literature. First we investigate precisely the oscillation length for periodic functions. Then, we establish the existence of a uniform oscillation lentgh for any scalar almost periodic function with mean-value 0. After that, we investigate upper bounds of the oscillation length for some almost periodic functions which appear in the study of continuum mechanics, and we show the existence of a finite oscillation length with respect to time in any open subset of some vibrating continuous media. The paper finally recalls a long standing open problem on the vibrations of a square membrane with fixed edge.

While unconditional uniqueness for the cubic nonlinear Schrödinger equation (NLS) on $\Bbb R^3$ is known in the scaling-subcritical Sobolev spaces $\dot H^s(\Bbb R^3)$ for $1/2<s<1$, the critical case $s=1/2$ has remained open. We resolve this problem by adapting Kato's bootstrap argument to the critical space $\dot B^{-1}_{\infty,\infty}$. The main difficulty at this endpoint is that the natural solution class does not provide the coefficient regularity required to close the bootstrap in $\dot B^{-1}_{\infty,\infty}$. To overcome this issue, we perform a second Duhamel iteration of Born type and estimate the resulting double Duhamel integral as a whole. This yields improved estimates that allow the bootstrap to close in $\dot B^{-1}_{\infty,\infty}$ using only the available regularity of the coefficients.

We classify the regularity of the origin for the $p$-parabolic equation in the soda-can domains $$\Theta_{l,\theta}=\{(x,t)\in\mathbb{R}^n\times\mathbb{R}:0 < -t < \theta |x|^l < \theta\},\text{ where }l,\theta>0.$$ The classification covers every $p>1$ in dimensions $n\ge2$. At the endpoint $l=p$, the origin is regular if and only if $p\ge2n/(n+1)$. We also prove that the origin admits a traditional barrier for every $p>2$ and $l>0$. When $2 < p < n$ and $l < p$, the origin is nevertheless irregular. This answers negatively the well-known open question whether one barrier suffices to characterize boundary regularity in the degenerate range $p>2$.

We provide a complete study of the optimal control problem associated to the so-called Dyson equation, which describes the mean field evolution of the spectrum of large random matrices. We treat both the case of regular and singular terminal data. We construct admissible controls and prove the existence of optimal ones. We then proceed to give a thorough mathematical analysis of the associated Hamilton-Jacobi-Bellman equation by means of the theory of viscosity solutions in the space of measures. In particular, we characterize the value function of the optimal control problem as the unique viscosity solution to this infinite dimensional HJB equation.

We study a one dimensional nonlinear Klein-Gordon equation with a potential, a localized quadratic nonlinearity, and a non-localized quartic nonlinearity. We assume that the linearized operator has a single discrete eigenvalue below the continuous spectrum, corresponding to an ”internal mode”. This eigenvalue generates localized, time-periodic solutions for the linear equation. Assuming the natural Fermi Golden Rule, we give a global description of the dynamics of the amplitude of the internal mode through the radiation damping mechanism, and prove scattering for the radiation. Our approach is based on the distorted Fourier transform and a collection of refined dispersive decay and smoothing estimates. The corresponding cubic problem is closely related to the question of global-in-space dispersive asymptotics for perturbations of kink solutions in the classical $\phi^4$ model, which remains a challenging open question. From the perspective of nonlinear estimates, a quartic (non-localized) interaction is essentially sharp for our analysis, and closing the estimates with such a low degree nonlinear term requires a careful and delicate treatment. As a direct application of our analysis, we prove a result on the asymptotic stability of kinks for classes of scalar field models that can be seen as perturbations of the $\phi^4$ model.

We consider weak solutions $u \in W^{1,n}(U,\mathbf{R}^{n+1})$ of the $H$-system $- \operatorname{div} ( |\mathrm{D} u|^{n-2} \mathrm{D} u ) = (H \circ u) \, \mathbf{J} u$, where $n \ge 2$, $U$ is the open unit ball in $\mathbf{R}^n$, and $H$ is bounded. We show that $u$ is locally bounded if its distortion $|\mathrm{D} u|^n / |\mathbf{J} u|$ is bounded on the set where $|u|$ is large. By a result of Fusco, Kristensen, Leone, and Verde, it follows that $u$ is locally Hölder continuous if $H$ is continuous, and continuously differentiable with locally Hölder continuous derivative if $H$ is locally Lipschitz. The proof pushes forward $|\mathrm{D} u|^{n-2} \mathrm{D} u \circ \mathrm{D} u^*$ by $u$ to a linear functional on continuous compactly supported maps of $\mathbf{R}^{n+1}$ into $\operatorname{Hom}(\mathbf{R}^{n+1},\mathbf{R}^{n+1})$, whose values on derivatives are controlled by the equation. When the distortion is bounded, this functional satisfies the hypotheses of an inequality of Michael-Simon type for measures satisfying a first-order partial differential equation, due to De Philippis, Gennaioli, Pigati,and Rindler. The inequality yields lower bounds for the mass ratios of the push-forward of $|\mathrm{D} u|^n \mathscr{L}^n$ by $u$, and these imply boundedness.

We consider nonlocal divergence-form operators of the type $$PV\int_{\mathbb{R}^n}(u(x)-u(y))K(x, y) dy=0\quad \text{in} \; B_2,$$ where the kernel $K$ is symmetric ($K(x, y) = K(y, x)$). We first show that there exists a kernel satisfying the conditions of \cite[Open Question 2.1]{FRRO24} for which this problem admits a bounded discontinuous weak solution, answering that question negatively. We then impose the stronger one-sided lower bound suggested in \cite{CS20,IS20}, together with an $L^p$ bound of $K$ over annuli, and prove a coercivity estimate and interior Hölder regularity of weak solutions. In particular, this establishes the coercivity conjecture of \cite{CS20,IS20} under the additional $L^p$ hypothesis for $p>1$.

We establish global energy compactness for rough three-dimensional axisymmetric Euler flow without swirl and use it to construct hydrodynamic limits of collisional kinetic equations beyond finite-time Euler regularity breakdown. For relative vorticity in $L^1\cap L^p$, regular approximate Euler flows are globally compact in $C_tL_x^2$ for $p\ge4/3$, without a sign condition or any spatial-moment assumption. Below $4/3$, the same conclusion holds under uniform control of the absolute impulse. We identify the sharp radial dynamics governing this confinement: the absolute impulse is propagated without a sign condition for $p\ge5/3$, while for $1<p<5/3$ it is controlled under one-sided $L^{p^\sharp}$ integrability, where $p^\sharp= {4p} / (3p-1)$, and this exponent is sharp. In particular, for one-sign finite-impulse vorticity the resulting global energy compactness holds throughout the full range $p>1$. For the Coulomb Landau equation and a class of non-cutoff soft-potential Boltzmann equations, we construct strong solutions on lifespans $T^\varepsilon\to\infty$ realizing prescribed rough finite-energy Euler initial data and obtain energy-conserving axisymmetric Euler limits on every fixed time interval. If the corresponding regular Euler evolution loses regularity in finite time, the hydrodynamic fields converge to the regular Euler solution up to the singular time and have a common strong trace there, while subsequences yield global energy-conserving Euler continuations past that time.

We study the position and shape of invasion fronts for spatially periodic Fisher–KPP equations on higher-dimensional lattices, starting from localized initial data. Periodic spectral geometry determines a directional logarithmic upper bound. When the linear growth rate is independent of the propagation coordinate, we prove the matching axial Bramson correction and identify the critical pulsating front approached by the solution. The selected profile includes the local periodic cell and a bounded phase whose derivative tends to zero. This description also determines the times at which prescribed levels reach distant lattice sites and the asymptotic time gaps between levels or neighboring cells. The proof combines a periodic transverse heat kernel, a scalar lower comparison, and critical-tail analysis. For growth rates varying in every coordinate, a positive Dirichlet quotient comparison reduces the matching lower bound to two explicit estimates on the linear seed and its accumulated nonlinear loss.

We study higher moments of quantum fluctuations and microscopic correlation energy in the three-dimensional Gross–Pitaevskii regime. For a smooth, compactly supported, nonnegative radial interaction and an $N$-independent normalized initial condensate wave function in $H^6$, we propagate the third spectral moment of a positive correlated energy along the exact many-body evolution. A uniform initial bound controls the third moment of the depletion number and sixth moments of collective one-particle fluctuations on every compact time interval, independently of $N$. For a nonzero interaction and a smooth, compactly supported initial condensate, we also identify the excitation-energy measure at scale $N^2$, weighted by the first energy. This limit holds for norm-convergent pure-state families with a uniform bound on the initial third correlated-energy moment. Its positive-energy part is a spectral measure of the relative scattering operator $-2\Delta+v$, multiplied by a time-dependent dressed contact. We compute the radial density and prove strict positivity of the contact for nonzero references with finite kinetic energy and finite mean excitation number. The same evolving states have bounded physical fluctuation moments and excitation-energy moments of order $p>1$ bounded below by a positive multiple of $N^{2p-2}$, in the extended spectral sense. Inverse powers of the positive initial energy construct admissible pure and mixed families. For the norm-convergent pure-state class, we determine the scaled interaction-energy moments for $1\le p<2$ and bound the second moment by $O(N^2)$. A finite cubic approximation has norm error $O(N^{-3/16})$ on the energy form domain and $O(N^{-5/4})$ on a stronger initial domain of its generator.

We study positive solutions of a discrete Hénon equation on the integer lattice graph and on the half-space, with zero Dirichlet boundary conditions in the half-space case. The nonlinearity is governed by a Hénon-type potential with positive exponent. For sufficiently large nonlinearity exponents, we prove the existence of positive solutions both on the whole lattice in dimensions at least three and on the half-space in dimensions at least two. Thanks to the Hénon potential, the standard variational compactness is unavailable to obtain the positive solution. Our proofs combine weighted Banach spaces, two-sided Green kernel estimates and the Schauder fixed point theorem.

This paper introduces a robust strategy to establish large deviation principles for mean field interacting particle systems in T d with vanishing noise. Viewed as a Freidlin-Wentzell type problem, our approach relies on recent techniques for partial differential equations (in short PDE) on the space of probability measures [5]. We first outline the proof strategy for the PDE approach to large deviations in a self-contained manner. As the number of particles goes to infinity, we then show that the transfer function of the system converges to the unique viscosity solution of a limiting Hamilton-Jacobi equation on the space of probability measures. Finally, we explicitly derive the rate function by formulating the solution of this limiting PDE as the value function of an optimal control problem.

We study the equations $\Delta_p^H u=u^{-q}$, $u>0$, and $-\Delta_p^H u=e^u$ in $\mathbb{R}^N$, where $\Delta_p^H u=\operatorname{div}(H(\nabla u)^{p-1}\nabla H(\nabla u))$ is the Finsler $p$-Laplacian associated with a uniformly convex norm $H$, and $p\ge2$. We prove nonexistence of solutions stable outside a compact set for the negative-power equation when $p\le N<\frac{p(p+3)}{p-1}$ and $q$ lies above the sharp critical threshold, and for the exponential equation when $p<N<\frac{p(p+3)}{p-1}$. Consequently, neither equation admits solutions of finite Morse index in these ranges. In the borderline case $N=p$, we prove that the solutions of $-\Delta_N^H u=e^u$ that are stable outside a compact set are precisely the finite-mass solutions classified by Ciraolo and Li. A main ingredient is independent of stability: if $N>q\tau$, where $\tau=\frac{p}{q+p-1}$, a positive exterior solution of $\Delta_p^H u=u^{-q}$ cannot remain above $\theta$ times the singular solution near infinity for any $\theta>1$. An analogous obstruction holds for the exponential equation. Both obstructions are proved by comparison with explicit radial barriers having frozen right-hand sides. We also prove sharpness when $N>p$, and for the negative-power equation when $N=p>2$. Finally, for operators $\operatorname{div}(B'(H(\nabla u))\nabla H(\nabla u))$ with $p$-growth, we establish the integral estimates needed for the stability arguments with fully justified truncation and absorption procedures. These arguments also repair gaps in earlier proofs for the corresponding isotropic and anisotropic problems.

We investigate the conditions under which a Hessian operator satisfies the integral comparison principle. We establish a rather unexpected connection with the notion of a null Lagrangian from the calculus of variations. Under mild conditions we obtain a complete classification, showing that the only operators for which the integral comparison principle holds are linear combinations of $k$-Hessians and, in the homogeneous case, precisely the $k$-Hessians themselves. An analogous result is established for the complex Hessian operator, revealing in particular substantial obstructions to the development of a genuine pluripotential theory for the $\mathcal{J}$-equation.