Papers

We offer an alternative approach to the asymptotic rigidity of codimension-1 isometric immersions via quantitative rigidity estimates. We show that an immersion between compact manifolds $M$ and $N$ of dimensions $d$ and $d + 1$, respectively, with small stretching plus bending energy is close to an isometric immersion. In this way, we recover the results of Alpern, Kupferman, and Maor. In contrast to their intrinsic approach, we reduce the problem to the equidimensional Euclidean setting and apply the Friesecke-James-Müller rigidity estimate to obtain quantitative results. This yields an elementary proof based on Euclidean techniques. The rigidity estimates are of independent interest.

In this paper we consider the $\beta$-plane equation with a smooth external force which is a quasi-periodic traveling wave of large amplitude $O(\lambda^{\alpha - 1})$, $1 < \alpha < 2$, and with large speed of propagation of size $O(\lambda)$. In a previous paper, the second and the third author proved the existence of quasi-periodic traveling wave solutions of large amplitude of order $O(\lambda^{\theta})$, for some $\theta > 0$. The purpose of this paper is to analyze the long time dynamics for smooth initial data close to these traveling wave solutions. In particular, we shall prove that, for initial data sufficiently close to a fixed traveling wave solution (in the $H^s$ topology), the corresponding solution remains close to the traveling wave solution for arbitrary long time (independent of the size of the traveling wave solution). As a consequence, we prove that there are open sets of large initial for which one has almost global existence, namely such that the corresponding solution remains of the same size of the initial datum for arbitrary long time (independent of the size of the initial data). The proof combines several ingredients: an analysis of the linearized PDE at any traveling wave solution via normal form methods, a sharp analysis of the transformed nonlinear problem under the change of coordinates that diagonalizes the linearized equation and energy estimates.

We study real source-free Maxwell fields on slowly rotating, weakly charged Kerr-Newman exteriors and set up a finite-energy scattering theory after removal of the stationary Coulomb sector. The conserved electric and magnetic fluxes account exactly for the two-dimensional stationary non-decaying part, giving a natural decomposition of the Maxwell Cauchy space into stationary and charge-free radiative parts. For the radiative field, the paper develops a finite-order transfer mechanism from regular spin-one curvature variables back to the Maxwell tensor field, combining red-shift control, far-field hierarchy, trapped-set analysis, a Fredholm argument ruling out real-frequency modes, and same-order reconstruction of the middle components. Under the stated slow-weak master estimates, this gives uniform boundedness, integrated local energy decay, radiation fields, wave operators, and asymptotic completeness for the stationary-subtracted Maxwell evolution, with the Kerr case recovered as a special subcase and the charged rotating case reduced to explicit geometric and analytic estimates.

We show the existence of self-similar solutions with constant finite mass to the time-fractional Porous-Medium Equation for all spatial dimensions $d \ge 1$ and all exponents $m>m_c=(d-2)_+/d$. This range is optimal. We find two types of solution depending on the exponent: compactly supported solutions in the slow-diffusion range $m > 1$ and positive solutions with heavy tails in the sub-critical fast-diffusion range $m_c < m < 1$. The self-similar solutions in the linear case $m=1$ were already known explicitly obtained by the Fourier transform, and we discuss their properties in our settings and the limit $m \to 1$.

This paper is devoted to the investigation of the backward problem for a multi-term time-fractional diffusion equation. Backward problems for fractional diffusion equations are typically studied using regularization methods due to their ill-posedness in the sense of Hadamard; that is, a small change in u(T) may lead to large changes in the initial data. Nevertheless, we show that if sufficiently smooth current data are considered, then the solution exists, is unique, and is stable. A principal difficulty in the analysis of the backward problem stems from the structure of the solution, in which the multinomial Mittag-Leffler function appears in the denominator. Accordingly, a precise characterization of the asymptotic behavior of this function is required. Such asymptotic properties are nontrivial and have been rigorously established in the authors' recent work, which serves as a fundamental basis for the present study. In addition, we investigate the conditional stability of the backward problem. It is shown that, although the problem is ill-posed in general, stability can be restored under an appropriate a priori bound imposed on the initial data. The main novelty of the paper lies in proving the best smoothing property of the solution, showing that it belongs to the domain of the operator A for any positive time.

The present article studies solutions to the compressible Navier-Stokes equations for ideal gases in one dimension when thermal conductivity is present but very weak, while viscosity is positive and constant. The main novelty is the establishment of bounds that do not explode when the conductivity coefficient approaches zero. The conductivity coefficient is assumed to be constant and the framework is that of ”à la Hoff” solutions. More precisely, the velocity is initially assumed to be regular, while the density and temperature are only in L^infini and far from zero. A new proof of a stability result for cases without conductivity is given. Then, the proof of the zero-conductivity limit to the Navier-Stokes system without conduction is established in the ”à la Hoff” framework.

In this paper, we obtain some important inequalities for a class of Hessian quotient type operators $\frac{\sigma_k(\Lambda(D^2u))}{\sigma_l(\Lambda(D^2u))}$, which can be regarded as a generalization of the classical Hessian quotient operators. As an application, we establish global a priori estimates and prove an existence theorem for the Neumann problem of the corresponding degenerate Hessian quotient type equation, in which the admissible range of $k$ is extended to $0< k \leq C^\mathbf{p}_n$ with $1 \leq \mathbf{p} \leq n-1$.

This paper investigates the boundedness of bilinear pseudo-differential operators with symbols in the Hörmander class $BS_{\varrho,\delta}^m(\mathbb{R}^n)$ in the previously unexplored regime $0 \leq \varrho < \delta < 1$. We establish boundedness from $H^p(\mathbb{R}^n) \times H^q(\mathbb{R}^n)$ to $L^r(\mathbb{R}^n)$ (with $L^r$ replaced by $\mathrm{BMO}$ when $p=q=r=\infty$) under the probably optimal condition on the order $$m \leq m_\varrho(p,q) - \frac{n\max\{\delta-\varrho,0\}}{\max\{r,2\}},$$ where $m_\varrho(p,q)$ is the critical order in the case $0\leq\delta\leq\varrho<1.$ Furthermore, we develop refined pointwise estimates via sharp maximal functions, establishing that for $m \leq -n(1-\varrho)(\frac{1}{\min\{r_1,2\}}+ \frac{1}{\min\{r_2,2\}})$ with $1<r_{1},r_{2}<\infty$, the bilinear operators satisfy $$M^\sharp T_a(f_1,f_2)(x) \lesssim \mathcal{M}_{\vec{r}}(f_1,f_2)(x).$$ This extends the parameter range from the restrictive condition $0 \leq \delta \leq \varrho < 1$ to the general setting $0 \leq \varrho \leq 1$, $0 \leq \delta < 1$ with $\delta > \varrho$ permitted, and generalizes previous results of Park and Tomita to distinct exponent pairs. Consequently, we obtain weighted norm inequalities for bilinear pseudo-differential operators under multilinear $A_{\vec{p},(\vec{r},\infty)}$ weights.

Whether the 3D compressible Navier-Stokes-Poisson equations admit global classical solutions for general large initial data has long been a challenging open problem. In this paper, we provide an affirmative answer to this question under spherical symmetry on solid balls . Specifically, we consider the initial-boundary value problem for the multi-dimensional compressible equations with density-dependent viscosity coefficients satisfying the BD-type entropy equality, namely, assuming $\mu=\rho^{\alpha},\ \lambda=(\alpha-1)\rho^{\alpha}$ with $N=2, \alpha\in (\frac{1}{2},1]$ and $N=3, \alpha\in (\frac{5}{6},1]$, we establish the global existence of spherically symmetric classical solutions to the compressible Navier-Stokes-Poisson equations for both gaseous stars and plasmas with arbitrarily large initial data on solid balls. Our key observation lies in successfully handling the singularity at the center of the ball. By controlling the growth orders of the density and the gravitational potential at the central singularity, leveraging the structural advantages of the BD entropy and spherical symmetry, and fully exploiting the coupling between the effective velocity and the velocity, we establish $L^\infty$ estimates for the key quantities, which in turn yield upper and lower bound estimates for the density. This can be regarded as the first result on the existence of global classical solutions for arbitrarily large initial data to the compressible Navier-Stokes-Poisson equations in a truly multi-dimensional domain with high-dimensional features.

We study the Wasserstein projection of a compactly supported probability measure onto the class of measures whose density ratio is bounded, and we place this projection in a broader program connecting generative modeling, optimal transport, and nonlinear diffusion. The paper proves existence and uniqueness of the sampleability projection, uniqueness of the Brenier map at the minimizer, path independence of the quadratic Wasserstein generation loss, and the diffusion-threshold picture for the heat semigroup. The porous medium equation is then analyzed as a candidate forward regularizer. We prove the two rigorous properties that make the equation attractive for this purpose, namely finite propagation of compact support and an explicit Wasserstein cost bound obtained from dissipation of the Rényi entropy. We then identify a structural obstruction inherent to any porous-medium version of the sampleability theory. Every nontrivial compactly supported whole-space porous-medium profile has vanishing essential infimum on any compact set containing its support, hence infinite density ratio in the original sense, and the assertion that the porous medium flow reaches the same density-ratio sampleable class while preserving compact support is false. To isolate the mathematically valid content of the nonlinear-diffusion program, we also prove an endpoint-constrained Benamou-Brenier principle for the sampleability projection and derive the corrected spectral picture near a strictly positive equilibrium on a fixed compact domain. In that regime the leading-order damping is exponential, with quadratic mode coupling in the first nonlinear correction. The Hele-Shaw and mesa-limit interpretation is therefore presented here as a conjectural variational extension rather than as a proved theorem.

We investigate the well-posedness of scalar conservation laws whose flux depends on the solution both pointwise and nonlocally through integral averages. Our analysis is based on a fixed-point formulation, in which the nonlocal dependence is incorporated as a space- and time-dependent component of the flux, together with classical stability estimates for entropy solutions. This framework unifies and extends several models previously considered in the literature and applies, in particular, to conservation laws with memory effects (nonlocality in time) or delay. We prove the existence and uniqueness of weak entropy solutions on a sufficiently short time horizon and show that under additional assumptions, existence and uniqueness can be obtained on any finite time horizon. In addition, we present numerical simulations to illustrate the qualitative effects of memory on the solution dynamics.

We derive and analyze a relativistic quantum hydrodynamic (RQHD) system on the Heisenberg group. Starting from the Klein–Gordon–Poisson system, we apply the Madelung transformation to obtain a fluid-type model in which the relativistic and quantum parameters are explicitly separated. The Heisenberg-group structure gives rise to an additional geometric term in the momentum equation, reflecting the underlying noncommutative structure. A central analytical difficulty is the possible appearance of vacuum, where the phase function and the quantum potential become singular. To address this issue, we reformulate the RQHD system as an extended hyperbolic–elliptic system with auxiliary variables. For this extended system, we establish uniform higher-order energy estimates on $\mathbb H^1$ by combining the Banach algebra property of sub-elliptic Sobolev spaces with noncommutative Fourier analysis. We then prove that the extended system is equivalent to the original RQHD system at the level of classical solutions. As a consequence, we obtain the local-in-time existence and uniqueness of non-vacuum classical solutions to the RQHD system on $\mathbb H^1$. The result also provides a framework for the study of related singular limits, including the semiclassical and non-relativistic limits on nilpotent Lie groups.

In this article, we mathematically justify (globally in time) a Baer-Nunziato type system from the non-isentropic compressible Navier-Sokes equations for heat conducting ideal gases posed over the torus and in one space dimension. The breakthrough in this paper is to define and prove the global existence of solutions in a framework intermediate between weak and strong solutions and then to derive the system through homogenization and Young measures characterization. Note that the main difficulty is to derive a priori uniform bounds on appropriate unknowns in the presence of piecewise constant coefficients (viscosity and adiabatic constants) exhibiting rapid oscillations between two positive values.

Given a closed Riemannian Spin manifold $(M,g)$ of dimension greater or equal than four, we consider a generalized conformally invariant equation involving the Dirac operator with a non-linearity of convolution type. We show that the Aubib-type inequality corresponding to the problem is always strict, unless $(M,g)$ is conformal to the round sphere. In particular, this result provides an existence result for a ground state to the conformal Dirac-Einstein problem in dimension four. We point out that aside from some perturbative or special cases, this presents the first general existence result for the conformal Dirac-Einstein equations in dimension four.

In this paper we consider a quasilinear singular anisotropic elliptic problem with a p-sublinear perturbation. We prove uniqueness of weak solutions and we provide necessary and sufficient conditions for the existence of weak solutions in the same spirit of the celebrated paper by Lazer and McKenna.

In this paper, we present a general framework for constructively proving the existence of stationary localized solutions, spatially periodic solutions, and branches of spatially periodic solutions in the 1D Thomas model. Specifically, we develop the necessary analysis to compute explicit upper bounds required in a Newton–Kantorovich approach. Given an approximate solution $\bar{\mathbf{u}}$, this approach relies on establishing that a well-chosen fixed point map is contracting on a neighborhood $\bar{\mathbf{u}}$. For this matter, we construct an approximate inverse of the linearization around $\bar{\mathbf{u}}$, and establish sufficient conditions under which the contraction is achieved. This provides a framework for which computer-assisted analysis can be applied to verify the existence and local uniqueness of solutions in a vicinity of $\bar{\mathbf{u}}$, and control the linearization around $\bar{\mathbf{u}}$. Furthermore, as the Thomas model has a non-polynomial nonlinearity, we will need to use different techniques to handle it during our analysis. Our contributions are to provide a partial answer to how one can approach rigorously verifying results in the Thomas model, to adapt and combine previously developed techniques to apply to the Thomas model, and to perform the computer-assisted analysis to obtain such results. The code to perform the rigorous proofs is available on Github.

We construct a family of solutions $(u,B)$ of the incompressible magnetohydrodynamic (MHD) system, the $L^\infty$ norm of which blows up instantaneously at the critical rate. The solutions remain smooth except at the blowup time. An inverse energy cascade mechanism and a convex integration scheme along a time sequence are the main ingredients of the construction, inspired by our recent work [CDP25] for the Navier-Stokes equations. The challenge of the construction for the MHD system stems from the coupling and the necessity of preserving the same ansatz of the principal solution at every iterative step while implementing convex integration. Existing convex integration schemes for MHD can treat the coupling but fail to produce the same ansatz of the principal solution recursively. To achieve the goal, we introduce a coupled geometric lemma that decomposes a symmetric tensor and a skew-symmetric tensor simultaneously. We emphasize that such coupled geometric lemma is new and of independent interest.

In this paper, we study the long-time behavior of a stochastic heat equation with a class of infinite-dimensional multiplicative noises and localized control. We first illustrate through simple examples of the uncontrolled dynamics that mean-square and almost sure exponential stability may hold under different conditions on the parameters, reflecting the interplay between the drift and the multiplicative noise. We then construct a finite-dimensional feedback control acting on a measurable subset of positive measure, built from finitely many Fourier modes of the solution. In particular, we show that the number of controlled modes determines the decay rate and allows for arbitrarily fast stabilization in the mean-square sense. As a consequence, almost sure exponential stability is recovered via a probabilistic argument, so that both notions of stability are achieved within the same framework and with the same guaranteed decay rate. We also establish null-controllability at any positive time through an iterative construction of adapted controls from finite-dimensional feedback laws. The construction relies on the quantitative stabilization estimates and requires no observability inequality for the adjoint equation.

We study the large-scale behavior of solutions to the Allen-Cahn reaction-diffusion equation with Gaussian initial data. We consider the case of short-range dependence in the associated supercritical regime with spatial dimension $d \ge 3$. Under diffusive rescaling, the non-linearity formally vanishes on large scales in this case. Accordingly, we prove a central limit theorem for the rescaled solution, more precisely, that it converges to the solution of the heat equation started from a white noise. These initial conditions for the limit depend non-trivially both on the source of randomness and on the non-linearity. Our proof uses estimates obtained by a combination of comparison principles and Malliavin calculus, initiated by Castillo and Dunlap in arXiv:2509.06260 in the critical case. However, the result there is not a fluctuation result but rather an $L^2_\mathbf{P}$ comparison to a McKean-Vlasov problem with Gaussian solutions. Hence the mechanism behind the Gaussianity of the limit differs, and the proof requires new ideas that should be further applicable to other supercritical problems.

The inverse energy cascade in turbulent Taylor-Couette flow is studied in line with the results of the large eddy simulation. The simulation results show that the inverse energy cascade first occurs within the core region of the flow channel of the Taylor-Couette flow at higher Reynolds number. It is uncovered that this phenomenon is induced by the pulsed zero shear stress resulting from the singularities of the Navier-Stokes equation. In the core area between the two cylinders, the shear stress is nearly zero at higher Reynolds number. The turbulence generated there has high turbulent energy due to discontinuity of the tangential velocity. Since the energy transfer between the fluid layers is inhibited due to the low shear stress, the turbulent energy cannot be transferred along the radial direction, and small-scale vortices with high turbulent energy are produced. These small-scale vortices are located with the large-scale vortices and cannot be dissipated owing to low shear stress. A peak in the energy spectrum at middle frequency (or wave number) is formed due to the concentration of the small-scale vortices. As the number of the singular points of the Navier-Stokes equation increases with the increasing Reynolds number, the region with zero shear stress expands along the radial direction, intensifying nonlinear instability and energy accumulation. This, in turn, leads to more prominent peaks in the energy spectrum, resulting in a more pronounced inverse energy cascade.

We establish a two-weight fractional Poincaré-Sobolev sandwich, consisting of a two-weight fractional Poincaré-Sobolev inequality and a two-weight embedding from the first-order Sobolev space to a Triebel-Lizorkin space defined via a difference norm. Our constants are asymptotically sharp as the fractional parameter approaches $1$. Our results are new even in the one-weight case. For each inequality we give explicit quantitative dependence on Muckenhoupt weight characteristics and treat both subcritical and critical regimes, the former via elementary methods and the latter via sparse domination. As one of our main tools, we establish a new sparse domination result for Triebel-Lizorkin difference norms. Our methods unify, simplify and significantly extend various earlier approaches.

We consider two-dimensional periodic gravity water waves with constant nonzero vorticity $\gamma$, in infinite depth and with periodic boundary conditions. We prove that, if the characteristic wave number $\frac{\gamma^2}{g}$ is rational, the system admits smooth small-amplitude solutions whose high Sobolev norms grow arbitrarily large while lower-order norms remain arbitrarily small, thereby exhibiting a genuine transfer of energy toward high frequencies. This yields the first rigorous construction of weakly turbulent solutions for a quasilinear hydrodynamic wave system, in a regime where the flow remains smooth. Moreover, the growth occurs simultaneously in the free surface and in the vertical component of the velocity at the interface, showing that the instability involves the full hydrodynamic evolution. The proof relies on a new mechanism for generating energy cascades in quasilinear dispersive PDEs with sublinear dispersion and a nonlinear transport structure. A central ingredient is to exploit quasi-resonances from 2-wave interactions to produce a transport operator that drives energy to high modes and causes Sobolev norm growth. A virial-type argument then shows that the resulting instability affects both the free surface elevation and the velocity field.

The so-called JKO scheme, named after Jordan, Kinderlehrer and Otto, provides a variational way to construct discrete time approximations of certain partial differential equations (PDEs) appearing as gradient flows in the space of probability measures equipped with the Wasserstein metric. The method consists of an implicit Euler scheme, which can be implemented numerically. Yet, in practice, evaluating the Wasserstein distance can be numerically expensive. To address this problem, a common strategy introduced by Peyré in 2015 and which has been shown to produce faster computations, is to replace the Wasserstein distance with its entropic regularization, also known as the Schrödinger cost. In 2026, the first author, Hraivoronska and Santambrogio, proved that if the regularization parameter $\varepsilon$ is proportional to the time step $\tau$, that is, $\varepsilon = \alpha \tau$ for some $\alpha > 0$, then as $\tau \to 0$, this change results in adding to the limiting PDE the additional linear diffusion term $\frac{\alpha}{2} \Delta \rho$. Our goal in this article is to provide a convergence rate under convexity assumptions between the entropic JKO scheme and the solution of the initial PDE as both $\alpha$ and $\tau$ tend to zero. This will appear as a consequence of a new bound between the classical and entropic JKO schemes.

In this paper, we study the dynamics of a finite number of spherical bubbles in a compressible fluid within a bounded open domain of R 3 . The fluid-bubble interaction is described by a system of nonlinear partial differential equations (PDEs) and ordinary differential equations (ODEs) coupling the fluid's density, velocity and pressure to the bubble's translational, rotational and radial velocities. We prove the existence of weak solutions for this model until the collision or collapse of the bubbles. The formulation of the fluid-bubble system, along with the techniques used for the existence proof, is inspired by penalization methods developed for fluid-solid interaction. The main contribution of this work is the addition of a radial expansion-contraction mode in the bubble motion, which introduces new nonlinear terms in the momentum equations that need to be treated carefully in the compactness arguments.

We develop a nonlinear evolution framework for nonlinear parabolic equations with unbounded drift terms formulated in Lorentz spaces. The main contribution lies in the construction of uniformly m-accretive operators based on Lorentz-Sobolev embeddings, which allows us to apply the Crandall-Liggett generation theorem for nonlinear evolution equations. Within this framework, we establish existence, uniqueness, and stability of mild solutions. We further show that these mild solutions coincide with weak solutions, ensuring consistency with the variational formulation. Finally, we investigate the long-time asymptotic behavior of solutions.

Polynomial threshold functions (PTFs) are an important low-complexity class of Boolean functions, with strong connections to learning theory and approximation theory. Recent work on learning and testing PTFs has exploited structural and isoperimetric properties of the class, especially bounds on average sensitivity, one of the central themes in the study of PTFs since the Gotsman–Linial conjecture. In this work we study PTFs through the lens of the Boolean surface area (or Talagrand boundary) \[ \mathbf{BSA}[f]=\mathbb{E}|\nabla f|=\mathbb{E}\sqrt{s_{f}(x)}, \] a natural measure of vertex-boundary complexity on the discrete cube. Our main result is that every degree-$d$ PTF has polylogarithmic Boolean surface area: \[ \mathbf{BSA}[f]\le C_d(\log(en))^{C_d}. \] The proof is based on the PTF Restriction Lemma of Kabanets, Kane, and Lu \cite{KKL2017} and proceeds through a tail bound for the pointwise sensitivity. In particular, it controls all subcritical fractional moments of the sensitivity. We also record a random block partition principle for Boolean surface area and an alternative recursive argument following Kane's work \cite{DK} on average sensitivity, which independently yields the weaker bound \[ \mathbf{BSA}[f]\le \exp(C_d\sqrt{\log n}). \]

In this paper, we study a general class of inhomogeneous kinetic models that unifies fundamental models in both the statistical physics of particles and of waves, namely the kinetic Boltzmann equations and the kinetic wave equations, in both classical (non-relativistic), relativistic and quantum settings. We formulate this unified equation into the GENERIC (General Equation for Non-Equilibrium Reversible-Irreversible Coupling) framework. We then derive the grazing (small-angle) limit in two-body interaction systems, which leads to Landau-type equations. Finally, we show that these limiting systems can also be formulated as GENERIC systems.

We present $\phi-$DeepONet, a physics-informed neural operator designed to learn mappings between function spaces that may contain discontinuities or exhibit non-smooth behavior. Classical neural operators are based on the universal approximation theorem which assumes that both the operator and the functions it acts on are continuous. However, many scientific and engineering problems involve naturally discontinuous input fields as well as strong and weak discontinuities in the output fields caused by material interfaces. In $\phi$-DeepONet, discontinuities in the input are handled using multiple branch networks, while discontinuities in the output are learned through a nonlinear latent embedding of the interface. This embedding is constructed from a one-hot representation of the domain decomposition that is combined with the spatial coordinates in a modified trunk network. The outputs of the branch and trunk networks are then combined through a dot product to produce the final solution, which is trained using a physics- and interface-informed loss function. We evaluate $\phi$-DeepONet on several one- and two-dimensional benchmark problems and demonstrate that it delivers accurate and stable predictions even in the presence of strong interface-driven discontinuities.

We study the Cauchy problem for a generalized time-fractional Kuramoto–Sivashinsky equation on \(\mathbb R\) with the regularized Caputo derivative in the Schwartz space. The linear problem is solved by the Fourier transform and a Mittag–Leffler representation, with preservation of the Schwartz class. For the nonlinear problem, local solvability is established by successive approximations with convergence in the full Schwartz topology. A continuation principle accounting for the complete fractional memory is developed: the history term is estimated uniformly with respect to the continuation point, which yields global solvability on every finite time interval. An \(L^2(\mathbb R)\) stability estimate is also obtained, implying uniqueness.

We construct left, right and bilateral fundamental solutions for generalized steady Stokes' operators $S$ with smooth coefficients coefficients, associated with the de Rham complex of differentials on differential forms over a domain $X$ in ${\mathbb R}^n$. The investigated operators are Douglis-Nirenberg elliptic under reasonable assumptions. As an immediate corollary we produce a homotopy formula for regular solutions to this operator.