Michael Christ, Georgi Karadzhov
A necessary and sufficient condition for local solvability is presented for the linear partial differential operators $-X^2-Y^2+ia(x)[X,Y]$ in $\bold R^3=\{(x,y,t)\}$, where $X=\partial_x,\; Y=\partial_y+x^k\partial_t$, and $a\in C^{\infty}(\bold R^1)$ is real valued, for each positive integer $k$.
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The classical solvability of the initial-boundary problem for the Davey-Stewartson-II type system of equations is proved.
We prove results on solvability of nonlinear elliptic partial differential systems of principle type of second order. They are consequences of existence of non-radial solutions for nonlinear partial differential systems of Poisson type. As applications to geometry, we prove the exsitence of local harmonic maps with given tangent plane at a point between any Riemannan manifolds. More generally geometric objects defined by Beltrami-Laplace always exist locally.
We provide sufficient conditions of local solvability for partial differential operators with variable Colombeau coefficients. We mainly concentrate on operators which admit a right generalized pseudodifferential parametrix and on operators which are a bounded perturbation of a differential operator with constant Colombeau coefficients. The local solutions are intended in the Colombeau algebra $\G(\Om)$ as well as in the dual $\LL(\Gc(\Om),\wt{\C})$.
Systems of parabolic, possibly degenerate parabolic SPDEs are considered. Existence and uniqueness are established in Sobolev spaces. Similar results are obtained for a class of equations generalizing the deterministic first order symmetric hyperbolic systems.
We discuss linear algebra of infinite-dimensional vector spaces in terms of algebraic (Hamel) bases. As an application we prove the surjectivity of a large class of linear partial differential operators with smooth ($\mathcal C^\infty$-coefficients) coefficients, called in the article \emph{regular}, acting on the algebraic dual $\mathcal D^*(Ω)$ of the space of test-functions $\mathcal D(Ω)$. The surjectivity of the partial differential operators guarantees solvability of the corresponding partial differential equations within $\mathcal D^*(Ω)$. We discuss our result in contrast to and compar…
We provide a self-contained proof of the solvability and regularity of a Hodge-type elliptic system, wherein the divergence and curl of a vector field are prescribed in an open, bounded, Sobolev-class domain, and either the normal component or the tangential components of the vector field are prescribed on the boundary. The proof is based on a regularity theory for vector elliptic equations set on Sobolev-class domains and with Sobolev-class coefficients.
We prove locally in time the existence of the unique smooth solution (including smooth interface) to the multidimensional free boundary problem for the thin film equation in the case of partial wetting. We also obtain the Schauder estimates and solvability for the Dirichlet and the Neumann problem for a linear degenerate parabolic equation of fourth order. The final expanded version of this paper is available at AIMS Journals, Discrete and Continuous Dynamical Systems - A at http://aimsciences.org/article/doi/10.3934/dcds.2017156
The paper addresses questions of existence and regularity of solutions to linear partial differential equations whose coefficients are generalized functions or generalized constants in the sense of Colombeau. We introduce various new notions of ellipticity and hypoellipticity, study their interrelation, and give a number of new examples and counterexamples. Using the concept of $\G^\infty$-regularity of generalized functions, we derive a general global regularity result in the case of operators with constant generalized coefficients, a more specialized result for second order operators, and a …
We consider the Cauchy problem for the Hardy parabolic equation $\partial_t u-Δu=|x|^{-γ}u^p$ with initial data $u_0$ singular at some point $z$. Our main results show that, if $z\neq 0$, then the optimal strength of the singularity of $u_0$ at $z$ for the solvability of the equation is the same as that of the Fujita equation $\partial_t u-Δu=u^p$. Moreover, if $z=0$, then the optimal singularity for the Hardy parabolic equation is weaker than that of the Fujita equation. We also obtain analogous results for a fractional case $\partial_t u+(-Δ)^{θ/2} u=|x|^{-γ}u^p$ with $0<θ<2$.
In this paper we will first present some results about the local solvability property of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration (which in turn is a generalization of the Kannai operator) exhibits a degeneracy due to the interplay between the singularity associated with the characteristic set of a system of vector fields and the vanishing of a function. Afterward we will also discuss some local solvability results for two classes of degenerate second order linear partial differential operators with non-smo…
Let $(u,v)$ be a nonnegative solution to the semilinear parabolic system \[ \mbox{(P)} \qquad \cases{ \partial_t u=D_1Δu+v^p, & $x\in{\bf R}^N,\,\,\,t>0,$\\ \partial_t v=D_2Δv+u^q, & $x\in{\bf R}^N,\,\,\,t>0,$\\ (u(\cdot,0),v(\cdot,0))=(μ,ν), & $x\in{\bf R}^N,$ } \] where $D_1$, $D_2>0$, $0<p\le q$ with $pq>1$ and $(μ,ν)$ is a pair of nonnegative Radon measures or nonnegative measurable functions in ${\bf R}^N$. In this paper we study sufficient conditions on the initial data for the solvability of problem~(P) and clarify optimal singularities of the initial functions for the solvability.
Equivalence between solvability of the Dirichlet and Regularity problem under an $L^1$ Carleson condition on $\partial_t A$
本研究针对上半空间椭圆算子，在$|\u2202_t A|$满足$L^1$-Carleson条件下证明了Dirichlet问题与正则性问题可解性的等价性，扩展了$L^p$正则性问题可解的算子类。
The Dirichlet boundary value problem for the Stokes operator with $L^p$ data in any dimension on domains with conical singularity (not necessary a Lipschitz graph) is considered. We establish the solvability of the problem for all $p\in (2-\varepsilon,\infty]$ and also its solvability in $C(\overline{D})$ for the data in $C(\partial D)$
The non-local problem is considered for the partial differential equation of mixed-type with Bessel operator and fractional order. An explicit solution is represented by Fourier-Bessel series in the given domain. It is established the connection between the given data and the unique solvability of the problem.
On the existence of stationary solutions for certain systems of integro-differential equations with the double scale anomalous diffusion
本文研究双尺度反常扩散积分微分方程组平稳解的存在性，通过不动点技术并利用无界域非Fredholm椭圆算子可解性条件证明解的存在。
In this article we investigate the solvability of infinite-dimensional differential algebraic equations. Such equations often arise as partial differential-algebraic equations (PDAEs). A decomposition of the state-space that leads to an extension of the Hille-Yosida Theorem on Hilbert spaces for these equations is described. For dissipative partial differential equations the famous Lumer-Phillips generation theorem characterizes solvability and also boundedness of the associated semigroup. An extension of the Lumer-Phillips generation theorem to dissipative differential-algebraic equations is …
This is a the first in a series of two articles devoted to the question of local solvability of doubly characteristic second order differential operators. For a large class of such operators, we show that local solvability at a given point implies "essential dissipativity" of the operator at this point. By means of Hoermander's classical necessary condition for local solvability, the proof is reduced to the following question, whose answer forms the core of the paper: Suppose that $Q_A$ and $Q_B$ are two real quadratic forms on a finite dimensional symplectic vector space, and let $Q_C:=\{Q_A,…
Global Hypoellipticity and Solvability with Loss of Derivatives on the Torus
本文完整刻画了环面上傅里叶乘子算子带导数损失的全局亚椭圆性与可解性，并揭示其与数论的联系：向量场可解性的导数损失由无理测度决定，波动算子的损失显式依赖于参数。
We prove a general existence theorem for nonlinear partial differential systems of any order in one complex variable. A special case of first order contains a well-known theorem of Nijenhuis and Woolf concerning local existence of J-holomorphic curves on almost complex manifolds. As an application to differential geometry, we prove that non constant harmonic map always exists locally from a Riemann surface to a Riemannian manifold with a prescribed tangent vector at a given point.
This is a the first in a series of two articles devoted to the question of local solvability of doubly characteristic differential operators $L,$ defined, say, in an open set $\Om\subset \RR^n.$ Suppose the principal symbol $p_k$ of $L$ vanishes to second order at $(x_0,ξ_0)\in T^*\Om\setminus 0,$ and denote by $Q_\H$ the Hessian form associated to $p_k$ on $T_{(x_0,ξ_0)}T^*\Om.$ As the main result of this paper, we show (under some rank conditions and some mild additional conditions) that a necessary condition for local solvability of $L$ at $x_0$ is the existence of some $θ\in\RR$ such that …
Local solvability and non-solvability are classified for left-invariant differential operators on the Heisenberg group H_1 of the form L=P_n(X,Y)+Q(X,Y) where the P_n are certain homogeneous polynomials of order n greater than or equal to 2 and Q is of lower order with X= \partial_x, Y=\partial_y+x\partial_w on R^3. We extend previous studies of operators of the form P_n(X,Y) via representations involving ordinary differential operators with a parameter.
We will show a local solvability result for a class of degenerate second order linear partial differential operators with a complex subprincipal symbol. Due to the form of the operators in the class the subprincipal symbol is invariantly defined and we shall give sufficient conditions for the local solvability to hold involving the real and the imaginary part of the latter. Under suitable conditions we will prove that the class under consideration is L^2 to L^2 locally solvable.
On the local solvability and stability of the partial inverse problems for the non-self-adjoint Sturm-Liouville operators with a discontinuity
本文研究带复系数和间断点的非自伴Sturm-Liouville算子部分逆问题，在已知部分势函数和参数的前提下从特征值恢复剩余信息，得到其局部可解性、稳定性及两个新唯一性定理。
We study a time-fractional semilinear heat equation $$\partial^α_t u -Δu = u^{p},\ \ \mbox{in}\ (0,T)\times\mathbb{R}^N,\ \ u(0)=u_0\ge0$$ with $u_0\in L^{1}(\mathbb{R}^N)$ and $p=1+2/N$. Here $\partial_t^α$ denotes the Caputo derivative of order $α\in (0,1)$. Since the space $L^1(\mathbb{R}^N)$ is scale critical with $p=1+2/N$, this type of equation is known as a doubly critical problem. It is known that the usual doubly critical equation $\partial_t u-Δu=u^p$ does not have nonnegative global-in-time solutions, while the time-fractional problem does. Moreover, there exists a singular initial …
We consider parabolic operators of the form $$\partial_t+\mathcal{L},\ \mathcal{L}=-\mbox{div}\, A(X,t)\nabla,$$ in $\mathbb R_+^{n+2}:=\{(X,t)=(x,x_{n+1},t)\in \mathbb R^{n}\times \mathbb R\times \mathbb R:\ x_{n+1}>0\}$, $n\geq 1$. We assume that $A$ is a $(n+1)\times (n+1)$-dimensional matrix which is bounded, measurable, uniformly elliptic and complex, and we assume, in addition, that the entries of A are independent of the spatial coordinate $x_{n+1}$ as well as of the time coordinate $t$. For such operators we prove that the boundedness and invertibility of the corresponding layer potent…
This paper provides necessary as well as sufficient conditions on the Hurst parameters so that the continuous time parabolic Anderson model $\frac{\partial u}{\partial t}=\frac{1}{2}\frac{\partial^2 u}{\partial x^2}+u\dot{W}$ on $[0, \infty)\times {\bf R}^d $ with $d\geq 1$ has a unique random field solution, where $W(t, x)$ is a fractional Brownian sheet on $[0, \infty)\times {\bf R}^d$ and formally $\dot W =\frac{\partial^{d+1}}{\partial t \partial x_1 \cdots \partial x_d} W(t, x)$. When the noise $W(t, x)$ is white in time, our condition is both necessary and sufficient when the initial dat…
Solvability of Vekua-type periodic operators and applications to classical equations
该研究刻画了常系数下Vekua型周期算子可解性与全局亚椭圆性的充要条件，并将其应用于经典波动、热与Laplace方程对应的Vekua型算子的全面表征。
We show that for a uniformly elliptic divergence form operator $L$, defined in an open set $Ω$ with Ahlfors-David regular boundary, BMO-solvability implies scale invariant quantitative absolute continuity (the weak-$A_\infty$ property) of elliptic-harmonic measure with respect to surface measure on $\partial Ω$. We do not impose any connectivity hypothesis, qualitative or quantitative; in particular, we do not assume the Harnack Chain condition, even within individual connected components of $Ω$. In this generality, our results are new even for the Laplacian. Moreover, we obtain a converse, un…
Local Solvability of Quasilinear Pseudodifferential Operators of Real Principal Type
该研究证明了具实主型齐次主象征的拟线性伪微分算子局部可解，将二阶拟线性偏微分算子结果推广至更一般情形，并以微局部化到一阶模型算子完成证明。
We study the Cauchy problem for the quasilinear wave equation $ \partial^2 _t u = u^{2a} \partial^2_x u + F(u) u_x $ with $a \geq 0$ and show a result for the local in time existence under new conditions. In the previous results, it is assumed that $u(0,x) \geq c_0>0$ for some constant $c_0$ to prove the existence and the uniqueness. This assumption ensures that the equation does not degenerate. In this paper, we allow the equation to degenerate at spacial infinity. Namely we consider the local well-posedness under the assumption that $u(0,x)>0$ and $u(0,x) \rightarrow 0$ as $|x| \rightarrow \…
Let $(u,v)$ be a solution to a semilinear parabolic system \[ \mbox{(P)} \qquad \begin{cases} \partial_t u=D_1Δu+v^p\quad & \quad\mbox{in}\quad{\bf R}^N\times(0,T),\\ \partial_t v=D_2Δv+u^q\quad & \quad\mbox{in}\quad{\bf R}^N\times(0,T),\\ u,v\ge 0 & \quad\mbox{in}\quad{\bf R}^N\times(0,T),\\ (u(\cdot,0),v(\cdot,0))=(μ,ν) & \quad\mbox{in}\quad{\bf R}^N, \end{cases} \] where $N\ge 1$, $T>0$, $D_1>0$, $D_2>0$, $0<p\le q$ with $pq>1$ and $(μ,ν)$ is a pair of Radon measures or nonnegative measurable functions in ${\bf R}^N$. In this paper we study qualitative properties of the initial trace of the…
Solvability of the Dirichlet problem using a weaker Carleson condition in the upper half plane
该研究针对上半平面的椭圆算子Dirichlet问题，提出一种关于矩阵横向导数的$L^1$ Carleson条件，拓展了满足$A_\u221e$权性质、即$L^p$ Dirichlet问题可解的椭圆算子范围。
Extrapolation of solvability of the regularity and the Poisson regularity problems in rough domains
本文研究无连通性假设的 Ahlfors 正则粗糙边界区域，建立椭圆正则性问题在 Hajłasz-Sobolev 空间 $M^{1,1}$ 中的可解性向 $M^{1,p}$ 的外推，并推广至帐篷空间上的 Poisson 正则性问题及 Hardy-Sobolev 可解性。
Sufficient Conditions for Solvability of Operators of Subprincipal Type
本文研究了实亚主型线性伪微分算子局部可解性的充分条件，并探讨了实主型二次偏微分算子的局部可解性。
Periodic Solutions to Dissipative Hyperbolic Systems. I: Fredholm Solvability of Linear Problems
本文研究了双曲系统中周期解的可解性，提出了两个非共振条件以保证线性问题的Fredholm可解性和$C^1$-解的正则性，并展示了这些条件的鲁棒性。
Local solvability is analyzed for natural families of partial differential operators having double characteristics. In some families the set of all operators that are not locally solvable is shown to have both infinite dimension and infinite codimension.
Solvability of the Poisson-Dirichlet problem in domains with boundaries of mixed dimension
该研究将齐次Dirichlet问题的L^p可解性刻画推广到含混合维边界的区域中一类退化椭圆算子，拓展了可解性理论的适用范围。
Global Hölder solvability of second order elliptic equations with locally integrable lower-order coefficients
本文证明了一类具有低阶项、系数受边界距离负幂控制的椭圆方程全局Hölder连续解的存在性，改进了经典定理，并采用Ancona策略结合新Morrey型空间进行证明。
Existence of stationary solutions for some integro-differential equations with the double scale anomalous diffusion
本文针对R³中含不同分数阶负拉普拉斯算子之和的双尺度反常扩散积分-微分方程，借助不动点技术及无界域无Fredholm性质椭圆算子的可解性条件，证明了其平稳解的存在性。
In this work we study three exterior extension problems for strongly elliptic partial equations: the Cauchy problem (in a special statement), the "analytical" continuation problem and the so called "inner" Dirichlet problem in the scale of the Sobolev spaces over a domain with relatively smooth boundaries. We consider the existence of solutions to these problems, the dense solvability and conditional well-posedness of these problems for a wide class of strongly elliptic systems. We also consider the approximation of solutions to these problems by a single layer potential and by a linear combin…
From zero solutions with partially bounded supports to solvability and Runge approximation for partial differential equations
研究齐次偏微分方程具规定支撑性质非平凡解的存在性，为常系数偏微分算子建立有部分有界支撑的光滑零解存在性。通过多种方法获偏微分方程可解性与逼近相关结果，包括\(P\)-凸性特征、龙格逼近等，还开发龙格逼近框架并应用于相关系统。
The aim of this paper is to prove that the well known non solvable Mizohata type partial differential equations have Colombeau generalized solutions which are distributions if and only if they are solv- able in the space of Schwartz distributions. Therefore the Colombeau generalized solvability includes both a new solution concept and new mathematical objects as solutions.
On solvability of a time-fractional semilinear heat equation, and its quantitative approach to the classical counterpart
本文研究了时间分数阶半线性热方程在Fujita临界指数下的可解性，揭示了当分数阶导数阶数趋于1时，其全局与局部时间可解性发生坍塌的定量现象。
Let $Ω$ be a Lipschitz domain in $\mathbb R^n,n\geq 3,$ and $L=\divt A\nabla$ be a second order elliptic operator in divergence form. We will establish that the solvability of the Dirichlet regularity problem for boundary data in Hardy-Sobolev space $\HS$ is equivalent to the solvability of the Dirichlet regularity problem for boundary data in $H^{1,p}$ for some $1<p<\infty$. This is a "dual result" to a theorem in \cite{DKP09}, where it has been shown that the solvability of the Dirichlet problem with boundary data in $\text{BMO}$ is equivalent to the solvability for boundary data in $L^p(\pa…
We study the solvability of the second boundary value problem for a class of highly singular fourth order equations of Monge-Ampère type. They arise in the approximation of convex functionals subject to a convexity constraint using Abreu type equations. Both the Legendre transform and partial Legendre transform are used in our analysis. In two dimensions, we establish global solutions to the second boundary value problem for highly singular Abreu equations where the right hand sides are of $q$-Laplacian type for all $q>1$. We show that minimizers of variational problems with a convexity constr…
It is well known that the Serrin condition is a necessary condition for the solvability of the Dirichlet problem for the prescribed mean curvature equation in bounded domains of $\mathbb{R}^n$ with certain regularity. In this paper we investigate the sharpness of the Serrin condition for the vertical mean curvature equation in the product $ M^n \times \mathbb{R} $. Precisely, given a $\mathscr{C}^2$ bounded domain $Ω$ in $M$ and a function $ H = H (x, z) $ continuous in $\overlineΩ\times\mathbb{R}$ and non-decreasing in the variable $z$, we prove that the strong Serrin condition $(n-1)\mathcal…
We consider the following chemotaxis model %fully parabolic Keller-Segel system with logistic source $$ \left\{\begin{array}{ll} u_t=\nabla\cdot(D(u)\nabla u)-χ\nabla\cdot(u\nabla v)+μ(u-u^2),\quad x\in Ω, t>0, \disp{v_t-Δv=-uv },\quad x\in Ω, t>0, %\disp{τw_t+δw=u },\quad %x\in Ω, t>0, \disp{(\nabla D(u)-χu\cdot \nabla v)\cdot ν=\frac{\partial v}{\partialν}=0},\quad x\in \partialΩ, t>0, \disp{u(x,0)=u_0(x)},\quad v(x,0)=v_0(x),~~ x\in Ω\end{array}\right. $$ on a bounded domain $Ω\subset\mathbb{R}^N(N\geq1)$, with smooth boundary $\partialΩ, χ$ and $μ$ are positive constants. Besides appropria…
Solvability of the Dirichlet problem for a new class of elliptic operators
针对上半空间椭圆算子Dirichlet问题，通过引入仅依赖横向导数的混合L¹-L^∞型Carleson条件，改进原有t独立性条件，证明该条件下ω∈A∞(σ)，上半平面情形下进一步得到更强的L¹型Carleson条件结果。
In this work, we investigate a unique solvability of a direct and inverse source problem for a time-fractional partial differential equation with the Caputo and Bessel operators. Using spectral expansion method, we give explicit forms of solutions to formulated problems in terms of multinomial Mittag-Leffler and first kind Bessel functions.
The Dirichlet problem for double divergence form elliptic equations with measures as boundary conditions
研究低正则系数双散度型椭圆方程以一般Borel测度为边界条件的Dirichlet问题，证明可解性并应用于平稳Fokker-Planck-Kolmogorov方程解的性质。
In this paper, we consider a doubly nonlinear parabolic equation $ \partial _t β(u) - \nabla \cdot α(x , \nabla u) \ni f$ with the homogeneous Dirichlet boundary condition in a bounded domain, where $β: \mathbb{R} \to 2 ^{ \mathbb{R} }$ is a maximal monotone graph satisfying $0 \in β(0)$ and $ \nabla \cdot α(x , \nabla u )$ stands for a generalized $p$-Laplacian. Existence of solution to the initial boundary value problem of this equation has been investigated in an enormous number of papers for the case where single-valuedness, coerciveness, or some growth condition is imposed on $β$. However…
This monograph presents a comprehensive treatment of second order divergence form elliptic operators with bounded measurable t-independent coefficients in spaces of fractional smoothness, in Besov and weighted Sobolev classes. We establish mapping properties for the double and single layer potentials, as well as the Newton potential, on Besov spaces. We prove extrapolation-type solvability results: that is, we show that solvability of the Dirichlet or Neumann boundary value problem at any given L^p space automatically assures their solvability in an extended range of Besov spaces. We also esta…
Stochastic wave equation with additive fractional noise: solvability and global Hölder continuity
本文研究了带有加性分数噪声的随机波方程的可解性与全局Hölder连续性，确定了Hurst参数的可解范围，并在特定参数下获得了解的增长率和连续性结果。
Let $Ω\subset \mathbb{R}^{n+1}$ be an open set, not necessarily connected, with an $n$-dimensional uniformly rectifiable boundary. We show that $\partialΩ$ may be approximated in a "Big Pieces" sense by boundaries of chord-arc subdomains of $Ω$, and hence that harmonic measure for $Ω$ is weak-$A_\infty$ with respect to surface measure on $\partialΩ$, provided that $Ω$ satisfies a certain weak version of a local John condition. Under the further assumption that $Ω$ satisfies an interior Corkscrew condition, and combined with our previous work, and with recent work of Azzam, Mourgoglou and Tolsa…
Solvability of a Mixed Problem for a Time-Fractional PDE with Time-Space Degenerating Coefficients
本文研究了具有时间-空间退化系数的分数偏微分方程混合边界值问题的唯一解存在性，通过引入新算子和分离变量法，证明了谱问题的特征值和特征函数存在性，并揭示了退化对分数扩散过程的影响。
In this note we investigate the partial Fourier series on a product of two compact Lie groups. We give necessary and sufficient conditions for a sequence of partial Fourier coefficients to define a smooth function or a distribution. As applications, we will study conditions for the global solvability of an evolution equation defined on $\mathbb{T}^1\times\mathbb{S}^3$ and we will show that some properties of this evolution equation can be obtained from a constant coefficient equation.
Global solvability and hypoellipticity for evolution operators on tori and spheres
该文研究$\bT^{r+1}\times(\bS^3)^s$上带零阶扰动的类演化微分算子的全局性质，借助分量式傅里叶分析，得到与丢番图不等式、系数水平集连通性等相关的充要条件，可复现既有结果并给出新结论。
We are concerned with the solvability of linear second order elliptic partial differential equations with nonlinear boundary conditions at resonance, in which the nonlinear boundary conditions perturbation is not necessarily required to satisfy Landesman-Lazer conditions or the monotonicity assumption. The nonlinearity may be unbounded. The nonlinearity interact, in some sense with the Steklov spectrum on boundary nonlinearity. The proofs are based on a priori estimates for possible solutions to a homotopy on suitable trace and topological degree arguments.
Global solvability and cohomology of tube structures on compact manifolds
本文利用部分 Fourier 级数研究紧流形与环面乘积上的管结构，刻画复管结构的整体可解性及中间上同调有限性，并推广实管结构的上同调同构与闭值域条件。
Consider $A(x,D):C^{\infty}(Ω,E) \rightarrow C^\infty(Ω,F)$ an elliptic and canceling linear differential operator of order $ν$ with smooth complex coefficients in $Ω\subset \mathbb{R}^{N}$ from a finite dimension complex vector space $E$ to a finite dimension complex vector space $F$ and $A^{*}(x,D)$ {its} adjoint. In this work we characterize the (local) continuous solvability of the partial differential equation $A^{*}(x,D)v=f$ (in the distribution sense) for a given distribution $f$; more precisely we show that any $x_0\inΩ$ is contained in a neighborhood $U\subset Ω$ in which its continuo…
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