Pascal Auscher, Andreas Axelsson, Alan McIntosh
This paper has been withdrawn, and is replaced with paper "Solvability of elliptic systems with square integrable boundary data" by the same authors.
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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。
We consider second order elliptic divergence form systems with complex measurable coefficients $A$ that are independent of the transversal coordinate, and prove that the set of $A$ for which the boundary value problem with $L_2$ Dirichlet or Neumann data is well posed, is an open set. Furthermore we prove that these boundary value problems are well posed when $A$ is either Hermitean, block or constant. Our methods apply to more general systems of PDEs and as an example we prove perturbation results for boundary value problems for differential forms.
We consider a locally integrable real-analytic structure, and we investigate the local solvability in the category of Gevrey functions and ultradistributions of the complex d' naturally induced by the de Rham complex. We prove that the so-called condition Y(q) on the signature of the Levi form, for local solvability of d' u=f, is still necessary even if we take f in the classes of Gevrey functions and look for solutions u in the corresponding spaces of ultradistributions.
We present a general method for obtaining conservation laws for integrable PDE at negative regularity and exhibit its application to KdV, NLS, and mKdV. Our method works uniformly for these problems posed both on the line and on the circle.
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。
A PDE system consisting of the momentum balance, mass balance, and energy balance equations for displacement, capillary pressure, and temperature as a model for unsaturated fluid flow in a porous viscoelastoplastic solid is shown to admit a solution under appropriate assumptions on the constitutive behavior. The problem involves two hysteresis operators accounting for plastic and capillary hysteresis.
Almost global existence for some Hamiltonian PDEs on manifolds with globally integrable geodesic flow
该研究针对具有全局可积测地流的黎曼流形上的半线性哈密顿PDE，基于正规形方法证明了小光滑解的几乎全局存在性抽象结果，并将其应用于带卷积势的NLS方程、非线性梁方程，同时证明了NLS基态的$H^s$稳定性。
We show that advection-diffusion equations with porous media type diffusion and integrable initial data are globally solvable under very mild conditions. Some generalizations and related results are also given.
Solvability of the Zakharov-Shabat systems with meromorphic potentials by quadrature
本文研究了具有亚纯势的二维Zakharov-Shabat系统的可解性，证明了在势函数绝对可积条件下，该系统可通过求积法求解当且仅当势函数无反射。
In this work, we study the Neumann initial-boundary value problem for a two-species chemotaxis system with Lotka-Volterra competition and signal production. Under a rather weak and clean condition of sub-quadratic type damping and merely integrable initial data, we establish the global existence of generalized solutions in an N dimensional bounded and smooth domain.
We survey and compare, mainly in the two-dimensional case, various results obtained by IST and PDE techniques for integrable equations. We also comment on what can be predicted from integrable equations on non integrable ones.
We give different proofs and prove new results on the non complete solvability of some systems of complex first order p.d.e.'s, especially related to the analysis on CR manifolds.
Existence & Regularity of Weak Solutions of Degenerate Parabolic PDE Models for the Pricing of Security Derivatives
针对亚洲期权定价中出现的退化抛物型偏微分方程，通过有限差分数值格式证明边界退化模型的广义可解性，并研究解的正则性。
In this paper, we study the formal solution space of a nonlinear PDE in a fiber bundle. To this end, we start with foundational material and introduce the notion of a pfd structure to build up a new concept of profinite dimensional manifolds. We show that the infinite jet space of the fiber bundle is a profinite dimensional manifold in a natural way. The formal solution space of the nonlinear PDE then is a subspace of this jet space, and inherits from it the structure of a profinite dimensional manifold, if the PDE is formally integrable. We apply our concept to scalar PDEs and prove a new cri…
Solvability of elliptic homogeneous linear equations with measure data in weighted Lebesgue spaces
本文研究了在加权Lebesgue空间中，椭圆齐次线性微分方程在给定测度数据下的可解性条件，通过控制测度的Riesz势提出充分条件，并在p=∞时引入抵消条件。
Solvability of the Korteweg-de Vries equation under meromorphic initial conditions by quadrature
研究应用逆散射变换求解 Korteweg-de Vries 方程时，证明 Lax 对中的 Schrödinger 方程在亚纯势函数无反射且满足绝对可积条件时可通过求积法求解，并指出衰减的有理势函数若不满足该条件则不可积。
Formally Integrable Structures I. Resolution of Solution Sheaves
本文作为形式可积结构$L^2$理论系列首篇，为Hörmander超定系统构造解层分解，得全局正合性条件及局部解粘合技术，证明Levi形式消失（含Levi平坦）结构的Treves复形局部可解性。
We prove the solvability of the parabolic $L^p$ Dirichlet boundary value problem for $1 < p \leq \infty$ for a PDE of the form $u_t = \mbox{div} (A \nabla u) + B \cdot \nabla u$ on time-varying domains where the coefficients $A= [a_{ij}(X, t)]$ and $B=[b_i]$ satisfy a certain natural small Carleson condition. This result brings the state of affairs in the parabolic setting up to the elliptic standard. Furthermore, we establish that if the coefficients of the operator $A,\,B$ satisfy a vanishing Carleson condition and the time-varying domain is of VMO type then the parabolic $L^p$ Dirichlet bou…
Formalising Linear Elliptic PDE Theory in Lean 4
在 Lean 4 中形式化线性椭圆偏微分方程理论，包括 Dirichlet 问题可解性、Sobolev 空间及相关定理，实现无 sorry 的机器验证。
Keller-Osserman and Harnack type results for nonlinear elliptic PDE with unbounded ingredients
本文证明无界系数散度形式椭圆方程Keller-Osserman定理成立，解决强极大值原理开放问题，并推广Harnack不等式。
Solvability for non-smooth Schrödinger equations with singular potentials and square integrable data
本文为一阶算子$DB$发展全纯泛函演算，在奇异势和复椭圆系数条件下证明了薛定谔方程边值问题的适定性，并指出适定复系数集合是开集。
We present a framework to solve the open problem of formulating the inverse scattering method (ISM) for an integrable PDE on a star-graph. The idea is to map the problem on the graph to a matrix initial-boundary value (IBV) problem and then to extend the unified method of Fokas to such a matrix IBV problem. The nonlinear Schrödinger equation is chosen to illustrate the method. The framework unifies all previously known examples which are recovered as particular cases. The case of general Robin conditions at the vertex is discussed: the notion of linearizable initial-boundary conditions is intr…
In this paper, we consider a class of optimal control problems governed by state-equations of Kobayashi--Warren--Carter type. The control is given by physical temperature. The focus is on problems in dimensions less than equal to 4. The results are divided in four Main Theorems, concerned with: solvability and parameter-dependence of state-equations and optimal control problems; the first order necessary optimality conditions for these regularized optimal control problems. Subsequently, we derive the limiting systems and optimality conditions and study their well-posedness.
Spectrally accurate fully discrete schemes for some nonlocal and nonlinear integrable PDEs via explicit formulas
本文利用显式公式为Benjamin-Ono、Calogero-Sutherland DNLS和三次Szegő方程构造了时间精确、空间谱精度的全离散格式，证明其谱收敛且误差随时间线性增长，计算成本与最终时间无关，为研究可积方程长时间动力学提供了新工具。
The Geometry of Darboux Integrable Elliptic Systems
本文研究了通过复Pfaffian系统H的全纯解构造实椭圆微分系统，并证明其具有Darboux可积性。
In the following paper, one studies, given a bounded, connected open set $Ω$ $\subseteq$ R n , $κ$ > 0, a positive Radon measure $μ$ 0 in $Ω$ and a (signed) Radon measure $μ$ on $Ω$ satisfying $μ$($Ω$) = 0 and |$μ$| $κ$$μ$ 0 , the possibility of solving the equation div u = $μ$ by a vector field u satisfying |u| $κ$w on $Ω$ (where w is an integrable weight only related to the geometry of $Ω$ and to $μ$ 0), together with a mild boundary condition. This extends results obtained in [4] for the equation div u = f , improving them on two aspects: one works here with the divergence equation with mea…
Wall-chamber decompositions for generalised Monge-Ampère equations
本文研究了广义Monge-Ampère方程的求解问题，通过分析Nakai-Moishezon型准则失效时的代数性质，提出了有效的求解准则，并在高维紧Kähler流形中提供了首个存在性结果。
An overview of some recent results on the geometry of partial differential equations in application to integrable systems is given. Lagrangian and Hamiltonian formalism both in the free case (on the space of infinite jets) and with constraints (on a PDE) are discussed. Analogs of tangent and cotangent bundles to a differential equation are introduced and the variational Schouten bracket is defined. General theoretical constructions are illustrated by a series of examples.
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