Louis Dupaigne, Alberto Farina
The paper proves Liouville-type results for stable solutions of semilinear elliptic PDEs with convex nonlinearity, posed on the entire Euclidean space. Extensions to solutions which are stable outside a compact set are also presented.
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Very singular self-similar solutions of semilinear odd-order PDEs are studied on the basis of a Hermitian-type spectral theory for linear rescaled odd-order operators.
In this paper, we present output feedback boundary stabilization for a class of semilinear parabolic PDEs with a boundary measurement and an actuation located at the same place. The method uses backstepping transformations, where the state and error systems are proved to be locally exponentially stable in the $\mathbb{H}^4$ norm. The stability of the transformed systems are obtained by constructing a strict Lyapunov function. A numerical example using the FitzHugh-Nagumo equation shows the proposed control law stabilizes the system into its equilibrium solution.
We consider the Cauchy problem for semilinear parabolic equation in divergence form with obstacle. We show that under natural conditions on the right-hand side of the equation and mild conditions on the obstacle the problem has a unique solution and we provide its stochastic representation in terms of reflected backward stochastic differential equations. We prove also regularity properties and approximation results for solutions of the problem.
Forced Frequency Locking for Semilinear Dissipative Hyperbolic PDEs
本文研究了小时间周期性扰动对半线性耗散双曲偏微分方程时间周期解的影响，推导出标量方程以解释强制频率锁定现象。
Monotone heteroclinic solutions to semilinear PDEs in cylinders and applications
本文证明柱体内半线性椭圆方程存在严格单调异宿型解，由此构造出非一维的无临界点整体有界解及非剪切流的二维欧拉方程平稳解，完善了先前结果。
Global propagation of analyticity and unique continuation for semilinear conservative PDEs
本文提出了一种新方法，用于证明某些保守PDEs的全局唯一延续性，通过证明全局解析性传播，并应用于半线性波、板和薛定谔方程。
In this manuscript we consider semilinear PDEs, with a convex nonlinearity, in a sector-like domain. Using cylindrical coordinates $(r, θ, z)$, we investigate the shape of solutions whose derivative in $θ$ vanishes at the boundary. We prove that any solution with Morse index less than two must be either independent of $θ$ or strictly monotone with respect to $θ$. In the special case of a planar domain, the result holds in a circular sector as well as in an annular, and it can also be extended to a rectangular domain. The corresponding problem in higher dimensions is also considered, as well as…
We show that first order semilinear PDEs by stochastic perturbation are well-posedness for globally Holder continuous and bounded vector field, with an integrability condition on the divergence. This result extends the liner case presented in [2]. The proof is based on in the stochastic characteristics method and a version of the commuting Lemma.
This paper deals with symmetry phenomena for solutions of the Dirichlet problem involving semilinear PDEs on Riemannian domains. We shall present a rather general framework where the symmetry problem can be formulated and provide some evidence that this framework is completely natural by pointing out some results for stable solutions. The case of manifolds with density, and corresponding weighted Laplacians, is inserted in the picture from the very beginning.
Regularity of Time-Periodic Solutions to Autonomous Semilinear Hyperbolic PDEs
本文研究了自主半线性双曲PDEs时间周期解的正则性，证明了在特定条件下解的光滑性，并通过例子展示了非共振条件对解正则性的影响。
In this paper, we prove that there exists a unique weak solution to the mixed boundary value problem for a general class of semilinear second order elliptic partial differential equations with singular coefficients. Our approach is probabilistic. The theory of Dirichlet forms and backward stochastic differential equations with singular coefficients and infinite horizon plays a crucial role.
Classification results, rigidity theorems and semilinear PDEs on Riemannian manifolds: a P-function approach
该研究采用P函数方法，分析完备非紧黎曼流形上半线性椭圆方程解的分类与流形刚性，得到临界方程正解分类、流形刚性等结果，部分结论在欧氏情形下也具新颖性。
This paper proves the existence of viscosity solutions of path dependent semilinear PDEs via Perron's method, i.e. via showing that the supremum of viscosity subsolutions is a viscosity solution. We use the notion of viscosity solutions introduced by Ekren, Keller, Touzi and Zhang, in whose work all smooth processes which are tangent in mean are considered as test functions. We also provide a comparison result for semicontinuous viscosity solutions, by using a regularization technique. As an interesting byproduct, we give a new short proof for the optimal stopping problem with semicontinuous o…
The classical Feynman-Kac identity builds a bridge between stochastic analysis and partial differential equations (PDEs) by providing stochastic representations for classical solutions of linear Kolmogorov PDEs. This opens the door for the derivation of sampling based Monte Carlo approximation methods, which can be meshfree and thereby stand a chance to approximate solutions of PDEs without suffering from the curse of dimensionality. In this article we extend the classical Feynman-Kac formula to certain semilinear Kolmogorov PDEs. More specifically, we identify suitable solutions of stochastic…
We consider the Cauchy problem for semilinear parabolic equation in divergence form with obstacle. We show that under natural conditions on the right-hand side of the eqution and mild conditions on the obstacle a unique continuous solution of the problem admits a stochastic representation in terms of reflected backward stochastic differential equations. We derive also some regularity properties of so- lutions and prove useful approximation results.
We obtain existence results for the solution u of nonlocal semilinear parabolic PDEs on $\mathbb{R}^d$ with polynomial nonlinearities in $(u, \nabla u)$, using a tree-based probabilistic representation. This probabilistic representation applies to the solution of the equation itself, as well as to its partial derivatives by associating one of d marks to the initial tree branch. Partial derivatives are dealt with by integration by parts and subordination of Brownian motion. Numerical illustrations are provided in examples for the fractional Laplacian in dimension up to 10, and for the fractiona…
We study bounded, monotone solutions of$Δu=W'(u)$ in the whole of$\R^n$, where$W$ is a double-well potential. We prove that under suitable assumptions on the limit interface and on the energy growth, $u$ is $1$D. In particular, differently from the previous literature, the solution is not assumed to have minimal properties and the cases studied lie outside the range of $Γ$-convergence methods. We think that this approach could be fruitful in concrete situations, where one can observe the phase separation at a large scale and whishes to deduce the values of the state parameter in the vicinity o…
Optimal control of fractional semilinear PDEs
本文研究了具有谱和积分分数扩散算子的半线性分数PDEs的最优控制问题，通过证明解的有界性、非线性增长条件以及最优控制问题的解的存在性，推导了最优性条件。
We analyze the sensitivity of the extremal equations that arise from the first order necessary optimality conditions of nonlinear optimal control problems with respect to perturbations of the dynamics and of the initial data. To this end, we present an abstract implicit function approach with scaled spaces. We will apply this abstract approach to problems governed by semilinear PDEs. In that context, we prove an exponential turnpike result and show that perturbations of the extremal equation's dynamics, e.g., discretization errors decay exponentially in time. The latter can be used for very ef…
Boundary bilinear control of semilinear parabolic PDEs: quadratic convergence of the SQP method
本文研究了由半线性抛物型方程支配的双线性控制问题，通过SQP算法实现局部解的计算，并证明了在L^p和L^∞空间中的稳定性与二次收敛性。
We study and compare two concepts for weak solutions to semilinear parabolic path-dependent partial differential equations (PPDEs). The first is that of mild solutions as it appears, e.g., in the log-Laplace functionals of historical superprocesses. The aim of this paper is to show that mild solutions are also solutions in a viscosity sense. This result is motivated by the fact that mild solutions can provide value functions and optimal strategies for problems of stochastic optimal control. Since unique mild solutions exist under weak conditions, we obtain as a corollary a general existence re…
About semilinear low dimension Bessel PDEs
本文研究低维 $0<\delta<1$ 下由 Bessel 型生成元驱动的半线性 PDE，并在漂移仅为 Schwartz 分布时，于 $L^2$ 对偶弱意义下证明了解的存在唯一性。
The Gevrey class implicit mapping theorem with application to UQ of semilinear elliptic PDEs
本文建立 Gevrey 类隐映射定理，证明残差的 $s$-Gevrey 正则性可推出局部数据到解映射 Fréchet 导数的 $s$-Gevrey 界，并用于半线性椭圆型 PDE 的不确定性量化。
Some new Liouville-type results for fully nonlinear PDEs on the Heisenberg group
该研究针对Heisenberg群上由Heisenberg向量场构造的完全非线性方程，通过退化Hadamard三球面定理证明新的尖锐Liouville型性质，并将欧几里得框架下Pucci极值算子受扰情况的结果推广到Heisenberg群。
Semilinear fractional elliptic PDEs with gradient nonlinearities on open balls: existence of solutions and probabilistic representation
针对高维球域上带梯度非线性的分数阶半线性椭圆PDE，提出基于α-稳定分支过程的树状概率表示法，无需梯度非线性的多项式阶条件即可证明粘性解存在，且在10维下精度优于确定性有限差分法。
We prove the existence of solutions $u$ in $H^1(\mathbb{R}^N,\mathbb{R}^M)$ of the following strongly coupled semilinear system of second order elliptic PDEs on $\mathbb{R}^N$ \[ \mathcal{P}[u] = f(x,u,\nabla u), \quad x\in \mathbb{R}^N, \] whith pointwise constraints. We present the construction of the suitable topoligical degree which allows us to solve the above system on bounded domains. The key step in the proof consists of showing that the sequence of solutions of the truncated system is compact in $H^1$ by the use of the so-called tail estimates.
Multilevel Picard algorithm for general semilinear parabolic PDEs with gradient-dependent nonlinearities
针对系数非常数且非线性依赖梯度的一般半线性抛物型 PDE，提出多层 Picard 逼近算法，并借助 SFPE 证明解与梯度的表示、收敛性及复杂度，辅以 300 维数值验证。
This article is a survey on boundedness results for stable solutions to semilinear elliptic problems. For these solutions, we present the currently known $L^{\infty}$ estimates that hold for all nonlinearities. Such estimates are known to hold up to dimension 4. They are expected to be true also in dimensions 5 to 9, but this is still an open problem which has only been established in the radial case.
We construct a solution for a class of strongly perturbed semilinear heat equations which blows up in finite time with a prescribed blow-up profile. The construction relies on the reduction of the problem to a finite dimensional one and the use of index theory to conclude.
Let $2\le n\le 5$. We establish an apriori interior Hölder regularity of $C^2$-stable solutions to the semilinear equation $-Δu=f(u)$ in any domain of $R^n$ for any nonlinearity $f\in C^{0,1}(R) $.If $f $ is nondecreasing and convex in addition,we obtain an interior Hölder regularity, and hence the local boundedness, of $W^{1,2}(Ω)$-stable solutions by locally approximating them via $C^2(Ω)$-stable solutions. In particular, we do not require any lower bound on $f$.
For a long time it is well-known that high-dimensional linear parabolic partial differential equations (PDEs) can be approximated by Monte Carlo methods with a computational effort which grows polynomially both in the dimension and in the reciprocal of the prescribed accuracy. In other words, linear PDEs do not suffer from the curse of dimensionality. For general semilinear PDEs with Lipschitz coefficients, however, it remained an open question whether these suffer from the curse of dimensionality. In this paper we partially solve this open problem. More precisely, we prove in the case of semi…
Validated integration of semilinear parabolic PDEs
本文提出一种计算机辅助证明方法，通过傅里叶空间常数变易公式和牛顿-康托罗维奇论证，对半线性抛物型PDE进行严格时间积分，给出轨道存在性证明及误差界，并适用于多个经典方程。
We prove that flat ground state solutions ($i.e.$ minimizing the energy and with gradient vanishing on the boundary of the domain) of the Dirichlet problem associated to some semilinear autonomous elliptic equations with a strong absorption term given by a non-Lipschitz function are unstable for dimensions $N=1,2$ and they can be stable for $N\geq 3$ for suitable values of the involved exponents.
Constructive proofs for some semilinear PDEs on $H^2(e^{|x|^2/4},\mathbb{R}^d)$
本文提出了一种基于谱方法和牛顿-康托罗维奇论证的计算机辅助工具，用于在特定空间上研究半线性PDEs的构造性证明，并应用于多种非线性方程，获得自相似解。
Semilinear Feynman-Kac Formulae for $B$-Continuous Viscosity Solutions
本文用概率方法证明无穷维半线性 PDE 的 $B$-连续粘性解存在且唯一，并给出基于倒向随机微分方程的首个非线性 Feynman–Kac 表示框架。
We investigate existence, uniqueness and regularity for local solutions of rough parabolic equations with subcritical noise of the form $du_t- L_tu_tdt= N(u_t)dt + \sum_{i = 1}^dF_i(u_t)d\mathbf X^i_t$ where $(L_t)_{t\in[0,T]}$ is a time-dependent family of unbounded operators acting on some scale of Banach spaces, while $\mathbf X\equiv(X,\mathbb X)$ is a two-step (non-necessarily geometric) rough path of Hölder regularity $γ>1/3.$ Besides dealing with non-autonomous evolution equations, our results also allow for unbounded operations in the noise term (up to some critical loss of regularity …
In this work, we study the stable determination of four space-dependent coefficients appearing in a coupled semilinear parabolic system with variable diffusion matrices subject to dynamic boundary conditions which couple intern-boundary phenomena. We prove a Lipschitz stability result for interior and boundary potentials by means of only one observation component, localized in any arbitrary open subset of the physical domain. The proof mainly relies on some new Carleman estimates for dynamic boundary conditions of surface diffusion type.
Periodic Solutions to Dissipative Hyperbolic Systems. II: Hopf Bifurcation for Semilinear Problems
本文研究了半线性双曲系统中时间周期解的霍普夫分岔问题，推导了分岔方向的公式并讨论了耗散性对分岔的影响。
Multilevel Picard approximations overcome the curse of dimensionality in the numerical approximation of general semilinear PDEs with gradient-dependent nonlinearities
本文证明了Neufeld与Wu提出的用于求解含梯度依赖非线性项的一般半线性抛物型PDE的多级Picard（MLP）算法可克服维数灾难，其计算复杂度仅随维度和精度倒数呈多项式增长。
We consider stable solutions of a semilinear elliptic equation with homogeneous Neumann boundary conditions. A classical result of Casten, Holland [20] and Matano [44] states that all stable solutions are constant in convex bounded domains. In this paper, we examine whether this result extends to unbounded convex domains. We give a positive answer for stable non-degenerate solutions, and for stable solutions if the domain $Ω$ further satisfies $Ω$ $\cap$ {|x| $\le$ R} = O(R^2), when R $\rightarrow$ +$\infty$. If the domain is a straight cylinder, an additional natural assumption is needed. The…
Least total curvature solutions to steady Euler system and monotone solutions to semilinear equations in a strip
本文通过极小化程序为半线性椭圆 PDE 寻找单调异宿解，证明了带状区域内不可压缩 Euler 系统最小总曲率解的存在性，并构造出非凸上水平集方程的正且单调的稳定解，对文献[27]的广义问题给出了否定回答。
Invariant Manifolds for Capillary Waves and a Class of Quasilinear PDEs
本文研究了毛细波与一类拟线性偏微分方程的不变流形存在性、唯一性和光滑性，适用于多种非线性偏微分方程。
A rigorous integrator and global existence for higher-dimensional semilinear parabolic PDEs via semigroup theory
该研究提出结合半群理论与计算机辅助证明的构造性方法，通过切比雪夫级数展开与压缩不动点定理实现半线性抛物PDE解的严格验证，还应用于Swift-Hohenberg等方程证明全局存在性。
In this paper we extend the interior regularity results for stable solutions in [Cabré, Figalli, Ros-Oton, and Serra, Acta Math. 224 (2020)] to operators with variable coefficients. We show that stable solutions to the semilinear elliptic equation $a_{ij}(x)u_{ij} + b_i(x) u_i + f(u) = 0$ are Hölder continuous in the optimal range of dimensions $n \leq 9$. Our bounds are independent of the nonlinearity $f \in C^1$, which we assume to be non-negative. The main achievement of our work is to make the constants in our estimates depend on the $C^1$ norm of $a_{ij}$ and the $C^0$ norm of $b_i$, inst…
This paper is devoted to the study of semi-stable radial solutions $u\in H^1(B_1)$ of $-Δu=g(u) {in} B_1\setminus \{0\}$, where $g\in C^1(R)$ is a general nonlinearity and $B_1$ is the unit ball of $R^N$. We establish sharp pointwise estimates for such solutions. As an application of these results, we obtain optimal pointwise estimates for the extremal solution and its derivatives (up to order three) of the semilinear elliptic equation $-Δu=λf(u)$, posed in $B_1$, with Dirichlet data $u|_{\partial B_1}=0$, and a continuous, positive, nondecreasing and convex function $f$ on $[0,\infty)$ such t…
Stable solutions to fractional semilinear equations: uniqueness, classification, and approximation results
本文研究带一般 Dirichlet 外部条件的分数阶半线性方程稳定解，建立唯一性、分类及有界稳定解逼近结果，并用于证明低维半 Laplace 情形的内部正则性。
We prove the exact multiplicity of flat and compact support stable solutions of an autonomous non-Lipschitz semilinear elliptic equation of eigenvalue type according to the dimension N and the two exponents, $0<α<β<1$, of the involved nonlinearites. Suitable assumptions are made on the spatial domain $Ω$ where the problem is formulated in order to avoid a possible continuum of those solutions and, on the contrary, to ensure the exact number of solutions according to the nature of the domain $Ω$. Our results also clarify some previous works in the literature. The main techniques of proof are a …
We present a theory of hypoellipticity and unique ergodicity for semilinear parabolic stochastic PDEs with "polynomial" nonlinearities and additive noise, considered as abstract evolution equations in some Hilbert space. It is shown that if Hormander's bracket condition holds at every point of this Hilbert space, then a lower bound on the Malliavin covariance operator $M_t$ can be obtained. Informally, this bound can be read as "Fix any finite-dimensional projection $Π$ on a subspace of sufficiently regular functions. Then the eigenfunctions of $M_t$ with small eigenvalues have only a very sma…
A stable-compact method for qualitative properties of semilinear elliptic equations
本文提出 stable-compact 通用框架，用于分析带 Dirichlet 边界的单稳态反应扩散方程，给出保证稳态唯一的区域几何条件及非唯一性反例。
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。
Classification of the structures of stable radial solutions for semilinear elliptic equations in $\bf R^N$
本文研究了半线性椭圆方程在R^N中稳定径向解的结构分类，通过分析f'(u)F(u)的极限来判断解的存在性与非存在性，并探讨了奇异稳定解与解结构的关系。
We construct a periodic solution to the semilinear heat equation with power nonlinearity, in one space dimension, which blows up in finite time $T$ only at one blow-up point. We also give a sharp description of its blow-up profile. The proof relies on the reduction of the problem to a finite dimensional one and the use of index theory to conclude. Thanks to the geometrical interpretation of the finite-dimensional parameters in terms of the blow-up time and blow-up point, we derive the stability of the constructed solution with respect to initial data.
In the present paper, we investigate the regularity and symmetry properties of weak solutions to semilinear elliptic equations which are locally stable.
A simple rigorous integrator for semilinear parabolic PDEs
本文提出了一种简单且高效的严格积分方法，用于半线性抛物型PDEs的误差界推导，通过自适应时间步长策略和稳定性证明，适用于长时积分和非平凡动态捕捉。
Regularity of stable radial solutions to semilinear elliptic equations in MEMS problems
本文研究了MEMS问题中半线性椭圆方程稳定径向解的正则性，证明在特定条件下所有稳定径向解均为正则，解决了关于奇异非线性情况下稳定解正则性的开放问题。
The boundedness of stable solutions to semilinear elliptic equations with linear lower bound on nonlinearities
本文在非线性项满足线性下界条件下，证明了半线性椭圆方程 $C^2$-稳定解的内部 Hölder 正则性，并在 $f$ 非降凸时推广到 $W^{1,2}$-稳定解，且维数 $n\le9$ 最优。
Stable determination of damping and potential coefficients in a semilinear wave equation
针对半线性波动方程，通过二阶线性化与几何光学方法，首次建立了从Dirichlet-to-Neumann映射同时稳定恢复阻尼、线性势和非线性势系数的Hölder型估计。
In this paper we prove the following long-standing conjecture: stable solutions to semilinear elliptic equations are bounded (and thus smooth) in dimension $n \leq 9$. This result, that was only known to be true for $n\leq4$, is optimal: $\log(1/|x|^2)$ is a $W^{1,2}$ singular stable solution for $n\geq10$. The proof of this conjecture is a consequence of a new universal estimate: we prove that, in dimension $n \leq 9$, stable solutions are bounded in terms only of their $L^1$ norm, independently of the nonlinearity. In addition, in every dimension we establish a higher integrability result fo…
We classify stable and finite Morse index solutions to general semilinear elliptic equations posed in Euclidean space of dimension at most 10, or in some unbounded domains.
Interior Regularity for Stable Solutions of Semilinear Elliptic Equations with Sign-Changing Nonlinearities
该研究针对具变号非线性项的半线性椭圆方程稳定解，在不同维数下建立了内部正则性估计，解决了相关公开问题并推广了已有结果，还证明了对应Liouville型定理。
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