In the last three decades, powerful computer-assisted techniques have been developed in order to validate a posteriori numerical solutions of semilinear elliptic problems of the form $Δu +f(u,\nabla u) = 0$. By studying a well chosen fixed point problem defined around the numerical solution, these techniques make it possible to prove the existence of a solution in an explicit (and usually small) neighborhood the numerical solution. In this work, we develop a similar approach for a broader class of systems, including nonlinear diffusion terms of the form $ΔΦ(u)$. In particular, this enables us to obtain new results about steady states of a cross-diffusion system from population dynamics: the (non-triangular) SKT model. We also revisit the idea of automatic differentiation in the context of computer-assisted proof, and propose an alternative approach based on differential-algebraic equations.
Computer-Assisted Proof of the Main Theorem of 'The Classification of Branched Willmore Spheres in the $3$-Sphere and the $4$-Sphere'
我们使用免费的数学软件Sage对论文“The Classification of Branched Willmore spheres in the $3$ -Sphere and the $4$ -Sphere”（arXiv：1706.01405）中的主要定理4.11中的四次和八次亚纯微分的全纯性进行了计算机辅助证明。
A computer-assisted existence proof for Emden's equation on an unbounded L-shaped domain
我们证明了存在性，非退化，并在无穷远的非平凡的解决方案埃姆登的方程$-Δu = | u |^3$上的一个无界的$L$形域，Dirichlet边界条件的指数衰减。除了这个结果的直接价值，我们也把这个解决方案作为一个积木的解决方案扩大有界域的角落，建立在未来的工作。我们的证明大量使用计算机辅助：从一个数值近似解开始，我们使用一个不动点参数来证明附近的精确解的存在性。在证明过程中建立的特征值界也意味着解的非退化性。
Computer-assisted proof of heteroclinic connections in the one-dimensional Ohta-Kawasaki model
Institute of Computer Science and Computational Mathematics, Jagiellonian University（雅盖隆大学计算机科学与计算数学研究所）；Institute of Applied Mathematics and Mechanics, University of Warsaw（华沙大学应用数学与力学研究所）；Department of Mathematics, Rutgers, The State University of New Jersey（新泽西州立大学罗格斯大学数学系）；Department of Mathematical Sciences, George Mason University（乔治梅森大学数学科学系）
On the potential lack of response in a model of second-harmonic generation. A computer-assisted proof
School of Mathematical and Statistical Sciences, University of Texas Rio Grande Valley（德克萨斯大学里奥格兰德河谷分校数学与统计科学学院）；Department of Mathematics, Rutgers University（罗格斯大学数学系）；Department of Mathematics and Statistics, McGill University（麦吉尔大学数学与统计系）
