论文标题
模式和梯度估计的不存在
Non-existence of patterns and gradient estimates
论文作者
论文摘要
我们将模式称为具有Neumann边界条件的半线性椭圆方程的任何非恒定稳定解。 Casten，Holland [19]和Matano [49]的经典定理指出，凸域中不存在稳定模式。在本文中，我们表明，该定理中域的凸面和模式的稳定性的假设可以在多个方向上放松。特别是，我们提出了一个关于模式不存在的一般标准，可以处理可能的非凸域和不稳定的模式。我们的结果展开了域的几何形状，非线性的大小与模式的稳定性之间的相互作用。例如，我们提出了几种应用，例如，我们证明（在几何假设下）如果域缩小或非线性的幅度很小，则没有模式。我们还完善了Casten Holland和Matano的结果，并表明它在域和非线性的平滑扰动下是可靠的。此外，我们建立了（1）模式的几个梯度估计。我们证明了一般的非线性cacciopoli不平等（或反向poincar {é}不平等），表明溶液的梯度的L2-元素受F（u）的L2-norm的控制，其常数仅取决于域。这种不平等适用于非均匀方程。我们还给出了几个平坦的估计。我们的方法依赖于引入我们所谓的罗宾 - 曲面拉普拉斯人。该操作员是域的固有，并且包含有关域几何形状如何影响解决方案形状的大量信息。最后，我们将结果扩展到无限的域。它使我们能够改善以前论文的结果[53]，并将De Giorgi的猜想扩展到更大的域。
We call pattern any non-constant stable solution of a semilinear elliptic equation with Neumann boundary conditions. A classical theorem of Casten, Holland [19] and Matano [49] states that stable patterns do not exist in convex domains. In this article, we show that the assumptions of convexity of the domain and stability of the pattern in this theorem can be relaxed in several directions. In particular, we propose a general criterion for the non-existence of patterns, dealing with possibly non-convex domains and unstable patterns. Our results unfold the interplay between the geometry of the domain, the magnitude of the nonlinearity, and the stability of patterns. We propose several applications, for example, we prove that (under a geometric assumption) there exist no patterns if the domain is shrunk or if the nonlinearity has a small magnitude. We also refine the result of Casten Holland and Matano and show that it is robust under smooth perturbations of the domain and the nonlinearity. In addition, we establish several gradient estimates for the patterns of (1). We prove a general nonlinear Cacciopoli inequality (or an inverse Poincar{é} inequality), stating that the L2-norm of the gradient of a solution is controlled by the L2-norm of f(u), with a constant that only depends on the domain. This inequality holds for non-homogeneous equations. We also give several flatness estimates. Our approach relies on the introduction of what we call the Robin-curvature Laplacian. This operator is intrinsic to the domain and contains much information on how the geometry of the domain affects the shape of the solutions. Finally, we extend our results to unbounded domains. It allows us to improve the results of our previous paper [53] and to extend some results on De Giorgi's conjecture to a larger class of domains.
