We call pattern any non-constant solution of a semilinear elliptic equation with Neumann boundary conditions. A classical theorem of Casten, Holland [20] and Matano [50] 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 stability of patterns, and the $C^1$ norm of the nonlinearity.

In addition, we establish several gradient estimates for the patterns of (1). We prove a general nonlinear Cacciopoli inequality (or an inverse Poincaré inequality), stating that the $L^2$-norm of the gradient of a solution is controlled by the $L^2$-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 [54] and to extend some results on De Giorgi's conjecture to a larger class of domains.

我们将具有 Neumann 边界条件的半线性椭圆方程的任何非常数解称为模式。Casten、Holland [20] 和 Matano [50] 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性和模式稳定性的假设可以在几个方向上放宽。特别是，我们提出了模式不存在的一般标准，处理可能的非凸域和不稳定模式。我们的结果展示了域的几何形状、模式的稳定性和$C^1$ 非线性的范数。

此外，我们为（1）的模式建立了几个梯度估计。我们证明了一般非线性 Cacciopoli 不等式（或逆庞加莱不等式），说明${升}^2$- 溶液梯度的范数由 ${升}^2$-范数 $F(你)$，具有仅取决于域的常数。这种不等式适用于非齐次方程。我们还给出了几个平坦度估计。

我们的方法依赖于我们所说的Robin-curvature Laplacian的引入。该算子是域固有的，包含许多关于域的几何形状如何影响解的形状的信息。

最后，我们将结果扩展到无界域。它使我们能够改进我们之前论文 [54] 的结果，并将 De Giorgi 猜想的一些结果扩展到更大的领域。