锥上椭圆方程的正则性结果
Regularity results for elliptic equations on cones
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中文总结 AI 辅助
该研究针对N维欧氏空间球扇形上的椭圆问题，证明了泊松方程解梯度有界的最优特征值条件，还得到了p-拉普拉斯方程解的加权全局利普希茨连续性与二阶正则性。
中文摘要 AI 辅助
我们研究N维欧氏空间（N≥2）中半径R>0的球扇形S_{D,R}上Dirichlet或Neumann型椭圆问题解的全局正则性，其中D是单位球面S^{N-1}上张成该球扇形的有界区域。主要结果之一表明，当D上带Dirichlet或Neumann边界条件的拉普拉斯-贝尔特拉米算子-Δ_{S^{N-1}}的第一非平凡特征值λ₁(D)≥N-1时，泊松方程解的梯度有界。正如Maz'ya的例子所示，该特征值条件是最优的。对一般球扇形及p>1的p-拉普拉斯方程，我们证明了解的加权全局利普希茨连续性以及二阶正则性。
英文摘要
We study global regularity of solutions to Dirichlet or Neumann elliptic problems in spherical sectors $S_{D,R}$ of radius $R>0$ in $\mathbb{R}^N, N\ge 2$, where $D$ is the bounded domain on the unit sphere $\mathbb{S}^{N-1}$ which spans the spherical sector. One of the main results shows that boundedness of the gradient of the solutions of Poisson equations holds whenever $λ_1(D)\ge N-1$, where $λ_1(D)$ is the first nontrivial eigenvalue of the Laplace Beltrami operator $-Δ_{\mathbb{S}^{N-1}}$ on the domain $D$ with Dirichlet or Neumann boundary conditions on $\partial D$. As an example of Maz'ya shows, the condition on the eigenvalue is sharp. For general spherical sectors and for $p$-Laplacian equations, $p>1$ we prove weighted global lipschitzianity of the solutions, as well as second order regularity.
