论文标题
圆锥形域上椭圆方程的倾斜导数问题
Oblique Derivative Problems for Elliptic Equations on Conical Domains
论文作者
论文摘要
我们研究了锥体结构域均匀椭圆方程的斜导数问题。在溶液的公理对称性的假设下，我们在倾斜载体的角度上找到足够的条件，以使梯度的hölder规律性保持至锥体的顶点。规律性的证明是基于精心构造的屏障方法或通过扰动参数的应用。在这种规律性不达到的情况下，我们提供明确的反例。在没有公理对称性的情况下，我们还为规律性提供了反例。与等效的二维问题不同，梯度Hölder的规律性并不适用于所有Axi对称溶液，而是定性的规律性属性取决于锥形的开头和边界条件下倾斜矢量的角度。
We study the oblique derivative problem for uniformly elliptic equations on cone domains. Under the assumption of axi-symmetry of the solution, we find sufficient conditions on the angle of the oblique vector for Hölder regularity of the gradient to hold up to the vertex of the cone. The proof of regularity is based on the application of carefully constructed barrier methods or via perturbative arguments. In the case that such regularity does not hold, we give explicit counterexamples. We also give a counterexample to regularity in the absence of axi-symmetry. Unlike in the equivalent two dimensional problem, the gradient Hölder regularity does not hold for all axi-symmetric solutions, but rather the qualitative regularity properties depend on both the opening angle of the cone and the angle of the oblique vector in the boundary condition.
