标题： 梅耶指数规则支配二阶椭圆边值问题的一阶方法 显示英文标题

标题： Meyers exponent rules the first-order approach to second-order elliptic boundary value problems

摘要： 对于上半空间中具有横向独立复系数的二阶椭圆方程边值问题的一阶方法，将方程代数化为一阶系统，类似于平面调和函数与复分析中的柯西-黎曼系统的关联。 它依赖于扰动Dirac型算子的预解式在边界上的某些 p > 2 的全局 Lp -估计。 同时，这类方程局部弱解的梯度对于某些 p > 2 显示出更高的可积性，这通过弱逆Hölder估计来表达。 我们证明了这两种性质的最佳指数是一致的。 我们的证明依赖于一个简单但似乎被忽视的与切向方向上的算子值傅里叶乘子的联系。 显示英文摘要

摘要： The first-order approach to boundary value problems for second-order elliptic equations in divergence form with transversally independent complex coefficients in the upper half-space rewrites the equation algebraically as a first-order system, much like how harmonic functions in the plane relate to the Cauchy-Riemann system in complex analysis. It hinges on global Lp -bounds for some p > 2 for the resolvent of a perturbed Dirac-type operator acting on the boundary. At the same time, gradients of local weak solutions to such equations exhibit higher integrability for some p > 2, expressed in terms of weak reverse H{\"o}lder estimates. We show that the optimal exponents for both properties coincide. Our proof relies on a simple but seemingly overlooked connection with operator-valued Fourier multipliers in the tangential direction.