Morrey Potentials and PDE II

Abstract

First some examples of systems with isolated singular points; see [BF] and [G]. Let u(x) be given by

$$\displaystyle{ u(x)\; =\;\bigg (u^{1}(x),u^{2}(x),\cdots \,,u^{n}(x)\bigg)\; =\; \dfrac{x} {\vert x\vert ^{\gamma }}, }$$

where

$$\displaystyle{ x\; =\; (x_{1},x_{2},\cdots \,,x_{n})\mbox{ and }\gamma > 0. }$$

Then if we set

$$\displaystyle{ a_{ij}^{\alpha \beta }(x,u)\; =\;\delta _{ ij}\delta _{\alpha \beta } + (c\delta _{i\alpha } + d \cdot b_{i\alpha }(x,u))(c\delta _{j\beta } + d \cdot b_{j\beta }(x,u)) }$$

c and d two positive constants, then in 1968 DeGiorgi showed that the above u(x) solves

$$\displaystyle{ -(a_{ij}^{\alpha \beta }(x,u)\;u_{ x_{i}}^{\alpha })_{x_{j}}\; =\; 0,\;\;\beta = 1,2,\cdots \,,n }$$

(16.1)

(summation convention) in the ball B(0, 1) in the weak sense, when

$$\displaystyle{ b_{i\alpha }(x,u)\; =\; \dfrac{x_{i}\;x_{\alpha }} {\vert x\vert ^{2}}, }$$

(16.2)

with $\gamma = \frac{n} {2} (1 - 1/\sqrt{4(n - 1)^{2 } + 1})$ and with $c = n - 2,\;d = n$.