## Strong barrier function and generalized linear elliptic equation

Suppose is a domain in . satisfies the exterior sphere condition. That is there exists a ball such that . Then the function

is a barrier function for at .

For a strong barrier function at , we mean or , at such Define

Then

Assume and let

when and large enough.

Let be a solution of in a bounded domain . Suppose satisfy the exterior ball condition with . Moreover, assume

Suppose , . Show that satisfies a Lipschitz condition at

, .

where . If the sign is unrestricted, show that the same result holds provided also depends on .

When suppose is the strong barrier function of at , then there exists such that satisfies

on and

on and

By the maximum principle, we know that

.

When the sign of is undefined, let , apply the above procedure to , only need to adjust that .

Gilbarg Trudinger’s book. Chapter, exercise 3.6.

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