# Local “boundary comparison principle” for harmonic functions

Let $$u$$ be a positive solution of the elliptic equation $$\mathcal Lu = 0$$ on $$B^+_1 \subset \mathbb{R}^n$$ such that $$u$$ vanishes continuously on $$\{x_n = 0\}$$. To fix ideas, we may take $$\mathcal L = - \Delta$$, ie $$u$$ harmonic.

Then we can prove the comparison principle stated in STEP 1 below (Caffarelli, 1998 p. 40):

$$u\big|_{B^+_{1/2}} \le M.$$

I've some questions on the proof below (PROOF OF STEP 1).

• In particular, what does the (c) observation mean? What does the author mean by $$M>M_0$$ large enough? What is the "absurde assumption" that will lead to a contradiction?

• Why after proving that

"if $$M \ge M_0$$ large, we can construct a sequence of points, $$X_k$$ all contained in $$B^+_{3/4}$$, $$X_k → \{x_n = 0\}$$, and such that $$u(X_k)$$ goes to $$+\infty$$"

do we get that the proof of STEP 1 is complete?

• Iterate and consider a sequence of nested balls $B_{1/2^{n+1}}\subset B_{1/2^{n}}$, I presume? – leo monsaingeon Jan 28 at 9:39
• @leomonsaingeon My mistake: I meant the $(c)$ step. – user123456 Jan 28 at 9:57
• By Item (b), if $u(X_0)=M$, for $X_0\in B_{1/2}^+$ is large then one must have $x_n(X_0)$ small. This should yield the contradiction, I'm not sure why the points $X_k$ are said to lie in $B_{3/4}^+$ and not $B_{1/2}^+$. Is this this is because the balls are open? – RBega2 Feb 16 at 21:23
• Something must be wrong in the formulation. If $u$ is harmonic and positive in $B_1$, and if it vanishes at some interior point (here for $x_n=0$), then $u\equiv0$. – Denis Serre Feb 18 at 12:14
• @DenisSerre You're right. There is a typo in the first line: the function is harmonic on $B^+_1$, i.e. the part of the ball above $\{x_n=0\}$. – user123456 Feb 26 at 18:38