# Boundary regularity for the p-Laplace equation

Let $$f\in L^m(\Omega)$$ for some $$m>1$$, where $$\Omega$$ is a smooth bounded domain in $$\mathbb{R}^N$$ ($$N\geq 2$$). Consider the equation, $$\Delta_p u=f(x)$$for $$p=N$$, then $$u$$ is bounded in $$\Omega$$. Moreover, $$u$$ is continuous upto the boundary. Can anyone help me with the solution of this one. I have got this question while going through the paper : Lemma 3.7 of the paper below https://link.springer.com/content/pdf/10.1007%2Fs00030-016-0361-6.pdf

Thank you very much.

• What is $p$, $m$ and $N$? – Michał Miśkiewicz Nov 10 '18 at 23:09

I'm not sure what your assumptions are, but I assume $$p$$ is equal to the dimension of the domain.
In the special case $$f \equiv 0$$ the continuity of $$u$$ can be easily shown. Look up Peter Lindqvist's Notes on the $$p$$-Laplace equation, section 3.2. The proof can be adjusted to work in the case of sufficiently regular nonzero right-hand side $$f$$.
As for the continuity-up-to-the boundary, this obviously depends on the boundary data. If one solves the equation with non-continuous boundary data (chosen in the trace space for $$W^{1,n}$$), then the solution is not in $$C(\overline{\Omega})$$.
• I am extremely sorry for your confusion. Here $p>1$. $N\geq 2$ is the dimension of $\mathbb{R}^N$ where I assumed $p=N$. Moreover suppose $f\in L^m(\Omega)$ is nonnegative for some $m>1$. Then I want to prove that any weak solution $u\in W_0^{1,p}(\Omega)$ of the equation $-\Delta_p u=f$ in $\Omega$ belong to $C_0(\overline\Omega)$. – Mathlover Nov 11 '18 at 4:28
• And $f \in L^m$ with some $m > 1$ should be enough to adjust the proof I linked. If you show your efforts, I can comment on possible difficulties. – Michał Miśkiewicz Nov 11 '18 at 8:38