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Given the p-poisson equation ($1<p<\infty$): $$ -\Delta_p u=f \mbox{ in $B_r(0)\subset \mathbb R^N$} $$ with $f\in L^q(B_r)$ and $q>N$, I wish to show that:

$$ |\nabla u(0)|\leq C\left ( \frac{1}{r}\sup_{B_r}|u| + ||f||_{L^q(B_r)} \right ) $$.

Where $C$ is independent of $r$.

PS: I am talking about the equation (6.7) in M. Hayouni's Lipschitz Continuity of the state function in a shape Optimization Problem. Through Embedding Theorems, I could arrive up to uniform boundedness of gradient, but how to proceed for estimate (6.7). I expect it to be true for a general $p>1$

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  • $\begingroup$ What is $\Delta_p u$? $\endgroup$ – Robert Lewis Mar 3 '18 at 19:48
  • $\begingroup$ @RobertLewis It is the p-Laplacian: $\Delta_p u=:\textrm{div}(|\nabla u|^{p-2}\cdot \nabla u)$ $\endgroup$ – user522521 Mar 3 '18 at 19:51
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This is only a partial answer, or more of a suggestion. One standard trick to prove regularity for the $p$-Laplace equation is to regularize it in the following way

$$-\text{div} ( (|\nabla u|^2 + \epsilon^2)^{(p-2)/2}\nabla u) = f.$$

This operator is uniformly elliptic, and solutions are smooth. So you can differentiate the equation and try to use the maximum principle to bound the gradient. The idea is to look for estimates independent of $\epsilon>0$, since they hold for the $p$-Laplace equation as well. This should work to get gradient estimates, though they might have a bit of a different form from what you are asking.

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  • $\begingroup$ Thank you for the hint, I have few questions: are the solutions smooth because of the embedding theorem? (I mean, since $u$ solve the equation so $u\in W^{2,q}\hookrightarrow C^{1,\alpha}$). Or there is a better argument for the claim? $\endgroup$ – Harish Mar 4 '18 at 8:48
  • $\begingroup$ Moreover, I read $-\Delta_p u =f \in L^q$ imply $u\in W^{2,q}\hookrightarrow C^{1,\alpha}$. Hence $|\nabla u|\leq C ||u||_{2,q}$. But I am not sure how constant depends on the radius of ball. If it vanishes as $r\rightarrow 0$, I think the estimate in question should be irrelevant. $\endgroup$ – Harish Mar 4 '18 at 8:51
  • $\begingroup$ Solutions are smooth by elliptic regularity. Though you will have to regularize $f$ as well (say mollify it with bandwidth $\epsilon>0$). $\endgroup$ – Jeff Mar 4 '18 at 14:11
  • $\begingroup$ Thank you for the reply, can you comment on my previous question on how the constant of embedding depends in the size of domain? I expect it should be small as domain shrinks. But what are your thoughts in this? $\endgroup$ – Harish Mar 10 '18 at 6:47

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