Take $\Omega$ to be a bounded domain in $N$ dimensional Euclidean space with smooth boundary and we assume $\Omega$ contains the origin. I am interested is the following equation

$$ \Delta \phi(x) + \gamma \sum_{i,j=1}^N \frac{x_i x_j}{|x|^2} \phi_{x_i x_j}(x) = f(x)$$ in $\Omega$ with $ \phi=0$ on $ \partial \Omega$. Here $\gamma$ is such that the operator is elliptic. I would like to get some estimate of the form $$ \sup_{\Omega}| \nabla \phi(x)| \le C \sup_\Omega |f(x)|.$$ I guess there might be some issues at the origin. I have tried cutting the domain off and trying to get estimates independent of the domain cut off; but I was not successful. I guess part of my problem is that I don't truly understand the estimates I use; but normally I can perturb off the Lapclian and not really need to understand the main ideas.

QUESTION. Is this gradient estimate true and are there any standard methods to try and prove this fact?
The method I tried was the $C^{1,\alpha}$ blow up analysis method from Leon Simon; but I couldn't get it to work.


  • 1
    $\begingroup$ The operator in this problem is not in divergence form. This is not typical in problems that have geometric or physical significance. I have two suggestions. 1. Write the operator in polar/spherical coordinates. 2.Look at this paper of Stampacchia numdam.org/item/?id=SJL_1963-1964___3_1_0 It is about operators with discontinuous coefficients in divergence form and your operator is really close to one. Maybe you can glean some ideas in this paper. $\endgroup$ Mar 31 at 10:04
  • $\begingroup$ Everything should be contained in the last section of "Elliptic and parabolic problems for a class of operators with discontinuous coefficients", Annali SNS vol XiX (2019), by L. Negro, C. Spina and myself. We deal mainly with the whole space and and $W^{2,p}$ regularity but when some $p>N$ is allowed, then the gradient estimate follow from Sobolev embeddding. $\endgroup$ Mar 31 at 12:42
  • $\begingroup$ Thank you both for your comments. I will look through both mentioned references. If someone wants to make their comment an answer I can accept it. In anycase thank you for your very useful comments; its is much appreciated $\endgroup$
    – Math604
    Apr 1 at 5:17
  • $\begingroup$ @Giorgio. Is there access to this paper online somewhere? I searched a bit and can't seem to get access. Thanks $\endgroup$
    – Math604
    Apr 3 at 7:53
  • $\begingroup$ If you give me an email I can send to you in the afternoon. Please write to giorgio.metafune@unisalento.it $\endgroup$ Apr 3 at 8:33


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