# Estimate for the gradient of solutions in an elliptic differential equation in a Sobolev space

Let $$\Omega$$ be a bounded or unbounded domain in $$\mathbf R^{3}$$ with a smooth boundary $$S$$ and a normal vector given by $$n$$. Now, we consider the following second-order elliptic problem with Neumann condition (in the variable $$u$$):

\left \{\qquad \begin{aligned} \operatorname{div}(h \nabla u) & = f & \mbox{in}\, \Omega\\ h \tfrac{\partial u}{\partial n} & =g & \mbox{on} \, S \end{aligned}\right .

for some functions $$f,g,h$$ (it comes from studying a nonstationary problem for a nonhomogeneous incompressible fluid).

From a result of Agmon, Douglis and Nirenberg (in [$$*$$]) it is known that the problem is solvable in $$W^2_p(\Omega)$$ and the following estimate holds:

$$\Vert\nabla u(t)\Vert_{W_{p}^{1}(\Omega)}\leq K(\Vert h\Vert_{C^{1}\langle\overline{\Omega})})\bigl(\Vert f\Vert_{L_{p}(\Omega)}+\Vert g\Vert_{W_{p}^{1- 1/p}(S)}\bigr).$$

I do not understand the original proof in [$$*$$] for that inequality. Although it seems to be some sort of classical estimate for solutions of second-order elliptic partial differential equations, I couldn't find it in a modern book of elliptic equations, or derive it from other classical estimates. Is there any other recent reference for that estimate or does someone know how to deduce it?

Reference

[$$*$$] S. Agmon, A. Douglas and L. Nuremberg, "Estimates near the boundary for solution of elliptic partial differential equations satisfying general boundary conditions I", Communications on Pure and Applied Mathematics 12 (1959), 623-727, 10.1002/cpa.3160120405, MR0125307, Zbl 0093.10401.

• Grisvard's "Elliptic Problems in Nonsmooth Domains" has some similar a priori estimates in Chapter 2.3/2.4, maybe these are of help. (It says there that essentially the ADN results are reproduced.) Commented Sep 15, 2023 at 13:43
• Thanks a lot @Hannes for your suggestion. I am revising the chapters you mentioned. Commented Sep 15, 2023 at 21:15