Improvement of $W^{1,p}$ regularity of a elliptic equation solution.

Question

I'm looking for some reference for results like $$ \mbox{div}(A(x) \nabla u) = 0, \ \ u \in H^1=W^{1,2} \Rightarrow u \in W^{1,p}, p>2 $$ where $A(x)$ is elliptic, this is, $Id\lambda \le A(x) \le \Lambda Id $ with $0< \lambda<\Lambda$ are constant matrices and $Id$ is the identity matrix. I do not know if this is exactly true. Then if you know something like this, as adding hypothesis, please tell me or send me. I will be very grateful.

Answer 1

Have a look for this paper by Gröger. There, $W^{1,p}$-regularity for your problem is shown for a suitable domain $\Omega$ and mixed boundary conditions. This also applies to the case where the matrix $A$ is only $L^\infty(\Omega)$.

Answer 2

For the right hand side data is zero, the regularity of the solution depends on: the smoothness of the domain's boundary which is related to the smoothness of the boundary condition. Also certain properties of $A(x)$ (smoothness, monotonicity, and/or being in VMO/BMO).

Rewriting the equation as $-\mathrm{div}(A(x)\nabla u) = \mathrm{div}\mathbf{f}$ for some $\mathbf{f}$, we wanna check if there is such estimate: $$ \|\nabla u\|_{L^p(\Omega)} \leq c\|\mathbf{f} \|_{L^p(\Omega)}, \tag{1} $$ for some/all $p>2$, or some other similar estimates that involve the boundary data.

Caffarelli's paper is THE guide, if $A$ is sufficiently "close" to identity, then $u\in W^{1,p}$ for $p$ related to how close $A$ is to the identity.

If $A(x)$ is discontinuous, only some $p$ suffices for estimate (1), please see Theorem 1 to 3 in An $L^p$-estimate for the gradient of solutions of second order elliptic divergence equations.

For an example of discontinuous and not monotone $A(x)$ would result very singular solution, please refer to Kellogg's paper on elliptic equations with intersecting interfaces, see (3.2) for that example, we can manipulate the coefficients such that $u\in H^{1+\epsilon}$, and $\epsilon$ is very small, there is much space left for you to trade the differentiability for integrability.