$L^p$ regularity of elliptic operators

Question

Consider the Dirichlet problem $-\Delta u =f, u|_{\partial U}=0$ on a domain $U\subseteq \mathbb R^d.$ I am looking for a reference for the proof of the estimate: $$ \|u\|_{W^{1,q}(U)} \lesssim\|f\|_{W^{-1,q}(U)}. $$ The case $q=2$ is straightforward because it matches perfectly with the weak formulation. However I find the general case difficult to prove. In particular, if I use a Green's function $G$ to write $u(x)=\int_U G(x,y) f(y)dy,$ I am then lacking a way to estimate, say, the norm of a convolution involving $W^{-1,q}.$ Young's convolution estimate does not work for this purpose even when the domain is the entire $\mathbb R^n.$

So how can I prove this bound?

Answer 1

This is a non-trivial result of Calderón and Zygmund; classically this involves establishing $L^p$ estimates for the singular integral operator convolving with $D_{ij}\Gamma(x),$ where $\Gamma$ is the Newtonian potential in the full space. This provides estimates of the form $$ \lVert \nabla u\rVert_{L^p(\Bbb R^n)} \leq C(n,p) \lVert \Delta u \rVert_{\dot{W}^{-1,p}(\Bbb R^n)} $$ for all $u \in C^{\infty}_c(\Omega)$ and $1 < p < \infty.$ Note we cannot apply Young's inequality since the kernel scales like $|x|^{-n},$ so instead we need establish a weak(1,1) estimate (using the Calderón-Zygmund decomposition) and conclude via interpolation + duality. From here one extends this to the half-space by a reflection argument, then to variable-coefficient equations using a perturbation argument, and finally to bounded $C^1$ domains by a patching argument. The details can be found in Chapter 9 (Strong solutions) of the classical text of Gilbarg and Trudinger, namely:

Gilbarg, David; Trudinger, Neil S., Elliptic partial differential equations of second order, Grundlehren der mathematischen Wissenschaften. 224. Berlin-Heidelberg-New York: Springer-Verlag. X, 401 p. (1977). ZBL0361.35003.

Note that they prove $W^{2,p}$ estimates, but the argument for $W^{1,p}$ is similar.

There are a couple of other references however, and other approaches to prove this result. For this I mention an approach using the interpolation of Stampacchia; you can find the interior case in these lectures notes of Ambrosio. Another reference which may include more details is the following monograph:

Chen, Ya-Zhe; Wu, Lan-Cheng, Second order elliptic equations and elliptic systems. Transl. from the Chinese by Bei Hu, Translations of Mathematical Monographs. 174. Providence, RI: American Mathematical Society (AMS). xiii, 246 p. (1998). ZBL0902.35003.

There is a lot of literature of this subject, including alternative proofs and approaches (I think you can also estimate the Green's function directly as you try), but they all generally require this Calderón-Zygmund decomposition at some point. Because of this you do need a fairly strong regularity assumption on the domain - the $C^1$ condition can be relaxed, but this does fail for general Lipschitz domains.

Answer 2

I think the proof goes like this. Any $(L^q(\Omega))^d$ vector field can be decomposed as $f = h + \nabla g$, where $h$ is divergence free in the sense of distributions. We can also stipulate that $(g)_\Omega = \displaystyle \frac{1}{|\Omega|}\int_\Omega g = 0$ by subtracting a constant which vanishes with the gradient. The point is then that $g$ satisfies a Poincare inequality, so that $\|\nabla g\|_{L^q(\Omega)} \simeq \|g\|_{W^{1,q}(\Omega)}$. Also we have that $\|g\|_{W^{1,q}} + \|h\|_{L^q} \lesssim \|f\|_{L^q}.$ This is the Helhmoltz decomposition.

Now use duality. Given $f \in (C_c^\infty(\Omega))^d, \|f\|_{L^{q'}} = 1$ write $f = h + \nabla g$ where $h,g$ are as above. Since $h$ is divergence free, $\int \nabla u \cdot f = \int \nabla u \cdot \nabla g.$ Then $$\|\nabla u\|_{L^q(\Omega)} = \sup_{f \in C_c^\infty(\Omega), \|f\|_{L^{q'}} = 1} \left|\int_U \nabla u \cdot f\right| = \sup_{f \in C_c^\infty(\Omega), \|f\|_{L^{q'}} = 1} \left|\int_\Omega \nabla u \cdot \nabla g \right|$$ $$ = \sup_{f \in C_c^\infty(\Omega), \|f\|_{L^{q'}} = 1} | \langle \Delta u, g \rangle | \le \sup_{f \in C_c^\infty(\Omega), \|f\|_{L^{q'}} = 1} \|\Delta u \|_{W^{-1,q}(\Omega)} \|g\|_{W^{1,q'}(\Omega)}$$ $$ \lesssim \sup_{f \in C_c^\infty(\Omega), \|f\|_{L^{q'}} = 1} \|\Delta u \|_{W^{-1,q}(\Omega)} \|f\|_{L^{q'}(\Omega)}$$ $$ = \|\Delta u \|_{W^{-1,q}(\Omega)} .$$

Note: I don't have a proof for this decomposition in general bounded domains, but it is true. In the periodic case, you can directly decompose the Fourier series $S_Nu = h_N + \nabla g_N$ and apply Calderon-Zygmund estimates to the equation $$-\Delta g_N = -\text{div}(S_Nu)$$

which after differentiating, rewrites to

$$\partial_i g_N = - \partial_i \sum_j \partial_j (-\Delta)^{-1}S_N u = -\sum_j T_{ij}(S_N u)$$

where the $T_{ij}$ are the double Riesz transforms. So if anyone has a good reference for bounded domains that would be helpful for me too...