# Definition of an elliptic operator with measurable coefficients

Let $$\mathbb{k}$$ be one of $$\mathbb{R}$$ or $$\mathbb{C}$$. Say I'm given an $$m$$-th order linear partial differential operator $$L$$ in the form of a $$\mathbb{k}$$-linear operator $$L = \sum_{\substack{\alpha \in \mathbb{N}_0^n \\ |\alpha| \leq m}} a^{\alpha}\frac{\partial^{\alpha}}{\partial x^{\alpha}}: V_1 \rightarrow V_2$$ between a pair $$V_1,V_2$$ of (possibly negatively-indexed) Sobolev spaces, where all of the coefficient functions $$a^\alpha$$ belong to the space $$\mathsf{C}^k(\mathbb{k}^n;\mathbb{k})$$ or the Sobolev space $$\mathsf{W}^{k,p}(\mathbb{k}^n;\mathbb{k})$$, and where the symbols $$\frac{\partial^\alpha}{\partial x^\alpha}$$ are interpreted as weak/distributional derivatives. My question is simply, what are the most general choices of spaces I can make for $$V_1$$ and $$V_2$$ in this case?

For instance if all $$a^\alpha$$ belong to $$\mathsf{C}^\infty(\mathbb{k}^n;\mathbb{k})$$, then I think I can take $$V_1 = V_2 = \mathscr{D}'(\mathbb{k}^n;\mathbb{k})$$, the space of real-valued distributions on $$\mathbb{k}^n$$. But if $$a^\alpha$$ are less differentiable, possibly even only in some $$\mathsf{L}^p$$-space or $$\mathsf{W}^{k,p}$$-space, then I need to change $$V_1$$ and consequently $$V_2$$ in order for the multiplication $$a^\alpha\frac{\partial^{\alpha}u}{\partial x^{\alpha}}$$ to make sense, where $$u \in V_1$$.

Is there some explicit formula for this? (E.g. given that $$a^\alpha$$ are of class-$$\mathsf{W}^{k,p}$$, then we can take $$V_1 = \mathsf{W}^{k_1,p_1}(\mathbb{k}^n;\mathbb{k})$$ and $$V_2 = \mathsf{W}^{k_2,p_2}(\mathbb{k}^n;\mathbb{k})$$ where $$k_1 = (\text{function of k and p})$$ and similarly for $$k_2,p_1,p_2$$.) Any references would really be appreciated, thanks!

I'm not sure if I can answer what the most general spaces possible are, but I can maybe offer an answer about how this might arise in context. Often we are concerned with solving an equation $$Lu = f$$ in $$U \subset \mathbb{R}^n$$, bounded and open, and $$u = 0$$ on $$\partial U$$. Let's consider a specific example, when $$L$$ is a divergence-form elliptic operator: $$Lu = -\sum_{i, j = 1}^n \partial_{x_i}(a^{ij}(x)\partial_{x_j}u(x)) + \sum_{i = 1}^n b^i(x)\partial_{x_i}u(x) + c(x) u(x),$$ or $$Lu = -\nabla \cdot (A\nabla u) + B\cdot \partial u + cu,$$ where $$A = (a^{ij})_{i,j = 1}^n$$ is symmetric, positive definite. A priori this makes sense for $$A \in W^{1, p}$$ perhaps, and $$u \in W^{2, p}$$. Compare to Laplace's equation, $$-\Delta u = 0$$; this can be seen as a generalization of this type of PDE.

It also makes sense to talk mainly about $$p = 2$$ because of the nice structure of Sobolev spaces with $$p = 2$$. Therefore a priori the operator $$L$$ makes sense for $$A \in H^1, u \in H^2, b, c \in L^\infty(U)$$. So very little regularity is assumed on the $$B$$ and $$c$$. Note also that since we're solving the Dirichlet problem ($$u = 0$$ on the boundary) we want a solution $$u$$ to actually be in $$H^1_0$$ also, i.e. has trace zero.

However, by using integration by parts we can extend our considered class of solutions $$u$$ to $$H^1_0$$ instead of $$H^2$$ (and also relax the assumptions on $$A$$). If $$v \in C_c^\infty(U)$$ has compact support in $$U$$, note that integration by parts gives that, if $$Lu = f$$ and everything is smooth, \begin{align*} \int_U fv = \int_U (Lu)v\, dx &= \int_U \sum_{i,j}a^{ij}\partial_{x_j}u\partial_{x_i}v + \sum_i b^i \partial_{x_i} u v + cuv \, dx. \end{align*} In fact the above computation pushes through for $$v \in H_0^1(U)$$. Therefore we can define a bilinear form $$B[\cdot, \cdot] : H^1_0 \times H^1_0 \to \mathbb{R}$$ by $$B[u, v] = \int_U \sum_{i,j}a^{ij}\partial_{x_j}u\partial_{x_i}v + \sum_i b^i \partial_{x_i} u v + cuv \, dx.$$ Then we should call $$u$$ a weak solution to our boundary-value problem if $$u \in H^1_0$$ and $$B[u, v] = (f, v)_{L^2(U)} = \int_U fv\, d x$$ for all $$v \in H_0^1(U)$$. Therefore we have extended our notion of solution to $$Lu = f$$ to something that a priori only has one weak derivative, and may not lie in $$H^2$$.

Elliptic regularity theory gives that $$u$$ is in fact in higher order Sobolev spaces if the coefficients and $$f$$ are assumed to be more regular.

References: Ch. 6 of Evans for 2nd order elliptic operators. Ch. 7 does hyperbolic and parabolic operators.

• Thanks very much for your answer! I'll look further into Evans' book on that section.
– Alec
Feb 27, 2020 at 4:48