I'm studying elliptic equations in divergence form
$$-D_{j}(a_{ij}(x)D_{i}u) + c(x)u = f(x) \text { in a domain } \Omega \subset \mathbb{R}^{n}$$
I call a function $u \in H^{1}(\Omega)$ a weak solution if for every $\phi \in H^1_0(\Omega)$,
$$\int_\Omega (a_{ij}D_iuD_j\phi + cU\phi) = \int_\Omega f\phi$$
where
- The coefficients $a_{ij} \in L^{\infty}$ are uniformly elliptic; that is, there exists a positive $\lambda$ such that for all $x \in \Omega$, $\xi \in \mathbb{R}^n$,
$$\sum_{i,j} a_{ij}(x) \xi_i\xi_j \ge \lambda |\xi|^2$$ - The coefficient $c \in L^{\frac{n}{2}}(\Omega)$ and the inhomogeneous term $f \in L^{\frac{2n}{n+2}}(\Omega)$.
I have two questions:
- In what conditions a weak solution is a classical solution?
- How to find an example of elliptic pde with weak solution but not classical solution?
Thanks very much!
Best Answer
For $u$ to be a classical solution, both $u$ and the coefficients in the partial differential operator $P$ in the LHS of your first formula need to be regular enough so that $P$ does not map u away from $C^0(\Omega)$, for in this case you can integrate by parts the LHS of your second formula.
The path from weak solutions to classical solutions relies on two stepping stones (both are discussed in L.C. Evans's book "Partial Differential Equations"):
Elliptic regularity: if $P$ is an elliptic partial differential operator of order $k$ on $\Omega$ with coefficients in $C^{s+k-1}(\Omega)$, $s\in\mathbb{N}$, and $u\in\mathscr{D}'(\Omega)$ satisfies $Pu\in H^s_{loc}(\Omega)$, then $u\in H^{s+k}_{loc}(\Omega)$.
Sobolev's embedding theorem ($L^2-L^\infty$ case): $H^{s+k}_{loc}(\Omega)\subset C^k(\Omega)$ whenever $s>n/2$. If the boundary of $\Omega$ is "sufficiently well behaved" (e.g. locally Lipschitz) we even have $H^{s+k}(\Omega)\subset C^k_b(\Omega)$.
Both results above imply that if $a_{ij}$ and $c$ belong to $C^{s+1}(\Omega)$ with $s>n/2$ and $f\in H^s_{loc}(\Omega)$, then $u\in H^{s+2}_{loc}(\Omega)\subset C^2(\Omega)$ and therefore is a classical solution. Counterexamples stem from the fact that the Sobolev embedding theorem is sharp in the sense that if $s\leq n/2$, then one may find $u\in H^{s+2}_{loc}(\Omega)$ which is not in $C^2(\Omega)$. Since even in this case $Pu\doteq f$ still belongs to $H^s_{loc}(\Omega)$, $u$ is (by construction) a weak solution to $Pu=f$ which clearly cannot be a classical solution. The counterexample provided in Giuseppe Negro's answer is precisely of this kind.
Your integrability hypotheses on $f$ and $c$ may improve the integrability of weak solutions but do not really help in getting more regularity for $u$ all by themselves, due to the sharpness of the $L^p-L^q$ version of the Sobolev embedding theorem for $q=2,\infty$. More precisely:
To get $l$ weak derivatives in the $L^2$ sense from $k>l$ weak derivatives in the $L^p$ sense, one needs that $p=\frac{2n}{n+2(k-l)}$. Since you have picked $p=\frac{2n}{n+2}$ for $f$ so that $k-l=1$, we need $f$ to have all weak derivatives up to order $2s+1>n+1$ in $L^p(\Omega)$ (or at least in $L^p_{loc}(\Omega)$) in order to have $f\in H^s_{loc}(\Omega)$ with $s>n/2$;
As for $c$, we need instead that $p=\frac{n}{k-l}$ in order to get $l$ classical derivatives from $k>l$ weak derivatives in the $L^p$ sense. Since you have chosen $p=\frac{n}{2}$ for $c$ so that $k-l=2$, we need $c$ to have all weak derivatives up to order $2s+3>n+3$ in $L^p(\Omega)$ (or at least in $L^p_{loc}(\Omega)$) in order to have $c\in C^{s+1}(\Omega)$ with $s>n/2$.
The above hypothesis on $f$ cannot be circumvented, by the argument discussed before. I'm not sure if the above hypothesis on $c$ is also sharp, though.