I'm looking for a reference for a statement like:

Let $M$ be a $n$-dimensional smooth compact manifold with smooth boundary $\partial M$. In coordinates, let $\mathcal L$ have the form

$\mathcal L u(t,x) = \sum_{i,j=1}^n a_{i,j}(t,x)\partial_i \partial_j u(t,x) + \sum_{i=1}^n b_i(t,x) \partial_i u(t,x) + c(t,x)u(t,x)$

i.e. $\partial_t - \mathcal L$ should define a parabolic operator, assume that $\partial_t - \mathcal L$ is uniformly parabolic in the sense of being uniformly parabolic in some finite covering set of coordinate charts, and assume $a_{i,j},b_i,c \in C^\infty([0,1] \times \overline M)$, and $a_{i,j} = a_{j,i}$ (in short: as much regularity as possible, in fact I'm even ok with asking that $c \equiv 0$).

Then for sufficiently smooth $f$, the solution of the Dirichlet initial-boundary problem $ \partial_t u = \mathcal L u + f$ with initial value $ u(0,\cdot) = u_0 $ and Dirichlet boundary $u|_{\partial M} = 0$ has a unique solution, with $u \in C([0,1] \times M) \cap C^\infty((0,1] \times \overline M)$ (note that this includes boundary regularity), and moroever if $u_0|_\partial M = 0$ (in some reasonable sense) or if other compatibility conditions are fulfilled, then the solution becomes more smooth at initial time.

Having done an extensive search of the literature, I can't seem to find this statement anywhere. It seems like all the sources I can find treat this either as being too well-known to merit giving a reference (see e.g. Section 7, Remark 6.3 in Pazy's book for the case $M\subset \mathbb R^n$ ), or only consider the case where the coefficients do not depend on $t$, or reference things like Friedman/Amann's books which, if they contain such a statement at all, hide it behind an impressive mass of notation and machinery that hide the simple statement of the question. And this is just the problem with homogeneous Dirichlet boundary, taking homogeneous Neumann boundary (which I would be interested in also, but have given up trying to find a reference for at this point) is significantly more difficult.

Does anyone know of a reference that contains a statement like the above (and not of the form "just follow the the 10 page proof in [x] and make the trivial changes to adapt to time-dependence and the manifold setting, and the non-empty boundary at every step" )? I would like to use such a result, but would prefer not to spend several pages explaining how it follows from the well-known theory of parabolic PDEs ...


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.