Reference request and methods indication to the continuity of solutions to the problema $L_tu = F(u), ~t\in [0,1],$ and $L_t$ elliptic

Question

Let $M$ be a closed manifold and assume that is given a family of elliptic operators $L_t,~t\in [0,1]$ and a smooth function $F :[a,b] \to \mathbb{R}$ such that for each $t$ the elliptic problem $L_tu = F(u)$ has a classical solution $u : M \to \mathbb{R}$.

I would like to know which kind of methods and techniques could be employed to ensure that we can get a continuous family of solution in the parameter $t$ provided if $L_t$ is also continuous in this parameter.

If needed it can be assumed that the solutions $u$ are firstly obtained by variational methods (as some minimizers for a certain functional) and regularity is proved classicaly by standard procedures.

Can anyone give me some reference to this kind of analysis or, if possible, make some remarks on the general procedure?

Answer 1

This question is difficult to answer because the obstruction to continuous families of solutions is (usually) not technical but substantive, and has to do with the local uniqueness of solutions to the equation for a given $t$.

To see why, consider a simple example which illustrates the basic approach one might take: $F = 0$, $L_t(u) = \text{div} (A_t(x) \nabla u) - f(x)$, with $A_t$ uniformly elliptic and smooth in both $x$ and $t$ variables and $f$ a smooth function with $\int_M f = 0$. Then $L_t(u) = 0$ admits a unique smooth solution with mean $0$ which may be characterized by the minimization of $$ \int \frac{1}{2} A_t \nabla u \cdot \nabla u + f u. $$ Call this solution $u_t$: our goal is to show $u_t$ is continuous as a function of $t$.

The idea is that you plug $u_t$ in to the equation for $u_s$: $$ L_s(u_t) = \text{div} A_s \nabla u_t - f= \text{div} [A_s - A_t] \nabla u_t, $$ using that $L_t(u_t) = 0$ at the end. Then we just estimate the right-hand side: $$ |L_s(u_t)| \leq C \|A_s - A_t\|_{C^1}\|\nabla u_t\|_{C^1} \leq C |t - s|, $$ using that $A_t$ is smooth and the regularity of $u_t$. Then as $L_s(u_s) = 0$, $$ |\text{div} A_s \nabla (u_s - u_t)| \leq C |t - s|. $$ Now multiplying by $u_s - u_t$ and integrating by parts: $$ \int A_s \nabla (u_s - u_t) \cdot \nabla (u_s - u_t) = - \int (u_s - u_t)\text{div} A_s \nabla (u_s - u_t) \leq C | t - s|\int |u_s - u_t|. $$ Uniform ellipticity and Holder's inequality give $$ \|\nabla(u_s - u_t)\|_{L^2}^2\leq C |t - s| \|u_s - u_t\|_{L^2}, $$ so the Poincare inequality (recall $u_s, u_t$ have mean $0$) leads to $$ \|u_s - u_t\|_{W^{1,2}} \leq C |t - s|. $$ This is basically the continuity estimate we want. If the fact that it is in $W^{1,2}$ topology is concerning, we only need to go back to the equation $$ |\text{div} A_s \nabla(u_s - u_t)|\leq C |t - s| $$ and apply our elliptic regularity estimate of choice (De Giorgi-Nash, Schauder, whatever) to upgrade the norms. Many modifications and improvements are available, but this is the basic idea.

The most important ingredient in this argument was that for the solution $u_s$ of $L_s$, if $v$ has $|L_s(v) - L_s(u_s)|$ small, then $v$ is close to $u_s$. This was true here, as we checked, but this is likely to be the hardest step to generalize to nonlinear equations. Even here, note that if we did not assume that $u_s$ all have mean value $0$, the argument fails (indeed, any family $u_s + g(s)$ is still a solution to $L_s(u_s) = 0$).

If you have, e.g. unique minimizing solutions to the nonlinear problem, you can likely establish this kind of stability property for them and run a similar argument. If not, you can attempt instead to construct your family of solutions "all at once," for example viewing $L_t$ as a perturbation of $L_0$ and using fixed point arguments or implicit function theorems.