Elliptic Differential Equations with rough boundary data

Question

Question stated roughly: Consider the Dirichlet problem for an elliptic equation on a ball. How much can we say about regularity at the boundary of non-linear elliptic equations? Further, how can one approach solving a quasi-linear or non-linear elliptic Dirichlet problem with rough data?

Evidence: Existence of Poisson kernels guarantee existence of solutions to the Laplace equation even with rough boundary. In the interior nevertheless, we have infinite smoothness. It's true that laplacian has all the nice properties, nevertheless, one might hope to be able to obtain similar, but possibly weaker, results in more general classes of equations.

The case of linear equations: Let us consider the case of a ball, $B$. In case we have a linear equation in the `divergence' form, the usual Hilbert space approach theory we get existence of solutions so long as the boundary data lies in the image of the trace operator, $\operatorname{Tr}: H^1(B)\rightarrow L^2(\partial B)$. Equivalently, if the boundary data can be extended to an $H^1$ function in the interior, the standard theory yields a generalised solution in $H^1$.

Nevertheless, if the equation is not in divergence form, this approach will not work. Therefore, one might restrict oneself to the case where the standard Schauder theory works: divergence equations with co-efficients in a H\"older space. The problem with the conventional Schauder theory is that it only works when the boundary data is $C^{2,\alpha}$. What I have been able to find has been an article by Gilbarg and H\"ormander which extends the regularity and existence results to less regular boundary data (actually with less restriction even on the co-efficients).

Question: Are there major approaches other than weighted H\"older spaces to this problem? In case the equation is fully non-linear, say $F(x,D^2 u)=0$, is it reasonable to expect, under certain structural conditions on the operator, to be able to prove existence and regularity in appropriate weighted H\"older spaces? Are there any major results in these directions?

Answer 1

For constant coefficient linear equations, the question of regularity and existence is nicely answered by representation formulas such as the Poisson kernel, as you mentioned. These formulas tell us that on the interior, solutions are nice and smooth and we have interior derivative estimates which get worse as we approach the boundary; in fact, if we only have continuous boundary data we expect the $k^{th}$ derivatives of harmonic functions to blow up at worst like $dist(x,\partial\Omega)^{-k}$. Furthermore, if we have some regularity of boundary and boundary data, we can only expect to have this much regularity (plus 2 derivatives) for $u$ at the boundary.

I'll try to convince you that this phenomenon holds for a large class of equations:

Another approach is to separate existence theory from regularity theory. A natural method for proving existence of viscosity solutions to fully nonlinear elliptic equations $F(D^2u,Du,u,x)$ with merely continuous boundary data is Perron's method, which "algorithmically" produces a solution continuous up to the boundary by taking higher and higher subsolutions. Perron's method will work (roughly) for any nondivergence equation with a maximum principle. Of course, F must satisfy some structure conditions (probably uniform ellipticity, some regularity in $Du$ like Lipschitz, monotonicity in u and some regularity in x).

Once a solution is produced via Perron's method, if we have continuous boundary data again we can only hope for some interior estimates which get worse as we approach the boundary. In general, the best regularity one can hope for is $C^{1,\alpha}$ on the interior, which follows from the Krylov-Safonov Harnack inequality. If $F$ satisfies some structure conditions, we can do better; This is exactly the case for concave uniformly elliptic equations. The Evans-Krylov theorem gives an interior $C^{2,\alpha}$ estimate $\|u\|_{C^{2,\alpha}(B_1)} \leq C\|u\|_{L^{\infty}(B_2)}$ for viscosity solutions. One sees that this estimate gets worse near the boundary by rescaling it in smaller balls near the boundary. I believe Nadirashvili has produced a counterexample to $C^{1,1}$ regularity for general uniformly elliptic equations.

So, we can always produce a $C^{1,\alpha}$ solution to a uniformly elliptic fully nonlinear equation continuous up to the boundary. The next question is, if the boundary and boundary data have some regularity, does $u$ inherit this regularity up to the boundary? To answer this question, I only know the method of continuity: establish a $C^{2,\alpha}$ apriori estimate up the the boundary and solve the equation by first solving a simpler equation and perturbing towards the equation of interest, using the apriori estimate to make sure I can perturb the whole way. For concave equations, with $C^3$ boundary and $C^3$ boundary data, one can do this with the help of Evans-Krylov and a boundary Harnack inequality (see Fully Nonlinear Elliptic Equations by Caffarelli-Cabre for details).

It is very interesting that $C^3$ boundary data is actually optimal for the Monge-Ampere equation $\det(D^2u) = f$; there are examples of solutions with $C^{2,1}$ boundary data whose normal second derivatives blow up near the boundary!

Hope this helps!