User francois monard - MathOverflow

Answer by Francois Monard for Seeking reference on regularity theory for nonlinear elliptic PDE

2012-08-02T21:38:05Z

"Elliptic Partial Differential Equations of Second Order" by David Gilbarg and Neil S. Trudinger

Answer by Francois Monard for What is the simplest oscillatory integral for which sharp bounds are unknown?

2012-08-02T19:48:52Z

As soon as the Hessian is not full rank, the problem becomes quickly messy:

if the hessian has rank $n-1$, then one can treat the one direction separately since we have explicit bound for a one-dimensional integral where the taylor expansion of $\Phi$ near a critical point $x_0$ looks like $(x-x_0)^p$ for any $p\ge 2$, the other directions will always give you $\lambda^{-\frac{1}{2}}$.
when the rank is less, then one must first identify those directions where the phase isn't quadratic, and look at the next terms in the expansion. V.I. Arnold then classifies the simple jets of functions in terms of their corresponding maximal decay in $\lambda$ in the following paper:

V.I. Arnold, Remarks on the stationary phase method and coxeter numbers, Russian Math. Surveys, 28 (1973), p. 19

See also J.J. Duistermaat, Oscillatory integrals, Lagrange immersions and unfolding of singularities, CPAM vol XXVII, 207-281 (1974)

The classification is algebraic and does not rely on estimating integrals, so it never tells you how to obtain the estimate corresponding to that optimal decay. For the simpler classes of degenerate critical points, Popov has worked on estimating the oscillatory integrals:

D.A. Popov, Estimates with constants for some classes of oscillatory integrals, Russian Math. Surveys, 52, pp. 73–145.

D.A. Popov, Remarks on uniform combined estimates oscillatory integrals with simple singularities, Izv. Math., 72, pp. 793–816.

These papers helped me for the following problem, where the oscillatory integral I had to study had an interesting degenerate behavior.

F. Monard-G. Bal "Inverse transport with isotropic time-harmonic sources", SIAM J. Math. Anal., Vol. 44, No. 1, pp. 134-161 (2012).

Along the way, another paper I found interesting for direct estimates: G.I. Arkhipov, A.A. Karatsuba, and V.N. Chubarikov, Trigonometric integrals, Izv. Akad. Nauk SSSR Ser. Mat., 43 (1979), pp. 971–1003 (in Russian); Math. USSR-Izv., 15 (1980), pp. 211–239 (in English). (I think they also have a multidimensional counterpart).

Answer by Francois Monard for C^{2} estimates for elliptic equations

2012-07-26T18:53:32Z

In [Gilbarg-Trudinger], exercise 4.9 pp. 71-72 constructs (i) an example of continuous function $f$ such that the equation $\Delta u = f$ does not have a $C^2$ solution in any neighbourhood of the origin, and (ii) an example of $u$ such that $\Delta u \in C^{1}$ but $u$ is not in $C^{2,1}$ in any neighbourhood of the origin.

Looking at $\partial_1 u$ in example (ii) would perhaps give a negative answer to this question.

I am also interested in that kind of estimate for the case where the operator is in divergence form $\sum_{i,j=1}^n \partial_i (a_{ij}(x) \partial_j u) = 0$ with no restriction on smoothness of the boundary $\partial\Omega$. Assuming the ${a_{ij}}$'s to be Lipschitz, a recent post in arXiv (http://arxiv.org/pdf/1207.4236.pdf) claims that $u\in C^{1,1}$, referring to [G-T] with no further comments. This is exactly what I need yet I cannot find it in [G-T] and it seems to me that this is again a limiting case which may have counter-examples... Any help appreciated !

Comment by Francois Monard

2012-10-19T01:08:39Z

The last row is indeed weird because it enforces $x_n = \pm 1$, which cannot be possible if, say, you pick $\vec v_1 = \vec e_2$, ... $\vec v_{n-1} = \vec e_n$, in which case the only nonzero component of $x$ is the first one. There has to be a way of enforcing the right-hand rule, i.e. $\det (\vec v_1,...,\vec v_{n-1}, \vec v_1\times ...\times \vec v_{n-1}) \ge 0$, which is indeed the missing constraint. However, writing this condition as I just wrote it would amount to computing the components of the cross-product by determinants !

Comment by Francois Monard

2012-08-16T18:03:52Z

If of any help, you can write $\theta (x) = x + f(x)$ with $f$ a $2\pi$-periodic function.

Comment by Francois Monard

2012-08-06T19:11:58Z

seems the recurrence relation linking $u_n$ with $u_{n-1}$ has an $A_0$ on the left side instead of $A_1$. From this relation, if you can show that the operator mapping $u_{n-1}$ to $u_n$ is a contraction from an appropriate space to itself, then the series is justified.

Comment by Francois Monard

2012-08-03T16:16:02Z

$W^{2,\infty}$-estimates in dimension $n\ge 2$ are not always true without assuming further regularity on the other terms, so dimension $1$ might not generalize so nicely. As regards your problem, $\tilde\theta$ may only depends on the $\theta$ and $\frac{\partial\theta}{\partial n}$ at the boundary $\partial\omega$. The values inside $\omega$ don't matter so much as long as they match with the traces I mentioned.

Comment by Francois Monard

2012-08-01T16:40:02Z

I discussed with one of the authors of the above arXiv post and he agreed that the result mentioned does not hold true, so the ambiguity is gone.

Comment by Francois Monard

2012-07-26T18:54:57Z

the example above might not be convex, though