On the problem of maximal $L^q$-regularity for viscous Hamilton-Jacobi equations

For $q>2, \gamma>1$, we prove that maximal regularity of $L^q$ type holds for periodic solutions to $-\Delta u + |Du|^\gamma = f$ in $\mathbb{R}^d$, under the (sharp) assumption $q>d \frac{\gamma-1}\gamma$.


Introduction
We address here the so-called problem of maximal L q -regularity for equations of the form for all M > 0, there exists M > 0 (possibly depending on M, γ, q, d) such that Q being the d-dimensional unit cube (−1/2, 1/2) d . It is known that maximal L q -regularity cannot be expected in general (even for classical solutions) if see Remark 3.1. The validity of (M) has been conjectured to hold in the complementary regime q > d γ − 1 γ ( and q > 1 ), but, to the best to our knowledge, the problem has remained so far unsolved in general. P.-L. Lions has discussed this conjecture in a series of seminars (e.g. [6]), and during his lectures at Collège de France [5]. He indicated some special cases that can be successfully addressed. When γ = 2, the so-called Hopf-Cole transformation v = e −u reduces (1) to a semilinear equation, and (M) may be obtained for any q > d/2 using (maximal) elliptic regularity and the Harnack inequality. Ad-hoc treatments for the special cases d = 1 and q < d/(d − 1) have been discussed in [6] also. As a final suggestion, an integral version of the Bernstein method [4] could be implemented to prove (M) when q is close enough to d (see also [1] for further refinements of this technique), but the full regime (2) seems to be out of range using these sole arguments.
We develop here a new method to obtain (M) in generality, assuming only q > 2 (which is always satisfied under (2) if γ > d/(d − 2)). The proof is based on an crucial estimate of the form for any k ≥ 0, where ω {|Du| ≥ k} → 0 as k → ∞. Such an estimate is obtained starting from a classical idea of Bernstein [2], namely shifting the attention from the equation (1) for u to an equation for (a suitable power of) |Du| 2 . A strong degree of coercivity with respect to |Du| 2 itself in this equation, which stems from uniform ellipticity and the nonlinear term |Du| γ , turns out to be a key ingredient to derive (3).
Once (3)  can be then recovered up to k = 0. This second key step has been inspired by a very interesting argument that appeared in [3], which suggests that, despite the strong non-linear nature of |Du| γ in (1), some information can be extracted from the equation on sets {|u| ≥ k} (and on {|Du| ≥ k} in our case) for k large.
Our result reads as follows.

Proof of the main theorem
For the sake of brevity, we will often drop the x-dependance of u, Du, ..., and the d-dimensional Lebsesgue measure dx under the integral sign. (x) + = max{x, 0} will denote the positive part of x, and for any p > 1, p ′ = p/(p − 1). This section is devoted to the proof of Theorem 1.1, which will be based on the following lemma.
and for all k ≥ 1, We postpone the proof of the lemma, and show first how (4) yields the conclusion of Theorem 1.1.
Note that the function F : Since u ∈ C 3 (Q), the function k −→ Y k is continuous and tends to zero as k → ∞ (it eventually vanishes for k large). Hence we deduce that and finally The estimate on ∆u L p (Q) is then straightforward.
Having proven Theorem 1.1, we now come back to the main estimate (4).
2 , δ ∈ (0, 1) to be chosen later. Note that, for any δ ∈ (0, 1), g enjoys the following properties: for all s ≥ 0, Note also that . . , d and β > 1 to be chosen later as test functions in the HJ equation. First, integrating by parts and substituting w Moreover, again integrating by parts, Note also that in (8) integrating on Q and on Ω k is the same, by the presence of w k . We use first Cauchy-Schwarz inequality, the equation (1) and the inequality (a − b) 2 ≥ a 2 2 − 2b 2 for every a, b ∈ R to get Moreover, again by Cauchy-Schwarz inequality (be careful about g ′′ < 0) and (7), The above inequalities then yield Note that for γ > 1 it holds and hence, we are allowed to conclude This gives, back to (8) and substituting (1 + |Du| 2 ) where c 1 = c 1 (δ, d, γ) > 0.
We now estimate the five terms on the right hand side of the previous inequality. The first three terms are somehow similar: using Cauchy-Schwarz inequality and that 2sg ′ ≤ g ′′ , we have for some We now make some choices for the coefficients. Recalling that d γ ′ < q, we take Note d γ ′ < p < q. Assuming that p > 2 (which is always true when γ > d d−2 , otherwise see the remark at the end of the proof), we have β > 1 whenever δ is close enough to zero. Moreover, Therefore, we apply Hölder's inequality (with conjugate exponents p/2 and p/(p − 2)) and Young's inequality, and then w k ≤ w together with (12) to obtain where c 3 = c 3 (δ, d, γ, p) > 0. Plugging the previous inequality into (10) yields δ 2d The fourth term in (9) is a bit more delicate, we proceed as follows. Use first that s ∫ where c 4 = c 4 (δ, d, γ, β) > 0. Since k ≥ 1, w ≥ 1 on Ω k , hence, recalling also (12), ∫ We now focus on the fifth term in (9). By Young's inequality, Furthermore, letting Plugging the previous inequality into (16) and using again Young's inequality leads to for some c 5 = c 5 (δ, d, γ, p) > 0. Plug now (14), (15) and (17) into (9) to obtain Sobolev's inequality related to the continuous embedding of We finally choose δ > 0 small enough so that δ pq q−p < 1. Recall that p < q, so using repeatedly Hölder's and Young's inequalities we obtain Then, on B 1/2 := {|x| < 1/2}, and v ε = 0 on ∂ B 1/2 . Therefore, there exists M > 0, depending on c, d, γ only, such that Note that the example is meaningful only if γ > d d−1 , that is when d γ−1 γ > 1. Note also that though v ε is not periodic, being smooth on B 1/2 and vanishing on ∂ B 1/2 , it is straightforward to produce similar examples in the periodic setting. Finally, different choices of the truncation χ(|x|) = χ ε (|x|) lead to counterexamples in the regime q < d γ−1 γ . Remark 3.2. d = 1, 2. Theorem 1.1 is stated in dimension d ≥ 3, but the proof for d = 1, 2 follows identical lines. As it usually happens, the point is that in the latter case W 1,2 (Q) is continuously embedded into L p (Q) for all finite p ≥ 1, and not only into L 2d d−2 (Q). Remark 3.3. Less regularity of u.