1 Introduction

We address here the so-called problem of maximal \(L^q\)-regularity for equations of the form

$$\begin{aligned} -\Delta u(x) + |Du(x)|^\gamma =f(x)\quad \text { in } \mathbb {R}^d, \end{aligned}$$
(1)

where \(\gamma > 1\), \(f : \mathbb {R}^d \rightarrow \mathbb {R}\) is 1-periodic (i.e. \(f(x+z) = f(x)\) for all \(x \in \mathbb {R}^d\), \(z \in \mathbb {Z}^d\)), \(d \geqq 1\); that is,

figure a

Q being the d-dimensional unit cube \((-1/2,1/2)^d\). This regularity problem has been proposed by P.-L. Lions in a series of seminars and lectures (e.g. [31, 32]), where he conjectured its general validity under the assumption that

figure b

Some special cases have been addressed in these seminars, but the general problem has remained so far unsolved, to the best of our knowledge. We present here a proof of (M) + (A), under the sole restriction \(q > 2\) (which is always realized when \(\gamma > d/(d-2)\)).

Equations of the form (1) are prototypes of semilinear uniformly elliptic equations with superlinear growth in the gradient, and arise for example in ergodic stochastic control [4] and in the theory of growth of surfaces [25]. The study of regularity of their solutions has recently received a renewed interest in the theory of Mean Field Games [11, 28]. There is a vast literature on such equations and more general quasilinear problems. While the existence of classical (or strong) solutions was investigated first (see for example [3, 26, 29, 38]), the attention has since been largely focused on the existence (and uniqueness) of solutions \(u \in W^{1, \gamma }(Q)\) satisfying (1) in the weak or generalized sense (typically with Dirichlet boundary conditions; see, for example, [1, 6, 8,9,10, 14, 18, 22, 35] and more recent works [2, 7, 16, 17]. It has been observed that due to the superlinear nature of the problem, its (weak) solvability requires \(f \in L^q\), where

$$\begin{aligned} q \geqq d \frac{\gamma -1}{\gamma }. \end{aligned}$$

Such a condition has been improved in the finer scale of Lorentz-Morrey spaces, and end-point situations typically require additional smallness assumptions [19, 23]. It is worth observing that many results in the literature cover the case \(1 < \gamma \leqq 2\); that is, when the gradient term has at most natural growth. General results in the full range \(\gamma > 1\), based on methods from nonlinear potential theory, appeared quite recently in [34, 36, 37].

Roughly speaking, properties (M)+(A) say that if f belongs to a sufficiently small Lebesgue space, then solutions should enjoy much better regularity than \(W^{1,\gamma }\), namely, be in \(W^{1, q \gamma }(Q)\) (and even in \(W^{2,q}(Q)\), by standard Calderón-Zygmund theory). Still, additional gradient regularity is typically achieved via methods that require much stronger hypotheses on the summability of f, being based on the classical or weak maximum principle: viscosity theory indeed requires f to be bounded [24], while the Aleksandrov-Bakel’man-Pucci estimate needs \(f \in L^d\), as in [33]. The situation is even worse when \(\gamma > 2\), as one observes that general weak solutions are just Hölder continuous [15], so one has to select u in a suitable class.

Here, we look at solutions to (1) that can be approximated by classical ones. Therefore, we will prove (M) in the form of an a priori estimate. It is known that in such a form, (M) cannot be expected in general if

$$\begin{aligned} 1 < q \leqq d \frac{\gamma -1}{\gamma }, \end{aligned}$$

as described in Remark 1. On the other hand, P.-L. Lions indicated that (M)+(A) can be obtained in some particular cases. First, when \(\gamma = 2\), the so-called Hopf-Cole transformation \(v= e^{-u}\) reduces (1) to a linear elliptic equation, and one has the result employing (maximal) elliptic regularity and the Harnack inequality. Special cases \(d=1\) and \(\gamma < d / (d-1)\) can be also treated. As a final suggestion, an integral version of the Bernstein method [30] could be implemented to prove (M) when q is close enough to d (see also [27], and [5] for further refinements of this technique), but the full regime (A) seems to be out of range using these sole arguments.

The Bernstein method is the starting point of our work. It consists in shifting the attention from the equation (1) for u to the equation for a suitable function of \(|Du|^2\), i.e. \(w=g(|Du|^2)\); if g is properly chosen, the equation for w enjoys a strong degree of coercivity with respect to w itself, which stems from uniform ellipticity and the coercivity of the gradient term in (1). By a delicate combination of these two regularising effects, it is possible to produce a crucial estimate on superlevel sets of |Du|, i.e.

$$\begin{aligned} \left[ \int _{\{|Du| \geqq k\}} \Big (|Du| - k\Big )^{\gamma q}\right] ^{\frac{d-2}{d}} \leqq \omega \Big (\big | \{|Du| \geqq k\} \big |\Big ) + \int _{\{|Du| \geqq k\}} \Big (|Du| - k\Big )^{\gamma q}\nonumber \\ \end{aligned}$$
(2)

for any \(k \geqq 0\), where \(\omega (t) \rightarrow 0\) as \(t \rightarrow 0\). This inequality again reflects the superlinear nature of the problem, being the exponents in the two sides unbalanced. Nevertheless, it is possible to control on \(\Vert |Du|^\gamma \Vert _{L^q}\) as follows: (2) guarantees that \(\int _{\{|Du| \geqq k\}} \Big (|Du| - k\Big )^{\gamma q}\) is either belonging to a neighborhood of zero, or to an unbounded interval (for k large enough, but independent of \(\Vert |Du|^\gamma \Vert _{L^q}\)). By the fact that \(k \mapsto \int _{\{|Du| \geqq k\}} \Big (|Du| - k\Big )^{\gamma q}\) is continuous and vanishes as \(k \rightarrow \infty \), the second case can be ruled out, and boundedness of \(\int _{\{|Du| \geqq k\}} \Big (|Du| - k\Big )^{\gamma q}\) can be then recovered up to \(k = 0\). This second key step has been inspired by an interesting argument that appeared in [20] (see also [21]), where \(W^{1,2}\) estimates of (powers of) u are obtained arguing similarly on superlevel sets of |u|.

Our result reads as follows:

Theorem 1.1

Let \(f \in C^1(Q)\), \(d \geqq 3\), \(\gamma > 1 \) and

$$\begin{aligned} q> d \frac{\gamma -1}{\gamma }, \qquad q > 2 . \end{aligned}$$

For all \(M > 0\), there exists \(K = K(M, \gamma , q, d)> 0\) such that if \(u \in C^3(Q)\) is a classical solution to (1) and

$$\begin{aligned} \Vert f\Vert _{L^q(Q)} + \Vert Du\Vert _{L^1(Q)} \leqq M, \end{aligned}$$

then

$$\begin{aligned} \Vert \Delta u\Vert _{L^q(Q)} + \big \Vert |Du|^\gamma \big \Vert _{L^q(Q)} \leqq K. \end{aligned}$$

We stress that our approach is not perturbative, in the sense that the gradient term is not treated as a perturbation of a uniformly elliptic operator (which would be natural under the growth condition \(\gamma < 2\)), nor vice-versa. It applies also to equations that have the gradient term with reversed sign (since there are no sign constraints on f, just reverse \(u \mapsto -u\)), and to solutions in a strong sense (Remark 3). As far as periodicity is concerned, it is common in applications to ergodic control and Mean Field Games. The study of (M) in cases where u satisfies boundary conditions, or a local version of the estimate, will be matter of future work. We also conjecture that (M) holds in the limiting case \(q = d \frac{\gamma -1}{\gamma }\) under an additional smallness assumption on M, which controls the norm of \(\Vert f\Vert _q\). This would be coherent with known results on the existence of weak solutions. Nevertheless, it does not seem evident how to adapt our proof to cover this end-point case.

Finally, our technique does not apply to the parabolic counterpart of (M). In this direction, some results based on rather different duality methods developed in [12] to get Lipschitz regularity, have been obtained in [13].

2 Proof of the Main Theorem

\(\partial _i\), D, \(D^2\) will denote the partial derivative in the i-th direction, the gradient, and the Hessian operator respectively. For the sake of brevity, we will often drop the x-dependence of \(u, Du, \ldots \), and the d-dimensional Lebsesgue measure \(\mathrm {d} x\) under the integral sign. \((x)^+ = \max \{x,0\}\) will denote the positive part of x, and for any \(p > 1\), \(p' = p/(p-1)\). For any measurable and 1-periodic set \(\Omega \subseteq \mathbb {R}^d\), \(|\Omega |\) will be the Lebesgue measure of its representative set, i.e. \(|\Omega | = \int _{\Omega \cap Q} \mathrm {d}x\).

This section is devoted to the proof of Theorem 1.1, which will be based on the following lemma:

Lemma 2.1

There exists \(\delta \in (0,1)\) (depending on \(\gamma , q, d\)) and \(\omega : [0,+\infty ) \rightarrow [0,+\infty )\) (depending on \(M, \gamma , q, d\)) such that

$$\begin{aligned} \lim _{t \rightarrow 0} \ \omega (t) = 0, \end{aligned}$$

and for all \(k \geqq 1\),

$$\begin{aligned}&\left( \int _Q \left( \Big ( (1 + |Du|^2)^{\frac{1+\delta }{2} } - k \Big )^+\right) ^{\frac{q\gamma }{1+\delta }} \right) ^{\frac{d-2}{d}}\nonumber \\&\quad \leqq \omega \Big (|\{ 1 + |Du|^2 > k^{\frac{2}{1+\delta }} \}|\Big )+ \int _Q \left( \Big ( (1 + |Du|^2)^{\frac{1+\delta }{2} } - k \Big )^+\right) ^{\frac{q\gamma }{1+\delta }}. \end{aligned}$$
(3)

We postpone the proof of the lemma, and show first how (3) yields the conclusion of Theorem 1.1. Setting \(Y_k := \int _Q \left( \Big ( (1 + |Du|^2)^{\frac{1+\delta }{2} } - k \Big )^+\right) ^{\frac{q\gamma }{1+\delta }}\), then (3) reads as

$$\begin{aligned} Y_k^{\frac{d-2}{d}} \leqq Y_k + \omega \Big (|\{ 1 + |Du|^2 > k^{\frac{2}{1+\delta }} \}|\Big ) \qquad \text {for all } k \geqq 1. \end{aligned}$$
(4)

Note that the function \(F:Z\longmapsto Z^\frac{d-2}{d} - Z\) has a unique maximizer \(Z^*=\left( \frac{d-2}{d}\right) ^{\frac{d}{2}}\) whose corresponding value is \(F(Z^*)= F^* > 0\) (which depends on d only). For any \(0\leqq \omega < F^*\) the equation

$$\begin{aligned} F(Z)=\omega \end{aligned}$$

has two roots \(0< Z^-(\omega )<Z^*<Z^+(\omega )\). Since \(\lim _{t \rightarrow 0} \ \omega (t) = 0\), pick \(t^* = t^*(M, \gamma , p, d)\) such that \(\omega (t) < F^*\) for all \(t < t^*\). By Chebyshev’s inequality,

$$\begin{aligned} \sqrt{k^{\frac{2}{1+\delta }}-1}> \frac{\Vert Du\Vert _{L^1(Q)}}{t^*} \quad \Longrightarrow \quad |\{ 1 + |Du|^2 > k^{\frac{2}{1+\delta }} \}| < t^*, \end{aligned}$$

hence (4) yields the alternative

$$\begin{aligned} \forall k> \left( \frac{\Vert Du\Vert _{L^1(Q)}}{t^*} + 1\right) ^{\frac{1+\delta }{2}} =: k^*, \quad Y_k < Z^* \text { or }Y_k > Z^*. \end{aligned}$$

Since \(u \in C^3(Q)\), the function \(k\longmapsto Y_k\) is continuous and tends to zero as \(k\rightarrow \infty \) (it eventually vanishes for k large). Hence we deduce that

$$\begin{aligned} \forall k > k^*, \quad Y_k < Z^*, \end{aligned}$$

and finally,

$$\begin{aligned} \big \Vert |Du|^\gamma \big \Vert _{L^q(Q)}^{\frac{1+\delta }{\gamma }}= & {} \big \Vert |Du|^{1+\delta } \big \Vert _{L^{\frac{\gamma q}{1+\delta }}(Q)} \leqq \big \Vert \Big ((1+|Du|^2)^{\frac{1+\delta }{2}} - k^*\Big )^+ + k^*\big \Vert _{L^{\frac{\gamma q}{1+\delta }}(Q)} \\\leqq & {} (Z^*)^{\frac{1+\delta }{\gamma q}} + k^*. \end{aligned}$$

The estimate on \(\Vert \Delta u\Vert _{L^p(Q)}\) is then straightforward.

Having proven Theorem 1.1, we now come back to the main estimate (3).

Proof of Lemma 2.1

Let \(w(x):=g(|Du(x)|^2)\), where \(g(s) =g_\delta (s)= \frac{2}{1+\delta }(1+s)^{\frac{1+\delta }{2}}\), \(\delta \in (0,1)\) to be chosen later. Note that, for any \(\delta \in (0,1)\), g enjoys the following properties: for all \(s \geqq 0\),

$$\begin{aligned}&g'(s)s^{\frac{1}{2}} \leqq (1+s)^{\frac{\delta }{2}}, \end{aligned}$$
(5)
$$\begin{aligned}&g'(s) + 2sg''(s) \geqq \delta g'(s). \end{aligned}$$
(6)

Note also that

$$\begin{aligned} g'(|Du(x)|^2)=(1+|Du(x)|^2)^{\frac{\delta -1}{2}}=\left( \frac{\delta +1}{2}w \right) ^{\frac{\delta -1}{1+\delta }} \end{aligned}$$

(\(g, g', g''\) below will be always evaluated at \(|Du(x)|^2\)).

Define \(w_k=(w-k)^+\in W^{1,\infty }(Q)\) and set \(\Omega _k:=\{w>k\}\). We now use \(\varphi = \varphi ^{(j)} = -2\partial _j(g' \, \partial _j u \, w_k^\beta )\), \(j = 1, \ldots , d\) and \(\beta > 1\) to be chosen later as test functions in the Hamilton–Jacobi equation. First, integrating by parts and substituting \(\partial _ i w=2g'Du\cdot D\partial _i u\),

$$\begin{aligned} \sum _j \int _Q Du \cdot D\varphi= & {} -2 \sum _{i,j} \int _{Q} \partial _iu\cdot \partial _j \big (\partial _i(g'\partial _j u \,w_k^\beta ) \big )=2 \sum _{i,j} \int _{Q}\partial _{ij} u \, \partial _i(g'\partial _j u\, w_k^\beta ) \\= & {} 4\int _{Q}g'' \sum _j(Du\cdot D\partial _j u)^2w_k^\beta +\,2\int _{Q}|D^2u|^2g'w_k^\beta +\beta \int _{Q}w_k^{\beta -1}Dw_{k} \cdot D w . \end{aligned}$$

Moreover, again integrating by parts,

$$\begin{aligned} -2 \sum _j \int _{Q} |Du|^\gamma \partial _j(g'\partial _j u\, w_k^\beta )= & {} \gamma \sum _j \int _{Q}w_k^\beta |Du|^{\gamma -2} Du \cdot D\partial _j u\, 2g' \, \partial _j u \, w_k^\beta \\= & {} \gamma \int _{Q}|Du|^{\gamma -2}Du\cdot Dw \ w_k^\beta . \end{aligned}$$

Noting that \(w_k^{\beta -1} Dw = w_k^{\beta -1} Dw_k\) on Q, we end up with

$$\begin{aligned}&\beta \int _{Q}w_k^{\beta -1}|Dw_{k}|^2+\int _{Q}\Big (4g''\sum _j(Du\cdot D\partial _j u)^2+2g'|D^2u|^2\Big )w_k^\beta \nonumber \\&\qquad +\,\gamma \int _{Q}|Du|^{\gamma -2}Du\cdot Dw_k\ w_k^\beta \nonumber \\&\quad =-2\int _{Q}f\mathrm {div}(g'Du\, w_k^\beta ) . \end{aligned}$$
(7)

Note also that in (7) integrating on Q and on \(\Omega _k\) is the same, by the fact that \(w_k\) vanishes on \(Q {\setminus } \Omega _k\). We use first Cauchy–Schwarz inequality, the equation (1) and the inequality \((a-b)^2\geqq \frac{a^2}{2}-2b^2\) for every \(a,b\in \mathbb {R}\) to get

$$\begin{aligned} |D^2u|^2\geqq \frac{1}{d}(\Delta u)^2\geqq \frac{1}{2d}|Du|^{2\gamma }-\frac{2}{d}f^2. \end{aligned}$$

Moreover, again by Cauchy–Schwarz inequality (be careful about \(g'' < 0\)) and (6),

$$\begin{aligned} g'|D^2u|^2+2g''\sum _j(Du\cdot D\partial _j u)^2\geqq (g' + 2|Du|^2g'') |D^2u|^2 \geqq \delta g' |D^2u|^2. \end{aligned}$$

The above inequalities then yield

$$\begin{aligned} 2g'|D^2u|^2+4g''\sum _j(Du\cdot D\partial _j u)^2\geqq \delta g' |D^2u|^2+\frac{\delta }{2d}|Du|^{2\gamma }g'-\frac{2\delta }{d}f^2g'. \end{aligned}$$

Note that for \(\gamma >1\) it holds that

$$\begin{aligned} (1+|Du|^2)^\gamma \leqq 2^{\gamma -1}(1+|Du|^{2\gamma }), \quad \text {so} \quad |Du|^{2\gamma }\geqq \frac{(1+|Du|^2)^\gamma }{2^{\gamma -1}}-1, \end{aligned}$$

and hence, we are allowed to conclude

$$\begin{aligned} \frac{\delta }{2d}|Du|^{2\gamma }g' \geqq \frac{\delta }{2^\gamma d}(1+|Du|^2)^\gamma g'-\frac{\delta }{2d} g' = \frac{\delta }{2^\gamma d}(1+|Du|^2)^{\gamma + \frac{\delta -1}{2}} -\frac{\delta }{2d}g' . \end{aligned}$$

This gives, going back to (7) and substituting \((1+|Du|^2)^{\frac{1}{2}}=\left( \frac{\delta +1}{2}w\right) ^{\frac{1}{1+\delta }}\),

$$\begin{aligned}&\beta \int _{\Omega _k}w_k^{\beta -1}|Dw_{k}|^2+\delta \int _{\Omega _k} g'w_k^\beta |D^2u|^2+ c_1 \int _{\Omega _k}w^{\frac{2\gamma + \delta -1}{1+\delta }} w_k^\beta \nonumber \\&\quad \leqq \frac{\delta }{2d} \int _{\Omega _k} (1+4f^2)g ' w_k^\beta -2\int _{\Omega _k}f\Delta u \, g' w_k^\beta -4\int _{\Omega _k}fg''Du \cdot (D^2u \, Du ) w_k^\beta \nonumber \\&\qquad -2\beta \int _{\Omega _k} fg'Du\cdot Dw_k w_k^{\beta -1} - \gamma \int _{\Omega _k}|Du|^{\gamma -2}Du\cdot Dw_k\ w_k^\beta , \end{aligned}$$
(8)

where \(c_1 = c_1(\delta , d, \gamma ) > 0\).

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 \(2sg''\leqq g'\), we have for some \(c_2 = c_2(\delta ,d) > 0\) that

$$\begin{aligned}&\frac{\delta }{2d} \int _{\Omega _k} (1+4f^2)g ' w_k^\beta -2\int _{\Omega _k}f\Delta u \, g' w_k^\beta -4\int _{\Omega _k}fg''Du \cdot (D^2u \, Du ) w_k^\beta \nonumber \\&\quad \leqq \frac{\delta }{2d} \int _{\Omega _k} (1+4f^2)g ' w_k^\beta + (2d+2) \int _{\Omega _k}|f|\, |D^2u|g' w_k^\beta \nonumber \\&\quad \leqq \delta \int _{\Omega _k}|D^2 u|^2g'w_k^\beta + c_2 \int _{\Omega _k}(1+f^2)w^{\frac{\delta -1}{1+\delta }}w_k^\beta . \end{aligned}$$
(9)

At this stage, we make some choices for the coefficients. Recalling that \( \frac{d}{\gamma '} < q\), we take

$$\begin{aligned} p=\frac{2}{d} \frac{d}{\gamma '}+\frac{d-2}{d} q, \qquad \text {and} \quad \beta =\frac{1}{1+\delta }[\gamma (p-2)+1-\delta ]. \end{aligned}$$
(10)

Note that \(\frac{d}{\gamma '}< p < q\). Assuming that \(p > 2\) (which is always true when \(\gamma >\frac{d}{d-2}\), otherwise see the remark at the end of the proof), we have \(\beta > 1\) whenever \(\delta \) is close enough to zero. Moreover,

$$\begin{aligned} \frac{2\gamma +\delta -1}{1+\delta }= & {} \frac{\delta -1}{1+\delta }\frac{p}{p-2}+\beta \frac{2}{p-2}, \end{aligned}$$
(11)
$$\begin{aligned} (\beta +1)\frac{d}{d-2}= & {} \frac{\gamma q}{1+\delta }. \end{aligned}$$
(12)

Therefore, we apply Hölder’s inequality (with conjugate exponents p/2 and \(p/(p-2)\)) and Young’s inequality, and then \(w_k\leqq w\) together with (11) to obtain

$$\begin{aligned}&c_2 \int _{\Omega _k} (1+f^2)w^{\frac{\delta -1}{1+\delta }}w_k^\beta \leqq c_2 \left( \int _{\Omega _k}(1+f^2)^{\frac{p}{2}}\right) ^{\frac{2}{p}}\left( \int _{\Omega _k}w^{\frac{\delta -1}{1+\delta }\frac{p}{p-2}}w_k^{\beta \frac{p}{p-2}}\right) ^{1-\frac{2}{p}}\\&\quad \leqq c_3 \int _{\Omega _k}(1+|f|)^p+\frac{c_1}{3} \int _{\Omega _k}w^{\frac{\delta -1}{1+\delta }\frac{p}{p-2}} w_k^{\beta \frac{2}{p-2}} w_k^\beta \\&\quad \leqq c_3\int _{\Omega _k}(1+|f|)^p+ \frac{c_1}{3}\int _{\Omega _k}w^{\frac{\delta -1}{1+\delta }\frac{p}{p-2}+\beta \frac{2}{p-2}}w_k^\beta \\&\quad \leqq c_3 \int _{\Omega _k}(1+|f|)^p+ \frac{c_1}{3} \int _{\Omega _k}w^{\frac{2\gamma +\delta -1}{1+\delta }}w_k^\beta , \end{aligned}$$

where \(c_3 = c_3(\delta , d, \gamma , p) > 0\). Plugging the previous inequality into (9) yields

$$\begin{aligned}&\frac{\delta }{2d} \int _{\Omega _k} (1+f^2)g ' w_k^\beta -2\int _{\Omega _k}f\Delta u \, g' w_k^\beta -4\int _{\Omega _k}fg''Du \cdot (D^2u \, Du ) w_k^\beta \nonumber \\&\quad \leqq \delta \int _{\Omega _k}|D^2 u|^2g'w_k^\beta + c_3 \int _{\Omega _k}(1+|f|)^p+ \frac{c_1}{3} \int _{\Omega _k}w^{\frac{2\gamma +\delta -1}{1+\delta }}w_k^\beta . \end{aligned}$$
(13)

The fourth term in (8) is a bit more delicate. We first use that \(s^{\frac{1}{2}}g'(s)\leqq (1+s)^{\frac{\delta }{2}}\), in conjection with Hölder’s and Young’s inequality, to get

$$\begin{aligned}&2\beta \int _{\Omega _k}fg'Du\cdot Dw_k w_k^{\beta -1}\leqq 2\beta \int _{\Omega _k}|f|(1+|Du|^2)^{\frac{\delta }{2}}|Dw_k| w_k^{\beta -1} \\&\quad \leqq 2\beta \left( \int _{\Omega _k}w_k^{\beta -1}|Dw_k|^2\right) ^{\frac{1}{2}} \left( \int _{\Omega _k}|f|^q\right) ^{\frac{1}{q}}\\&\qquad \left( \int _{\Omega _k}(1+|Du|^2)^{\frac{\delta }{2}\frac{pq}{q-p}}\right) ^{\frac{q-p}{pq}} \left( \int _{\Omega _k}w_k^{(\beta -1)\frac{p}{p-2}}\right) ^{\frac{p-2}{2p}} \\&\quad \leqq \frac{\beta }{3}\int _{\Omega _k}w_k^{\beta -1}|Dw_k|^2+\frac{c_1}{3}\int _{\Omega _k} w_k^{(\beta -1)\frac{p}{p-2}} +c_4\left( \int _{\Omega _k}|f|^q\right) ^{\frac{p}{q}}\\&\qquad \left( \int _{\Omega _k}(1+|Du|^2)^{\frac{\delta }{2}\frac{p q}{q-p}}\right) ^{\frac{q-p}{q}}, \end{aligned}$$

where \(c_4 = c_4(\delta ,d,\gamma ,\beta ) > 0\). Since \(k \geqq 1\), we have \(w \geqq 1\) on \(\Omega _k\). Hence, recalling also (11),

$$\begin{aligned} \int _{\Omega _k} w_k^{(\beta -1)\frac{p}{p-2}}= & {} \int _{\Omega _k}w_k^{\beta \frac{2}{p-2}-\frac{p}{p-2}} w_k^{\beta } \leqq \int _{\Omega _k}w^{\beta \frac{2}{p-2}-\frac{p}{p-2}} w_k^{\beta } \leqq \int _{\Omega _k} w^{\beta \frac{2}{p-2}-\frac{1-\delta }{1+\delta }\frac{p}{p-2}} w_k^{\beta }\\= & {} \int _{\Omega _k}w^{\frac{2\gamma +\delta -1}{1+\delta }}w_k^\beta , \end{aligned}$$

so

$$\begin{aligned}&2\beta \int _{\Omega _k}fg'Du\cdot Dw_k w_k^{\beta -1} \nonumber \\&\quad \leqq \frac{\beta }{3}\int _{\Omega _k}w_k^{\beta -1}|Dw_k|^2+\frac{c_1}{3} \int _{\Omega _k}w^{\frac{2\gamma +\delta -1}{1+\delta }}w_k^\beta \nonumber \\&\quad \quad +c_4\Vert f\Vert _{L^q(Q)}^p \left( \int _{\Omega _k}(1+|Du|^2)^{\frac{\delta }{2}\frac{p q}{q-p}}\right) ^{\frac{q-p}{q}}. \end{aligned}$$
(14)

We now focus on the fifth term in (8). By Young’s inequality,

$$\begin{aligned} - \gamma \int _{\Omega _k}|Du|^{\gamma -2}Du\cdot Dw_k\ w_k^\beta \leqq \frac{3 \gamma ^2}{4 \beta } \int _{\Omega _k}|Du|^{2\gamma -2}\ w_k^{\beta +1} + \frac{\beta }{3} \int _{\Omega _k}|Dw_k|^2 \ w_k^{\beta -1}. \end{aligned}$$
(15)

Furthermore, letting

$$\begin{aligned} \eta =\frac{2\gamma +\delta -1}{1+\delta }, \qquad \Big (\, \text { so that } \beta + \eta = \frac{p\gamma }{1+\delta }\, \Big ), \end{aligned}$$

we get (it holds that \(s^{\frac{1}{2}} \leqq g^{\frac{1}{1+\delta }}\))

$$\begin{aligned} \int _{\Omega _k}|Du|^{2\gamma -2}\ w_k^{\beta +1}\leqq & {} \int _{\Omega _k}w^{\frac{2\gamma -2}{1+\delta }}w_{k}^{\beta +1}=\int _{\Omega _k} w^{\eta -1}w_{k}^{\beta /\eta '}w_k^{\beta /\eta +1}\\\leqq & {} \left( \int _{\Omega _k} w^\eta w_k^\beta \right) ^{\frac{1}{\eta '}}\left( \int _{\Omega _k} w_k^{\beta +\eta }\right) ^{\frac{1}{\eta }}. \end{aligned}$$

Plugging the previous inequality into (15) and using Young’s inequality again leads to

$$\begin{aligned} - \gamma \int _{\Omega _k}|Du|^{\gamma -2}Du\cdot Dw_k\ w_k^\beta\leqq & {} \frac{c_1}{3} \int _{\Omega _k} w^{\frac{2\gamma +\delta -1}{1+\delta }} w_k^\beta + c_5\int _{\Omega _k} w_k^{\frac{p\gamma }{1+\delta }} \nonumber \\&+ \frac{\beta }{3} \int _{\Omega _k}|Dw_k|^2 \ w_k^{\beta -1} \end{aligned}$$
(16)

for some \(c_5 = c_5(\delta , d, \gamma , p) > 0\).

Plug now (13), (14) and (16) into (8) to obtain

$$\begin{aligned} \frac{\beta }{3}\int _{\Omega _k}w_k^{\beta -1}|Dw_{k}|^2\leqq & {} c_3 \int _{\Omega _k}(1+|f|)^p +c_4\Vert f\Vert _{L^q(Q)}^p \left( \int _{\Omega _k}(1+|Du|^2)^{\frac{\delta }{2}\frac{p q}{q-p}}\right) ^{\frac{q-p}{q}}\nonumber \\&+ c_5\int _{\Omega _k} w_k^{\frac{p\gamma }{1+\delta }}. \end{aligned}$$
(17)

Sobolev’s inequality, related to the continuous embedding of \(W^{1,2}(Q)\) into \(L^{\frac{2d}{d-2}}(Q)\), gives (for \(c_6 = c_6 (d, \delta , \gamma , p)\))

$$\begin{aligned} \frac{\beta }{3}\int _{Q} w_k^{\beta -1}|Dw_{k}|^2\geqq c_6 \left( \int _{Q}w_k^{(\beta +1)\frac{d}{d-2}}\right) ^{\frac{d-2}{d}} - \frac{\beta }{3} \int _{Q}w_k^{\beta +1}, \end{aligned}$$

hence

$$\begin{aligned}&c_6 \left( \int _{\Omega _k}w_k^{(\beta +1)\frac{d}{d-2}}\right) ^{\frac{d-2}{d}} \\&\quad \leqq c_3 \int _{\Omega _k}(1+|f|)^p +c_4\Vert f\Vert _{L^q(Q)}^p \left( \int _{\Omega _k}(1+|Du|^2)^{\frac{\delta }{2}\frac{p q}{q-p}}\right) ^{\frac{q-p}{q}} \\&\quad \quad +\, c_5\int _{\Omega _k} w_k^{\frac{p\gamma }{1+\delta }} + \frac{\beta }{3} \int _{\Omega _k}w_k^{\beta +1}. \end{aligned}$$

We finally choose \(\delta > 0\) small enough so that \(\delta \frac{p q}{q-p} < 1\). Recall that \(p < q\), so, using Hölder’s and Young’s inequalities repeatedly, we obtain

$$\begin{aligned} c_3 \int _{\Omega _k}(1+|f|)^p&\leqq c_3 \Vert 1+|f|\Vert _{L^q(Q)}^p |\Omega _k|^{\frac{q-p}{q}}, \\ c_4\Vert f\Vert _{L^q(Q)}^p \left( \int _{\Omega _k}(1+|Du|^2)^{\frac{\delta }{2}\frac{p q}{q-p}}\right) ^{\frac{q-p}{q}}&\leqq c_4 \Vert f\Vert _{L^q(Q)}^p \Big \Vert \sqrt{1 + |Du|^2} \Big \Vert _{L^1(Q)}^{\delta p} |\Omega _k|^{(1- \delta \frac{p q}{q-p} )\frac{q- p}{q}}, \\ c_5\int _{\Omega _k} w_k^{\frac{p\gamma }{1+\delta }}&\leqq \frac{c_6}{2}\int _{\Omega _k} w_k^{\frac{q\gamma }{1+\delta }} + c_7 |\Omega _k|, \\ \frac{\beta }{3}\int _{\Omega _k}w_k^{\beta +1}&\leqq \frac{c_6}{2} \int _{\Omega _k}w_k^{(\beta +1)\frac{d}{d-2}} + c_8 |\Omega _k|. \end{aligned}$$

Recalling that \((\beta +1)\frac{d}{d-2} = \frac{q\gamma }{1+\delta }\) and \(\Vert f\Vert _{L^q(Q)} + \Vert Du\Vert _{L^1(Q)} \leqq M\), we obtain

$$\begin{aligned}&\left( \int _{Q}w_k^{\frac{q\gamma }{1+\delta }} \right) ^{\frac{d-2}{d}} \leqq \int _{Q} w_k^{\frac{q\gamma }{1+\delta }} +\frac{c_3}{c_6} \Vert 1+|f|\Vert _{L^q(Q)}^p |\Omega _k|^{\frac{q-p}{q}} \\&\qquad + \frac{c_4}{c_6} \Vert f\Vert _{L^q(Q)}^p \Big \Vert \sqrt{1 + |Du|^2} \Big \Vert _{L^1(Q)}^{\delta p} |\Omega _k|^{\left( 1 - \delta \frac{p q}{q-p} \right) \frac{q-p}{p}} + \frac{c_7+c_8}{c_6} |\Omega _k| \nonumber \\&\quad \leqq \int _{Q} w_k^{\frac{q\gamma }{1+\delta }} \quad + \quad \underbrace{ \frac{c_3}{c_6} (1+M)^p |\Omega _k|^{\frac{q-p}{q}} + \frac{c_4}{c_6} M^p (1+M)^{\delta p} |\Omega _k|^{\left( 1 - \delta \frac{p q}{q-p} \right) \frac{q-p}{p}} + \frac{c_7+c_8}{c_6} |\Omega _k| }_{=: \ \omega (|\Omega _k|)}. \end{aligned}$$

Replacing \(w_k\) by its definition provides the assertion (up to an additional constant in front of \(\omega \)).

If the choice of p in (10) does not satisfy \(p > 2\), just pick \({\tilde{p}}\) so that \(p< {\tilde{p}} < q\) and \({\tilde{p}} > 2\), and proceed in the same way. Then, (12) becomes

$$\begin{aligned} (\beta +1)\frac{d}{d-2} > \frac{\gamma q}{1+\delta }, \end{aligned}$$
(18)

so it suffices once again to apply Hölder’s and Young’s inequalities to get the same assertion (with an additional term in \( \omega \)). \(\square \)

3 Further Remarks

Remark. 1

General failure of (M) when \(q \leqq d \frac{\gamma -1}{\gamma }\). In the critical case \(q = d \frac{\gamma -1}{\gamma }\) one may consider the family of functions \(v_\varepsilon \) defined as follows for \(\varepsilon \in (0,1]\): let \(\chi \in C^\infty _0\big ((1,+\infty )\big )\) be a non-negative cutoff function, \(\chi \equiv 1\) on \([2,+\infty )\), and \(v_\varepsilon (x) = v_\varepsilon (|x|)\), where

$$\begin{aligned} v_\varepsilon (r) = c \int _r^{1/2} s^{-\frac{1}{\gamma -1}}\chi \Big (\frac{s}{\varepsilon }\Big )ds, \qquad |c|^{\gamma } = - \left( d-1 - \frac{1}{\gamma -1}\right) c. \end{aligned}$$

Then, on \(B_{1/2} := \{|x| < 1/2\}\),

$$\begin{aligned} -\Delta v_\varepsilon + |Dv_\varepsilon |^\gamma= & {} \frac{c}{\varepsilon }|x|^{-\frac{1}{\gamma -1}}\chi '(\varepsilon ^{-1}|x|) + |c|^\gamma \big (\chi ^\gamma (\varepsilon ^{-1}|x|)\\&- \chi (\varepsilon ^{-1}|x|)\big )|x|^{-\frac{\gamma }{\gamma -1}}\quad =:f_\varepsilon (x), \end{aligned}$$

and \(v_\varepsilon = 0\) on \(\partial B_{1/2}\). Therefore, there exists \({\overline{M}} > 0\), depending on \(c, d, \gamma , \chi \) only, such that

$$\begin{aligned}&\Vert f_\varepsilon \Vert _{L^{d \frac{\gamma -1}{\gamma }}(B_{1/2})} = {\overline{M}} \ \ \text { for all } \varepsilon \in (0,1/4],\\&\quad \text { but } \big \Vert |Du|^\gamma \big \Vert _{L^{d \frac{\gamma -1}{\gamma }}(B_{1/2})} \rightarrow \infty \ \ \text { as } \varepsilon \rightarrow 0. \end{aligned}$$

Note that the example is meaningful only if \(\gamma > \frac{d}{d-1}\), that is when \(d \frac{\gamma -1}{\gamma }> 1\). Note also that though \(v_\varepsilon \) is not periodic, being smooth on \(B_{1/2}\) and vanishing on \(\partial B_{1/2}\), it is straightforward to produce similar examples in the periodic setting. Finally, different choices of the truncation \(\chi (|x|) = \chi _\varepsilon (|x|)\) lead to counterexamples in the regime \(q < d \frac{\gamma -1}{\gamma }\).

Note however that existence of weak solutions to the viscous Hamilton–Jacobi equation (1) can be obtained when \(f \in L^q(Q)\) and \(q = d \frac{\gamma -1}{\gamma }\) (at least for the Dirichlet problem), provided that \(\Vert f\Vert _{L^q}\) is small, see e.g. [18, 20]. Therefore, we do not exclude that (M) holds even when \(q = d \frac{\gamma -1}{\gamma }\), under extra smallness assumptions on \(\Vert f\Vert _{L^q}\).

Remark. 2

\(d=1,2\). Theorem 1.1 is stated in dimension \(d \geqq 3\), 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 \(p \geqq 1\), and not only into \(L^{\frac{2d}{d-2}}(Q)\).

Remark. 3

Less regularity of u. Theorem 1.1 holds more in general for (strong) solutions \(u \in W^{2,q} \cap W^{1, \gamma q}(Q)\) of the equation. Indeed, consider a sequence \(\psi _\varepsilon \) of standard compactly supported regularizing kernels, and observe that \(u_\varepsilon = u \star \psi _\varepsilon \) satisfies

$$\begin{aligned} -\Delta u_\varepsilon + |Du_\varepsilon |^\gamma =f \star \psi _\varepsilon + |Du_\varepsilon |^\gamma - |Du|^\gamma \star \psi _\varepsilon . \end{aligned}$$

For \(0 < \varepsilon \leqq \varepsilon _0\),

$$\begin{aligned} \Vert f \star \psi _\varepsilon + |Du_\varepsilon |^\gamma - |Du|^\gamma \star \psi _\varepsilon \Vert _{L^q(Q)} + \Vert Du_\varepsilon \Vert _{L^1(Q)} \leqq M+1, \end{aligned}$$

so applying Theorem 1.1 to \(u_\varepsilon \) and passing to the limit \(\varepsilon \rightarrow 0\) yields

$$\begin{aligned} \Vert \Delta u\Vert _{L^q(Q)} + \big \Vert |Du|^\gamma \big \Vert _{L^q(Q)} \leqq K(M+1, \gamma , q, d ). \end{aligned}$$

More generally, Theorem 1.1 continues to hold for solutions that can be obtained as limits of smooth approximations.

Remark. 4

More general Hamiltonians. Theorem 1.1 can be easily generalized to more general equations of the form

$$\begin{aligned} -\Delta u + H(Du)=f, \end{aligned}$$
(19)

where \(H : \mathbb {R}^d \rightarrow \mathbb {R}\) satisfies, e.g.,

$$\begin{aligned} \Big | H(r) - c_1 |r|^\gamma \Big | \leqq c_2 \qquad \text {for all } r \in \mathbb {R}^d \end{aligned}$$

for some \(c_1, c_2 \in \mathbb {R}\) and \(\gamma > 1\). Indeed, any u solving (19) also solves

$$\begin{aligned} -\Delta u + c_1|Du|^\gamma =f + f_H, \qquad f_H = c_1|Du|^\gamma - H(Du). \end{aligned}$$

Since \(\Vert f + f_H\Vert _{L^q(Q)} \leqq \Vert f \Vert _{L^q(Q)} + c_2\), and \(c_2\) does not depend on u, it suffices to apply Theorem 1.1 (which is easily proven to hold for any \(c_1 \in \mathbb {R}\)) with \(f + f_H\).