Interior regularity of space derivatives to an evolutionary, symmetric $\varphi$-Laplacian

Abstract

We consider the Orlicz-growth generalization to the evolutionary p-Laplacian and to the evolutionary symmetric p-Laplacian. We derive the spatial second-order Caccioppoli-type estimate for a local weak solution to these systems. Our result is new even for the p-case.

Appendix: Orlicz growths

Here we gather some details on Orlicz structure. In principle, these are known facts, put together for reader’s convenience. The exceptions, in the sense that we could not find the references, may be: Proposition 1 (saying that the property ${\varphi }' (t) \sim t {\varphi }''(t)$ implies $\Delta _2$ for ${\varphi }$ and ${\varphi }^*$) and our version of Korn’s inequality (Lemma 6). It seems that these facts were never stated explicitly, even though they seem quite straightforward. By the end of this Appendix, we recall the standard examples for Orlicz growths that are admissible in our main results.

The standard sources for the theory of Orlicz spaces are monographs of Musielak [33] and of Rao and Ren [36, 37]. However for parts of the theory missing here it is more straightforward to follow monograph [31] by Málek, Nečas, Rokyta and Růžička and paper of Diening and Ettwein [12], because they are PDEs-oriented. Let us begin with

Definition 5

A real function ${\varphi }: \mathbb {R}_+ \rightarrow \mathbb {R}_+$ is an ${\mathcal {N}}$ -function iff there exists ${\varphi }'\!: \mathbb {R}_+ \rightarrow \mathbb {R}_+$

• that is right-continuous, non-decreasing,

• that satisfies ${\varphi }' (0) =0$, ${\varphi }' (t) >0$ for $t>0$ and ${\varphi }' (+ \infty ^-) =+ \infty$,

such that

$$\begin{aligned} {\varphi }(t) = \int _0^t {\varphi }' (s) \, ds \end{aligned}$$

(53)

We use here the nomenclature non-decreasing and increasing.

The preceding definition agrees with that of a Young function in Section 1.2.5 of [31] or with Definition 2.1, [15].

Now we introduce

Definition 6

A complementary function to an ${\mathcal {N}}$-function ${\varphi }$ is

$$\begin{aligned} {\varphi }^* (t) := \int _0^t ({\varphi }')^{-1} (s) \, ds \end{aligned}$$

Lemma 5

(good ${\varphi }$ growth and Young’s inequality) Assume that a family $\Psi$ of ${\mathcal {N}}$-functions and their conjugates $(\Psi = \Phi \cup \Phi ^* )$ satisfies common $\Delta _2$ condition. Then it holds for any $\varphi \in \Psi$ and any $t, a, b \in \mathbb {R}_+$

$$\begin{aligned} t \varphi ' (t) \sim \varphi (t), \end{aligned}$$

(54)

$$\begin{aligned} \varphi ^*( \varphi ' (t) ) \sim \varphi (t), \end{aligned}$$

(55)

where all the constants depend only on the magnitude of $\Delta _2 (\Psi )$, thus are uniform for the family $\Psi$.

Moreover, for any $\delta >0$

$$\begin{aligned} ab \le \delta {\varphi }(a) + C(\delta ) {\varphi }^* (b), \end{aligned}$$

(56)

where $C(\delta )$ depends only on $\delta$ an the magnitude of $\Delta _2 (\Psi )$, thus is uniform for the family $\Psi$.

The $\Delta _2$ condition for ${\varphi }$ does not imply the $\Delta _2$ for ${\varphi }^*$. So it is a little surprise that one has

Proposition 1

(good ${\varphi }'$ property implies other properties) Take ${\mathcal {N}}$-function ${\varphi }$ that has good ${\varphi }'$ property. Then ${\varphi }$ satisfies also the $\Delta _2$-condition and it holds $\Delta _2 ({\varphi }) \le C \left( G ({\varphi }') \right)$. Moreover, the complementary function ${\varphi }^*$ also enjoys good $({\varphi }^*)'$ property and $G (({\varphi }^*)') \le C( G ({\varphi }'))$. Hence also $\Delta _2 ({\varphi }^*) \le C \left( G ({\varphi }') \right)$.

Proof

First we show that $\Delta _2 ({\varphi }) \le C \left( G ({\varphi }') \right)$. Formula (1) and Definition 5 give for $s> 0$

$$\begin{aligned} \frac{ \varphi '' (s) }{\varphi ' (s)} \le \left( G ({\varphi }')s \right) ^{-1} \end{aligned}$$

so we have after integration over $(\tau , 2\tau )$

$$\begin{aligned} \ln \left( \varphi ' (2\tau ) \right) \le \ln \left( \varphi ' (\tau ) \right) + \ln 2^\frac{1}{G ({\varphi }') } = \ln \left( \varphi ' (\tau ) 2^\frac{1}{G ({\varphi }') } \right) . \end{aligned}$$

This by monotonicity of $\ln$ gives

$$\begin{aligned} \varphi ' (2 \tau ) \le \varphi ' ( \tau ) 2^\frac{1}{G ({\varphi }') } \end{aligned}$$

which after integration over (0, t) gives the non-trivial part of $\Delta _2$ for ${\varphi }$. Next we focus on the complementary function ${\varphi }^*$. It is an ${\mathcal {N}}$-function and by its definition, see Definition , $({\varphi }^*(t))' = ({\varphi }')^{-1} (t)$. The good ${\varphi }'$ property formula (1) shows that ${\varphi }''$ is positive for positive arguments. This, together with fact that $({\varphi }')^{-1} (t) =0$ only for $t=0$, gives for $t > 0$

$$\begin{aligned} ({\varphi }^*(t))'' = \left( ({\varphi }')^{-1} (t)\right) ' = \frac{1}{{\varphi }'' \left( ({\varphi }')^{-1} (t) \right) } :=I \end{aligned}$$

Next, let us use good ${\varphi }'$ property at $s = ({\varphi }')^{-1} (t)$ to get

$$\begin{aligned} I \sim \frac{ ({\varphi }')^{-1} (t)}{{\varphi }' \left( ({\varphi }')^{-1} (t) \right) } = \frac{ ({\varphi }^*(t))' }{t}, \end{aligned}$$

where for the equality we again use $({\varphi }^*(t))' = ({\varphi }')^{-1} (t)$. We put together both above expressions that contain I and obtain good $({\varphi }^*)'$ property and $G (({\varphi }^*)') \le C( G ({\varphi }'))$. Consequently, we can use first part of this proposition for ${\varphi }^*$ to write $\Delta _2 ({\varphi }^*) \le C \left( G (({\varphi }^*)')\right) \le C( G ({\varphi }'))$. The second inequality follows from the last-but-one sentence. $\square$

Remark 4

Proposition 1 implies that in Lemma 5, used for ${\mathcal {N}}$-functions with good ${\varphi }'$ property, we have

$$\begin{aligned} \Delta _2 (\Psi ) \le G (\{ {\varphi '} | \; \varphi \in \Phi \}). \end{aligned}$$

Hence we control in Lemma 5 the constants that depend on $\Delta _2 (\Psi )$, which involves ${\mathcal {N}}$-function and their conjugates, with $C \left( G (\{ {\varphi '} | \; \varphi \in \Phi \}) \right)$, where the conjugates are not present.

We will use the above remark as an inherent part of Lemma 5, when we deal with ${\mathcal {N}}$-functions with the good ${\varphi }'$ property, without referring to it directly.

From [12, Lemma 3] we see that for $P : \mathbb {R}\rightarrow Sym^{d \times d}$ and ${\mathcal V}\circ P : \mathbb {R}\rightarrow Sym^{d \times d}$ differentiable at s we have

$$\begin{aligned} {\varphi }'' (|P| ) |\partial _s P|^2 \sim \left| \partial _s\! \left( {\mathcal V}(P)\right) \right| ^2. \end{aligned}$$

(57)

This gives for ${\mathcal A}\circ P : \mathbb {R}\rightarrow Sym^{d \times d}$ differentiable at s

$$\begin{aligned} | \partial _s ({\mathcal A}(P)) | \le C (G ({\varphi }')) {\varphi }'' (|P|) |\partial _s P| \le C (G ({\varphi }')) \left( |\partial _s {\mathcal V}(P)|^2 + {\varphi }'' (|P|) \right) , \end{aligned}$$

(58)

and

$$\begin{aligned} | \partial _s ({\mathcal A}(P)) | |\partial _s P| \le C {\varphi }'' (|P|) |\partial _s P|^2 \le C (G ({\varphi }')) \left| \partial _s \left( {\mathcal V}(P)\right) \right| ^2, \end{aligned}$$

(59)

where we used in the first estimates Assumption 1.

Moreover, if an ${\mathcal {N}}$-function ${\varphi }$ is as in Definition 1 with almost increasing ${\varphi }''$ and ${\varphi }''(0)>0$, we see from (57) that

$$\begin{aligned} |\partial _s P|^2 \le \frac{C (G ({\varphi }'))}{{\varphi }'' (0)} | \partial _s ({\mathcal V}(P)) |^2. \end{aligned}$$

(60)

Let us recall the ingenious concept of a shifted ${\mathcal {N}}$-function. It was supposedly stated in an explicit form first by Diening and Ettwein in [12].

Definition 7

For an ${\mathcal {N}}$-function ${\varphi }$ and $a \ge 0$ the shifted ${\mathcal {N}}$ -function is given as

$$\begin{aligned} {\varphi }_a (t) := \int _0^t {\varphi }' (a+s ) \frac{s}{a+s} ds. \end{aligned}$$

(61)

We will need

Proposition 2

For an ${\mathcal {N}}$-function ${\varphi }$ with the good ${\varphi }'$ property and ${\varphi }''$ almost increasing, we have the implication

$$\begin{aligned} a \le b \implies {\varphi }_a (t) \le C (G ({\varphi }')) {\varphi }_b (t). \end{aligned}$$

Proof

Definition 7 of a shifted ${\mathcal {N}}$-function, the good ${\varphi }'$ property and the fact that ${\varphi }''$ is almost increasing give for $a \le b$

$$\begin{aligned} {\varphi }_a (t)= & {} \int _0^t \frac{ {\varphi }' (a+s )}{a+s} s ds \le C (G ({\varphi }')) \int _0^t {\varphi }'' (a+s ) s ds \\\le & {} C (G ({\varphi }')) \int _0^t {\varphi }'' (b+s ) s ds \le C (G ({\varphi }')) \int _0^t \frac{ {\varphi }' (b+s )}{b+s} s ds = {\varphi }_b (t). \end{aligned}$$

$\square$

1.1 Korn’s inequality

Lemma 6

Let $f \in W^{1,{\varphi }}_x (\Omega )$, where ${\varphi }$ has good ${\varphi }'$ property. Then there exists $C (\Delta _2 ({\varphi }, {\varphi }^*))$ such that

$$\begin{aligned} \int _{B_r} {\varphi }(|\nabla u|) \le C (G ({\varphi }')) \int _{B_r} {\varphi }(|\mathrm{D}u|) + {\varphi }\left( \frac{|u - (u)|}{r} \right) \end{aligned}$$

(62)

For Orlicz and weighted $L^p$ spaces, the standard source of Korn’s inequality is paper by Diening, Růžička and Schumacher [19]. It contains the needed by us inequality (62) in the case of weighted $L^p$ spaces (see formula (5.19) in Theorem 5.17 there), but lacks its Orlicz version (see Theorem 6.13 there). In [19], the step from weighted $L^p$ spaces to Orlicz growths follows from the (amazing) extrapolation technique that originates in [35] by Rubío de Francia. In the Orlicz context it says, roughly speaking, the following. If, for a fixed $p \in (1, \infty )$ and a certain family $\mathcal {F}$ of pairs $(f;g) \in \mathcal {F}$, an inequality holds in a weighted $L^p$ space for any weight in the Muckenhoupt class $A_p$, then this inequality holds for $(f;g) \in \mathcal {F}$ also for (sufficiently regular) Orlicz growths, see [19, Proposition 6.1]. Hence, the extrapolation-based step from weighted-$L^p$ formula (5.19) in Theorem 5.17 to its Orlicz counterpart would basically consist in choosing $\mathcal {F}$ appropriately. Below, we present a proof that does not involve extrapolation.

Proof

Using monotonicity, convexity and $\Delta _2$ condition, we have

$$\begin{aligned}&\int _{B_r} {\varphi }(|\nabla u|) \le C (\Delta _2 ({\varphi })) \int _{B_r} {\varphi }(|\nabla u - (\nabla u)|) ) + {\varphi }(| (\nabla u)|) \nonumber \\&\quad \le C (\Delta _2 ({\varphi }, {\varphi }^*)) \int _{B_r} ( {\varphi }(|\mathrm{D}u - (\mathrm{D}u)|) + {\varphi }(| (\nabla u)|), \end{aligned}$$

(63)

where the second inequality follows from Korn’s inequality for oscillations, see Theorem 6.13 of [19]. Now it suffices to deal with ${\varphi }(| (\nabla u)|)$. To this end let us define h as

$$\begin{aligned} h (t) := {\varphi }(t^\frac{1}{q}). \end{aligned}$$

We need two facts on h. The first one is that we can find such $q_0 > 1$ that h is convex, because for $s = t^\frac{1}{q}$

$$\begin{aligned} h'' (t)= & {} q^{-2} {\varphi }'' (s) s^{2-2q} + q^{-1} (q^{-1} -1) {\varphi }' (s) s^{1-2q} \nonumber \\\ge & {} {\varphi }'' (s) s^{2-2q} \left[ q^{-2} + C (G ({\varphi }')) q^{-1} (q^{-1} -1) \right] \end{aligned}$$

(64)

and the term in the square brackets above can be made positive by choosing $q = q_0$ sufficiently close to 1. The second one is that

$$\begin{aligned} t = {\varphi }((h^{-1} (t))^\frac{1}{q} ) \end{aligned}$$

(65)

since $s = {\varphi }( {\varphi }^{-1} (s)) = h ( ({\varphi }^{-1})^q (s))$. We are ready to deal with ${\varphi }(| (\nabla u)|)$ on r.h.s. of (63) as follows

$$\begin{aligned} {\varphi }(| (\nabla u)|)\le & {} {\varphi }\left( \left( \mathop {\int \!\!\!\!~\!~\!\!\!\!-}\nolimits _{B_r} |\nabla u|^{q_0} \right) ^\frac{1}{q_0} \right) \nonumber \\\le & {} {\varphi }\left( C \left( \mathop {\int \!\!\!\!~\!~\!\!\!\!-}\nolimits _{B_r} \left( |\mathrm{D}u|^{q_0} + \frac{|u - (u)|^{q_0}}{r^{q_0}} \right) \right) ^\frac{1}{q_0} \right) =: I, \end{aligned}$$

(66)

where the first inequality follows from Jensen’s inequality and monotonicity of ${\varphi }$, the second one from the Korn’s inequality for Lebesgue spaces (see for instance formula (5.19) in Theorem 5.17 with $w \equiv 1$). Next, we plug $h^{-1} \circ h \equiv 1$ into I and use convexity of h to get

$$\begin{aligned} I= & {} {\varphi }\left( C \left( h^{-1} \circ h \left( \mathop {\int \!\!\!\!~\!~\!\!\!\!-}\nolimits _{B_r} \left( |\mathrm{D}u|^{q_0} + \frac{|u - (u)|^{q_0}}{r^{q_0}} \right) \right) \right) ^\frac{1}{q_0} \right) \\\le & {} C {\varphi }\left( \left( h^{-1} \left( \mathop {\int \!\!\!\!~\!~\!\!\!\!-}\nolimits _{B_r} h \left( |\mathrm{D}u|^{q_0} + \frac{|u - (u)|^{q_0}}{r^{q_0}} \right) \right) \right) ^\frac{1}{q_0} \right) \\= & {} C \mathop {\int \!\!\!\!~\!~\!\!\!\!-}\nolimits _{B_r} h \left( |\mathrm{D}u|^{q_0} + \frac{|u - (u)|^{q_0}}{r^{q_0}} \right) \le C \mathop {\int \!\!\!\!~\!~\!\!\!\!-}\nolimits _{B_r} h \left( |\mathrm{D}u|^{q_0} \right) +h \left( \frac{|u - (u)|^{q_0}}{r^{q_0}} \right) , \end{aligned}$$

where the second equality is given by (65) and the last inequality follows from convexity and validity of the $\Delta _2$-condition for h (see its definition). Putting together the estimates involving I we have via definition of h

$$\begin{aligned} {\varphi }(| (\nabla u)|)\le & {} C \mathop {\int \!\!\!\!~\!~\!\!\!\!-}\nolimits _{B_r} h \left( |\mathrm{D}u|^{q_0} \right) +h \left( \frac{|u - (u)|^{q_0}}{r^{q_0}} \right) \nonumber \\= & {} C \mathop {\int \!\!\!\!~\!~\!\!\!\!-}\nolimits _{B_r} {\varphi }\left( |\mathrm{D}u| \right) + {\varphi }\left( \frac{|u - (u)|}{r} \right) , \end{aligned}$$

(67)

with C depending only on $G ({\varphi }')$. The above estimate used in (63) implies thesis. $\square$

1.2 Examples of admissible growths

$$\begin{aligned} {\mathcal A}^1 (Q) := (\mu + |Q|^{p-2} ) Q, \qquad {\mathcal A}^2 (Q) := (\mu + |Q|^2)^\frac{p-2}{2} Q \end{aligned}$$

with $\mu \ge 0$, $p > 1$, provide us with the (symmetric) p-Laplacian prototypes. Tensors ${\mathcal A}^1, {\mathcal A}^2$ are given by the following p-potentials

$$\begin{aligned} {\varphi }^1 (t) = \int _0^t \! (\mu + s^{p-2}) \, s \,ds, \qquad {\varphi }^2 (t) = \int _0^t \!(\mu + s^2)^\frac{p-2}{2} \, s \,ds \end{aligned}$$

The respective square root tensors ${\mathcal V}$ read

$$\begin{aligned} {\mathcal V}^1 (Q) = \sqrt{\mu + |Q|^{p-2}} Q , \quad {\mathcal V}^2 (Q) = (\mu + |Q|^2)^\frac{p-2}{4} Q \end{aligned}$$

An example of an admissible potential connected with a non-polynomial growth reads

$$\begin{aligned} {\varphi }^3 (t) = \int _0^t (\mu +s)^{p-2} s \ln (e+s) ds \end{aligned}$$

where the associated tensors are given by

$$\begin{aligned} {\mathcal A}^3 (Q)= (\mu + |Q|)^{p-2} \ln (e+|Q|) Q , \quad {\mathcal V}^3 (Q) = (\mu + |Q|)^\frac{p-2}{2} \sqrt{ \ln (e+|Q|)} Q. \end{aligned}$$

Unfortunately, already the $\Delta _2$-condition, widely used in the regularity theory for systems with generalized growths, excludes exponential or $L\log {L}$ growths, that are of some interest from the perspective of applications.