We consider non-autonomous functionals of the form , where , . We assume that grows at least as and at most as . Moreover, is Hölder continuous with respect to x and convex with respect to z. In this setting, we give a sufficient condition on the density that ensures the absence of a Lavrentiev gap.
We consider variational integrals of the form
where , , , is a Caratheodory function, and Ω is bounded and open. Moreover, we assume that, for exponents and constants , , we have
In the scalar case , when , the local minimizers of (1.1) are locally Hölder continuous, see  and [19, p. 361]. If and q is far from p, then the local minimizers might be unbounded, see [12, 17, 18, 15] and [19, Section 5]. We are concerned with higher integrability for the gradient of minimizers. More precisely, assume that makes the energy (1.1) finite. Then the left-hand side of (1.2) implies that . In addition to finite energy, we assume that u is a minimizer of (1.1) and we ask: Does the minimality of u boost the integrability of the gradient Du from to ? The answer is given in : We assume that (1.2) holds with , , we require that and, for constants and , we assume that the following hold:
The exponents p and q must be close enough, i.e.,
We recall that n is the dimension of the space where x lives, i.e., . Let us remark that (1.5) asserts that the smaller α is, the closer p and q must be. In addition, we assume that the Lavrentiev gap on u is zero:
for every ball . Such a Lavrentiev gap will be defined in the next section. Under (1.2)–(1.6), the local minimizers u of (1.1) enjoy higher integrability, i.e., Checking (1.6) is not easy for non-autonomous densities ; it has been done in  for some model functionals using some arguments due to . The aim of the present paper is to give a sufficient condition on the density that ensures the vanishing of the Lavrentiev gap (1.6), see Theorem 3.1 (4).
In the following Ω will be an open, bounded subset of , , and we will denote
where, unless differently specified, all the balls considered will have the same center. We assume that is a Caratheodory function verifying the -growth (1.2) with and . Moreover, we assume that is convex. Due to the non-standard growth behavior of f, we shall adopt the following notion of local minimizer.
A function is a local minimizer of if and only if and
for any with .
We consider functionals that are sequentially weakly lower semicontinuous (s.w.l.s.c.) on X, and we set
We have , and we define the Lavrentiev gap as follows:
when and if . Since is convex with respect to z, standard weak lower semicontinuity results give (see, for instance, [14, Chapter 4]).
The following lemma will be used in the proof of the main theorem (see ).
Let be a function such that . Then if and only if there exists a sequence such that
Let be a function satisfying the following conditions:
is convex for all x,
for , such that , and , there exists such that for and , we have .
Let such that and assume that
Then for all .
Let us now explain condition (4). For every fixed z, is continuous, so the minimization of when gives a minimizer y depending on and z. Condition (4) asks for independence on z, i.e., there exists a minimizer that works for every z. We will first give the proof of Theorem 3.1, and then we will show examples of densities satisfying condition (4).
Let us compare (1.5) with (3.1). When proving the absence of the Lavrentiev gap , the borderline case is allowed but we need strict inequality (1.5) when proving higher integrability of minimizers, see [9, p. 32].
Consider as in hypothesis (4), then and
Let us denote , the usual mollification, where , and define
By definition, it follows that
where . Moreover, by the Hölder continuity hypothesis (i.e., hypothesis 2), we have
Note that the left-hand side of hypothesis (1) gives
Now we observe that is possible to find such that
Therefore, using (3.5), we have
Finally, since strongly in , by recalling that in , and by using a well-known variant of Lebesgue’s dominated convergence theorem and Lemma 2.2, the proof is completed. ∎
Now we give some examples of functions for which Theorem 3.1 is valid.
with the following conditions:
and for all x,
b and c are convex functions such that
For instance, we can consider the following functions:
, where and verify the corresponding conditions of the previous example for all .
, where, in addition to the previous conditions, h is increasing, convex, Lipschitz and such that .
with the following conditions:
h is convex with respect to the second variable,
Hypothesis (4) was used during the proof of Theorem 3.1 in order to obtain the second increase in (3.6). Now we want to show an example of a function for which hypothesis (4) fails. Let us consider and the function such that
where, for ,
In this case the minimum point of the function changes depending on the choice of z. Indeed, let us consider , , . Then we deal with the two cases: and .
When , we have
and then the minimum value in is reached for .
If , then
and therefore, in this situation, is any point such that .
Let be a function verifying Theorem 3.1 and let be a function such that
where . We consider such that and assume . Then for all .
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