Local boundedness of variational solutions to evolutionary problems with non-standard growth

AbstractWe prove the local boundedness of variational solutions and parabolic minimizers to evolutionary problems, where the integrand f is convex and satisfies a non-standard p, q-growth condition with
$$
1  < p \leq q \leq p \tfrac{n+2}{n}.$$1<p≤q≤pn+2n.A function $${u\colon \Omega_T := \Omega \times (0,T) \to \mathbb{R}}$$u:ΩT:=Ω×(0,T)→R is called parabolic minimizer if it satisfies the minimality condition
$$\int_{\Omega_T} u \cdot \partial_t \varphi +f(x, Du) {\rm d} z \leq \int_{\Omega_T} f(x, Du + D \varphi) {\rm d}z
$$∫ΩTu·∂tφ+f(x,Du)dz≤∫ΩTf(x,Du+Dφ)dzfor every $${\varphi \in C^\infty_0(\Omega_T)}$$φ∈C0∞(ΩT). Moreover, we will show local boundedness for parabolic minimizers, if f satisfies an anisotropic growth condition.


Introduction
We are interested in the regularity of variational solutions, where the integrand f satisfies a non-standard p, q-growth condition for 1 < p ≤ q ≤ p n+2 n and 0 < ν ≤ L. The formal corresponding differential equation is ∂ t u − div ∂ ζ f (x, Du) = 0, but since we do not assume that f is differentiable, the PDE above may have no meaning at all. Moreover, such an equation would only be well-defined, if the weak solution u belongs to the space L q (0, T ; W 1,q (Ω)), but the theory does not ensure the existence of such weak solutions. To overcome these problems, we consider the notion of variational solutions, which was introduced by Lichnewsky and Temam [12] in the context of evolutionary parametric minimal surface equations. In the context of parabolic equations with p, q-growth, the notion of variational solutions has been introduced by Bögelein et al. in [1]. Therein they showed existence of variational solutions associated to a convex integrand f , only assuming that f fulfils a coercivity condition. In this paper we establish an L ∞ loc -bound for these solutions. In order to show this, we will use a parabolic version of the De Giorgi-classes, which was introduced by DiBenedetto in [6]. The analogous elliptic problem is treated in [8,14,16], where the convexity of f and a Δ 2 -condition is required. For integrands with p, q-growth, it is crucial that the gap between p and q is not too large. Otherwise, there exist examples of unbounded solutions (cf. [13]). In [14,16] local boundedness of minimizers to elliptic variational integrands is shown, if 1 < p ≤ q < p = np n − p holds. Here, the embedding W 1,p (Ω) → L q (Ω) is compact. In [8], this result is extended to the case q = p where the Sobolev embedding is only continuous. However, it is not possible to state an explicit L ∞ -bound in this case. In this paper, we prove boundedness of parabolic minimizers, provided the gap between p and q is The upper bound q ≤ p stems from the parabolic embedding. Just as in the elliptic setting, it is not possible to specify an explicit L ∞ -bound in the limit case q = p . Furthermore, we only need the convexity of the integrand f . This assumption is essential for proving a Caccioppoli inequality, since f satisfies only a non-standard growth condition. In the proof we have to handle the lack of regularity of parabolic minimizers in time, a problem which can, due to the growth condition, not be treated by a time regularization method like Steklov averages. But we will use the methods of [2] to show that ∂ t u ∈ L 2 (Ω T ) holds for a variational solution u, if it possess time-independent boundary data. Thus we can prove that these solutions are locally bounded. We also consider parabolic minimizers of functionals, where the integrand f satisfies an anisotropic growth condition of the form If we take p = min {p i } and q = max {p i }, we observe that this is a special case of p, q-growth. Here we can additionally allow a u-dependency for f and do not need a convexity assumption. This stems from the fact that we have more structure conditions for the integrand f . Furthermore, we do not require any information for the boundary data, since we are able to use Steklov averages to compensate the lack of regularity in time. The analogous result for parabolic equations with anisotropic growth conditions has been proved in [15]. Therein the assumption is needed for the exponents p i , which is exactly the same condition we need in this paper. In the elliptic setting, an L ∞ loc -bound for minimizers of integrands satisfying an anisotropic or respectively a p, q-growth condition is proven in [3], where p i ≤ p or respectively p ≤ p is needed. Analogous results for weak solutions of systems are proven in [4]. Of course, the coefficients must satisfy stronger assumptions in order to show regularity. For more details we refer to [3,5] and the references given there.

Variational integrands with p, q-growth
Now we formulate our results for variational solutions, where the integrand f satisfies and non-standard p, q-growth condition. Therefore, let Ω ⊂ R n be an open bounded domain and Ω T := Ω × (0, T ) describes the space-time cylinder for T > 0. The integrand f : Ω × R n → R ∪ {∞} is supposed to be a Carathéodory-function and to fulfil the following convexity and growth assumptions: for some 0 < ν ≤ L. For the initial and boundary datum u 0 we assume that We define variational solutions in the same way as in [1]: Here we used the shorthand notation W 1,p u0 (Ω) := u 0 + W 1,p 0 (Ω) and later on we will use the abbreviations with z 0 = (x 0 , t 0 ) ∈ R n+1 and x 0 ∈ R n . In this setting we will show: be a variational solution, where the variational integrand f satisfies (1.1) and the initial datum u 0 fulfills (1.2). If It is also possible to show a comparable result in the sub-critical case 1 < p ≤ 2n/(n + 2), but we have to assume some higher integrability for u.

Remark 1.5.
The assumption u ∈ L r loc (Ω T ) is already needed and sharp in the case of parabolic equations with p-growth (cf.

Anisotropic variational integrals
Here we consider local parabolic minimizers of evolutionary problems, where the integrand f satisfies an anisotropic growth condition of the form with p i > 1 and p > 1 where holds. Note that this is a special case of (1.1).
In this context, we define the anisotropic Sobolev space W 1,pi (Ω) as the closure of C ∞ (Ω) under the norm and we use the same definition of local parabolic minimizers as in [17]: and moreover, the following minimality condition holds true, whenever ϕ ∈ C ∞ 0 (Ω T ). The appearance of this definition is natural in the context of variational solutions. In [1,Proposition 3.2] it is shown that every variational solution u in the sense of Definition 1.1 also satisfies (1.6). Though the reverse statement is only true, if ∂ t ∈ L 2 (Ω T ) or ∂ t ∈ L p (0, T ; W −1,p (Ω)) holds. Now we formulate our results for anisotropic integrands.
be a parabolic minimizer, where the variational integrand satisfies (1.5). If 2n n + 2 < p and ) is a parabolic minimizer, where the variational integrand satisfies (1.5), then u is locally bounded in Ω T . Additionally, if p i < p holds, we have for any with λ r = n(p − 2) + rp and c = c(n, p i , q, ν, L, r).

Auxiliary tools
In this subsection we state several auxiliary tools, that will be needed throughout the paper. We start with a parabolic version of the Sobolev embedding (cf.
n . Furthermore, we will use the following well known Lemmata (cf. [ for h ∈ N 0 , where M, γ and b are positive constants and b > 1. Then: In particular, if , where C, b > 1 and α ∈ (0, 1) are given constants. Then holds.

Lemma 2.5. Let f (t) be a non-negative bounded function defined for
where A, B, α, θ are positive constants with θ < 1. Then there exists a constant c, depending only on α and θ such that for every

Time derivative
Now we prove the existence of the time derivative in L 2 (Ω T ) of variational solutions, if they possess time independent boundary values (cf. [2]). Therefore, we only need the convexity of the integrand f . To be more precise, we have: ) be a variational solution in the sense of Definition (1.1), where the initial datum u 0 satisfies (1.2) and f (x, ζ) is convex with respect to ζ. Then we have ∂ t u ∈ L 2 (Ω T ).
Proof. We will use the mollification in time Hence ∂ t u ∈ L 2 (Ω T , R N ) holds. Note that we used

L ∞ loc -bound for p, q-integrands
In this section we show the L ∞ loc -bound for variational solutions stated in Theorems 1.2 and 1.4. First, we will only consider the case q < p(n +2)/n and give an explicit L ∞ -bound for u. In Sect. 5 we will treat the case q = p(n + 2)/n. In order to prove our results, we want to argue on the level of parabolic minimizers. Therefore we use the following definition: Definition 3.1. A measurable map u : Ω T → R is termed parabolic minimizer associated to the variational integrand f and the Cauchy-Dirichlet datum u 0 if and only if u ∈ L p (0, T ; W 1,p u0 (Ω)) and moreover, the following minimality condition holds true, whenever ϕ ∈ C ∞ 0 (Ω T ). In [1,Proposition 3.2], it was shown, that every variational solution in the sense of Definition 1.1 is also a parabolic minimizer in the sense of Definition 3.1. The reverse statement is only true if ∂ t u ∈ L 2 (Ω T ) or ∂ t u ∈ L p (0, T ; W −1,p (Ω)) holds. Hence, it is not restrictive to use the definition of and z 0 = (x 0 , t 0 ) ∈ R n+1 and k, ρ, θ > 0. A crucial point in the proof is, that we can not use the "hole-filling technique" due to the growth conditions. To overcome this problem, we will use the convexity of f . A similar technique has been used in [7, Lemma 3.1] to prove a Caccioppoli inequality.
Proof. Since every variational solution is also a parabolic minimizer, we have for any For k > 0 we choose as testing function, where the functions ψ ∈ C 1 ((t 0 − ρ 2 , t 0 )) and ζ ∈ C 1 0 (B ρ ) are cut-off functions with 0 ≤ ψ, ζ ≤ 1. Additionally we can choose ζ such that ζ ≡ 1 on B ρ1 (x 0 ), ζ ≡ 0 outside of B ρ2 (x 0 ) and 0 ≤ |Dζ| ≤ 2/(ρ 2 − ρ 1 ) holds. For ψ we can assume that ψ ≡ 0 on (t 0 − ρ 2 , t 0 − τ 2 ), ψ ≡ 1 on (t 0 − τ 1 , t 0 ) and 0 ≤ ψ ≤ 2/(τ 2 − τ 1 ) holds. Last, the function χ ε is defined by for τ ∈ (t 0 − τ 1 , t 0 − ε) and 0 < ε 1. From Lemma 2.6 we know that ∂ t u ∈ L 2 (Ω T ) holds, because the solutions have time-independent boundary values. Hence we can take ϕ as testing function, since this function can be approximated by smooth functions with compact support in Ω T . Now we use 19 Page 10 of 23 T. Singer NoDEA the convexity of f to estimate the right hand side of (3.1) where we used the abbreviatioñ If we use this estimate in (3.1) and subtract spt ϕ (1 − ψ q )f (x, Du)dz on both sides, we achieve Note that spt ϕ f (x, Du) dz is finite, since u is a parabolic minimizer. Thus we were able to absorb this term. Next we estimate the right hand side, again using the convexity of f Putting this into (3.2) and subtracting spt ϕ ψ q (1 − χ ε ζ q )f (x, Du)dz on both sides, we obtain Note that we used the growth-condition (1.1). Now we consider the term, that involves the time derivative and compute spt ϕ If we let ε ↓ 0 in (3.3) and insert the last estimate, take the supremum over τ ∈ (t 0 − τ 1 , t 0 ) in the first term on the left hand side, let τ → t o − τ 1 in the second term on the left hand side and use the growth assumption (1.1), we get sup t∈(t0−τ1,t0) Bρ 1 (x0) with a constant c depending only on p, q, ν and L.
Now we are ready to start with the proof of Theorem 1.2. Here we will only show the L ∞ loc -bound for the case q < p . The limit case q = p will be treated in Sect. 5.
Now we observe that and and since k ≥ 1, we have |A(k h+1 , ρ h , τ h )| ≤ 4 (h+2)δ k δ J h ≤ 4 (h+2)δ J h . with c = c(n, p, q, L, ν). Now we have proved that u is locally bounded from above in Ω T . Moreover, −u is a local minimizer of the integrandf (x, ζ) := f (x, −ζ). Sincef satisfies the same growth conditions as f , we conclude that −u is locally bounded from above by the same bound and the proof is completed for q < p . Now we will prove Theorem 1.4. Therefore, we let 1 < p ≤ 2n/(n + 2). Since we are in the sup-critical case we need to assume that u is integrable with exponent r > n(2 − p)/p in order to show the L ∞ loc -bound. We also note, that we can use Lemma 3.2, since we do not require any assumptions on p there. Now we will only consider the case p < p , for the limit case we refer to Sect. 5.