Regularity for double-phase functionals with nearly linear growth and two modulating coefficients

1 Introduction

In this article, we establish the first Schauder-type result for a local minimizer of non-uniformly elliptic integral functionals

where Ω ⊂ R n is a bounded open subset, n ≥ 2 , s ∈ [ 0 , 1 ] , and q > 1 . Additionally, we may assume that two modulating coefficients a ( ⋅ ) and c ( ⋅ ) satisfy

where α , α 0 ∈ ( 0 , 1 ) , and Λ ≥ 1 . For simplicity, we denote

and

for x ∈ Ω , z ∈ R n and t ∈ [ 0 , ∞ ) . Here, according to [18], g is a non-decreasing, concave, and unbounded function from [ 0 , ∞ ) into [ 1 , ∞ ) such that t ↦ t g ( t ) is convex and for every ε > 0 ,

holds for some constant c g ( ε ) .

We would like to show that the gradient of a local minimizer of (1.1) is Hölder continuous, when a ( ⋅ ) and c ( ⋅ ) are only Hölder continuous. In fact, this is implied to proving the nonlinear version of Schauder estimates originally established by Hopf and Caccioppoli. In [24] and [32], it is introduced that a local minimizer of the classical nearly linear functional

with c ( x ) ≡ 1 satisfies the Schauder estimate and is therefore Hölder continuous. According to [18], the term “nearly linear” refers to integrand where growth in the gradient variable is superlinear, but it is still slower than that of z ↦ ∣ z ∣ p for any p > 1 . The functional ℒ ( ⋅ ) can be seen as a combination of such nearly linear growth integrand and q -power growth integrand and is derived from the well-studied double-phase functional

The functional P ( ⋅ ) with c ( ⋅ ) ≡ 1 was first introduced by Zhikov [38–41] as an example of the Lavrentiev phenomenon (see [20,23] also) and as a model explaining strongly anisotropic materials in the context of homogenization. Writing the function

the pointwise ellipticity ratio of G ( x , ⋅ )

and the global ellipticity ratio of G ( x , ⋅ )

for any ball B ⊂ Ω , we have

and

provided that { a ( x ) ≡ 0 } ∩ B is nonempty, and hence, ℛ G ( z ; B ) is unbounded with respect to the gradient variable z . Therefore, the functional P ( ⋅ ) is uniformly elliptic in the pointwise sense, but non-uniformly elliptic in the global sense. We refer to [34] for information on uniformly elliptic functionals and non-uniformly elliptic functionals. Such non-uniformly elliptic functional P has been well researched. To obtain sharp regularities for a local minimizer u of P , we may assume that

or

Indeed, they are optimal conditions for obtaining regularity results of minima [20,23]. Moreover, if either (1.7) or (1.8) holds, we can obtain from [6,12,13,28] that the gradient of minimizer u is Hölder continuous. Moreover, discussions on Harnack’s inequality and Hölder continuity can be found in [5,35,36], and Calderón-Zygmund estimates have been studied by Baasandorj et al. [2], De Filippis and Mingione [15], and Colombo and Mingione [14]. Additionally, global gradient estimates, gradient Hölder continuity, and gradient higher integrability have been discussed in [6,10,12,13]. Furthermore, looking at [8,9,11,21,22,26,27,37] provides various insights into the regularities of double-phase problems.

In this article, the main functional ℒ ( ⋅ ) is different from the aforementioned situation because z ↦ ℛ G ( x , ⋅ ) ( z ) is unbounded for G ( x , z ) = c ( x ) ∣ z ∣ log ( 1 + ∣ z ∣ ) + a ( x ) ∣ z ∣ q . In particular, the methods of De Giorgi and Moser cannot be used, and it is impossible to obtain gradient regularity by differentiating the Euler-Lagrange equation. For this reason, a new approach is needed to prove the Hopf-Caccioppoli-Schauder estimates. Indeed, De Filippis and Mingione [18] has proven the results by using a new approach involving nonlinear potentials and fractional Sobolev spaces. Furthermore, De Filippis and Piccinini [19] has recently studied multi-phase problems with nearly linear growth

where 1 < q , s < ∞ and a ( ⋅ ) , b ( ⋅ ) are nonnegative bounded coefficients. We also plan to obtain the results in a similar manner. However, a key difference from [18] is that c ( ⋅ ) can be zero. As a result, we may not be able to obtain the following ellipticity and growth conditions for F ( ⋅ ) :

for every x ∈ Ω , z , ξ ∈ R n , with 0 < ν ≤ 1 ≤ L being fixed constants. Actually, (1.9)₁ and (1.9)₃ do not hold. For this reason, we consider the condition

in (1.2), and we prove the growth and ellipticity conditions of F ( ⋅ ) (see Proposition 2.1). Before introducing the main theorem in this article, we state the definition of a local minimizer of ℒ ( ⋅ ) .

As we proceed with estimates, we are going to use constants that depend on various parameters. We denote these parameters using the notation “ data ”:

Now, the main theorem is as follows:

Finally, we provide an overview of the remaining parts. In the next section, we introduce notations. Moreover, we obtain the growth and ellipticity conditions of F . In Section 3, we explain auxiliary functions and eigenvalues and obtain their properties. In Section 4, we consider the frozen functional of ℒ ( ⋅ ) and prove the regularity for a minimizer of the frozen functional. Finally, in Section 5, after establishing the comparison estimate for frozen functionals and fractional Caccioppoli inequality, we prove the main theorem.

2 Preliminaries

3 Auxiliary functions and their properties

We fix an arbitrary ball B ⊂ Ω centered at x c and define

for ω ∈ [ 0 , 1 ] and σ ∈ ( 0 , 1 ) . Then, t ↦ H ω , σ , i ( t ) is an increasing function. Next, we define two functions λ ω , σ , Λ ω , σ : Ω × [ 0 , ∞ ) → [ 0 , ∞ ) by

for z ∈ R n and x ∈ Ω . Then, we see that

for all x , y ∈ Ω , z , z 1 , z 2 , ξ ∈ R n and for some c ≡ c ( data ) . First, (3.2)₁ can be obtained from (2.4) 2 and (1.4) with ε = q − 1 . Moreover, (3.2)₃ and (3.2) 5 hold from (2.4)₄ and (2.4) 5 , respectively. Similar to [18, Remark 2], we can show that if (3.2) 2 holds, then (3.2)₄ is satisfied. Therefore, we only need to show that (3.2) 2 holds. Indeed, we obtain from (2.4)₃ and (2.6) that for x ∈ Ω and z , ξ ∈ R n ,

holds for some constant c . Also, we easily check that when ω > 0 , the following properties are satisfied:

for all x , y ∈ Ω and z , ξ ∈ R n , where α ˜ = min { α , α 0 } and c ≡ c ( data , ω ) .

We similarly define

Then, by following a similar proof for Proposition 2.1, (3.2) and (3.3), and using (1.2), we obtain