Regularity Theory for Non-autonomous Partial Differential Equations Without Uhlenbeck Structure

Arch. Rational Mech. Anal. Digital Object Identifier (DOI) https://doi.org/10.1007/s00205-022-01807-y Regularity Theory for Non-autonomous Partial Differential Equations Without Uhlenbeck Structure Peter Hästö & Jihoon Ok Communicated by G. Dal Maso Abstract We establish maximal local regularity results of weak solutions or local minimizers of ˆ F(x, Du) dx, div A(x, Du) = 0 and min u  providing new ellipticity and continuity assumptions on A or F with general ( p, q)growth. Optimal regularity theory for the above non-autonomous problems is a long-standing issue; the classical approach by Giaquinta and Giusti involves assuming that the nonlinearity F satisfies a structure condition. This means that the growth and ellipticity conditions depend on a given special function, such as t p , ϕ(t), t p(x) , t p +a(x)t q , and not only F but also the given function is assumed to satisfy suitable continuity conditions. Hence these regularity conditions depend on given special functions. In this paper we study the problem without recourse to, special function structure and without assuming Uhlenbeck structure. We introduce a new ellipticity condition using A or F only, which entails that the function is quasi-isotropic, i.e. it may depend on the direction, but only up to a multiplicative constant. Moreover, we formulate the continuity condition on A or F without specific structure and without direct restriction on the ratio qp of the parameters from the ( p, q)-growth condition. We establish local C 1,α -regularity for some α ∈ (0, 1) and C α -regularity for any α ∈ (0, 1) of weak solutions and local minimizers. Previously known, essentially optimal, regularity results are included as special cases. 1. Introduction Research on regularity of weak solutions or minimizers of the problems ˆ F(x, Du) dx div A(x, Du) = 0 and min u  Peter Hästö & Jihoon Ok is a major topic in the partial differential equations and the calculus of variations. If there is no direct dependence on x (i.e., A(x, ξ ) ≡ A(ξ ) and F(x, ξ ) ≡ F(ξ )), these are called autonomous problems. The simplest non-linear model cases is the p-power function F(ξ ) = |ξ | p , 1 < p < ∞, and the corresponding Euler– Lagrange equation is the p-Laplace equation where A(Du) = |Du| p−2 Du. The maximal regularity of weak solutions of the p-Laplace equation is C 1,α for some α ∈ (0, 1) depending only on p and the dimension n (e.g., [14,17,31,39]). For non-autonomous problems, there is also direct x-dependence. To tackle this case, Giaquinta and Giusti [20,21] introduced the following p-type growth conditions: ⎧ 2 ⎪ ⎪ξ → F(x, ξ ) is C , ⎪ ⎪ ⎨ν|ξ | p  F(x, ξ )  L(1 + |ξ | p ), (1.1) p−2 p−2 ⎪ ν(μ2 + |ξ |2 ) 2 |λ|2  Dξ2 F(x, ξ )λ · λ  L(μ2 + |ξ |2 ) 2 |λ|2 , ⎪ ⎪ ⎪ ⎩|F(x, ξ ) − F(y, ξ )|  ω(|x − y|)(1 + |ξ | p ). Here, F is related to the perturbed case a(x)|ξ | p and has the same p-type growth at all points. Lieberman [32] generalized the model by replacing |ξ | p with Orlicz growth ϕ(|ξ |). Marcellini [34] introduced non-standard, so-called ( p, q)-growth where the exponent p on the right-hand side is replaced by q > p. In this situation, we need to assume that qp is close to 1, see, e.g., [3,4,12,35]. However, all these structure conditions fail to accommodate many kinds of energy functionals since the variability in the x- and ξ -direction are treated separately. For many years, it was thought that the only way to treat the x- and ξ directions together was through special cases. Consequently, a plethora of studies deal with the variable exponent case F(x, ξ ) = |ξ | p(x) . Over the past halfdozen years the double phase functional F(x, ξ ) = |ξ | p + a(x)|ξ |q , 1 < p  q and a  0, has attracted much attention. These models were first studied by Zhikov [40,41] in the 1980’s in relation to Lavrentiev’s phenomenon and have been considered in thousands of papers [36,38]. Moreover, various variants and borderline cases have been investigated, such as: perturbed variable exponent |ξ | p(x) log(e + |ξ |); Orlicz variable exponent ψ(|ξ |) p(x) or ψ(|ξ | p(x) ); degenerate double phase |ξ | p +a(x)|ξ | p log(e+|ξ |); Orlicz double phase ϕ(|ξ |)+a(x)ψ(|ξ |); triple phase |ξ | p + a(x)|ξ |q + b(x)|ξ |r ; double variable exponent |ξ | p(x) + |ξ |q(x) ; and variable exponent double phase |ξ | p(x) + a(x)|ξ |q(x) ; see [28] for references. We emphasize that all these special cases are covered by our results. In [28], we introduced a different approach which does not impose any direct restriction on qp . However, we were only able to prove maximal regularity for local minimizers when F(x, ξ ) = F(x, |ξ |) has so-called Uhlenbeck structure. In this article we extend the results to both minimizers and weak solutions and dispense with the Uhlenbeck restriction. We collect some conditions for A :  × Rn → Rn and F :  × Rn → R with an open set  ⊂ Rn (n  2), which determine our equation and minimization energy, respectively; see Sect. 2 for further definitions and notation, including the continuity assumption (wVA1), with is the other main assumption. Regularity Theory Without Uhlenbeck Structure Assumption 1.1. We say that A :  × Rn → Rn satisfies Assumption 1.1 if the following three conditions hold: (1) For every x ∈ , A(x, 0) = 0, A(x, ·) ∈ C 1 (Rn \ {0}, Rn ) and for every ξ ∈ Rn , A(·, ξ ) is measurable. (2) There exist L  1 and 1 < p < q such that the radial function t → |Dξ A(x, te)| satisfies (A0), (aInc) p−2 and (aDec)q−2 with the constant L, for every x ∈  and e ∈ Rn with |e| = 1. (The ( p, q)-growth condition) (3) There exists L  1 such that |Dξ A(x, ξ  )|  L Dξ A(x, ξ )e · e for all x ∈  and ξ, ξ  , e ∈ Rn with |ξ | = |ξ  | = 0 and |e| = 1. (The quasi-isotropic ellipticity condition) The (A0) condition in (2) means that a coefficient factor of A is nondegenerate and nonsingular, for instance a ≈ 1 when A(x, ξ ) = a(x)|ξ | p−2 ξ . The (aDec)q−2 and (aInc) p−2 conditions in (2) are equivalent to the function t → t 2 |Dξ A(x, te)| satisfying the 2 - and ∇2 -conditions, respectively. In particular, we note from (2) that Dξ A(x, ξ ) = 0 when ξ = 0. Uhlenbeck structure has been replaced by (3), which is a quasi-isotropy condition since different directions behave the same up to a constant. It is known that completely anisotropic equations do not necessarily have any regularity as solutions may even be locally unbounded [19]. We also note that if A(x, ξ ) = Dξ F(x, ξ ) the condition (3) means that the Hessian matrix Dξ2 F(x, ξ ) with ξ = 0 is positive definite and all its eigenvalues on each sphere for ξ are comparable uniformly in x and the radii of spheres, that is, 1 sup{eigenvalues of Dξ2 F(x, te) : |e| = 1} inf{eigenvalues of Dξ2 F(x, te) : |e| = 1}  L̃ for each x ∈  and t > 0, where L̃ depends only on L and n. Compare this to the p-growth condition in (1.1), which implies a stronger condition, where x is inside the supremum and infimum: 1 sup{eigenvalues of Dξ2 F(x, te) : x ∈ , |e| = 1} inf{ei (...truncated)