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

Abstract

We establish maximal local regularity results of weak solutions or local minimizers of $$\begin{aligned} \mathrm {div}A(x, Du)=0 \quad \text {and}\quad \min _u \int _\Omega F(x,Du)\,\mathrm{d}x, \end{aligned}$$ div A ( x , D u ) = 0 and min u ∫ Ω F ( x , D u ) d x , 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 $$t^p$$ t p , $$\varphi (t)$$ φ ( t ) , $$t^{p(x)}$$ t p ( x ) , $$t^p+a(x)t^q$$ 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 $$\frac{q}{p}$$ q p of the parameters from the (p, q)-growth condition. We establish local $$C^{1,\alpha }$$ C 1 , α -regularity for some $$\alpha \in (0,1)$$ α ∈ ( 0 , 1 ) and $$C^{\alpha }$$ C α -regularity for any $$\alpha \in (0,1)$$ α ∈ ( 0 , 1 ) of weak solutions and local minimizers. Previously known, essentially optimal, regularity results are included as special cases.

Regularity theory for non-autonomous problems with a priori assumptions

“…It can be easily seen that (VA1-s)=⇒(wVA1-s)=⇒(A1-s) and (VA1-s ′ )=⇒(VA1-s) if s ′ s, similarly for (wVA1-s) and (A1-s). In the case M = N = 1, these conditions are somewhat differently formulated than in earlier papers, but we showed in [42] that the formulations are are equivalent to previous versions under natural assumptions on G.…”

“…Rather than study each case individually, we introduced an approach based on generalized Orlicz spaces in [41] and proved maximal regularity for minimizers when F (x, ξ) = F (x, |ξ|) has so-called Uhlenbeck structure. In [42] we extended the results to weak solutions and minimizers of problems with (p, q)-growth without the Uhlenbeck restriction. In both articles we only considered the assumption u ∈ W 1,ϕ (Ω) corresponding to case (ap1).…”

“…It implies the Hölder continuity of solutions and (quasi)minimizers [11,36,37]. For higher regularity, we introduced in [41] a "vanishing (A1)" condition, denoted (VA1), and a weak vanishing version, (wVA1) and generalized them to the quasi-isotropic situation in [42]. The anisotropic condition was further studied in [13,40].…”

“…The crucial step of the proofs is approximating the original problem (divA) and (min F ) with a suitable autonomous problem and obtaining a comparison estimate between solutions to the original problem and the autonomous problem. For the approximation we use tools from [42], but the comparison is achieved quite differently from our earlier papers. In this paper, we use harmonic approximation in Lemma 4.13 generalizing the double phase case from [8].…”

Bloch estimates in non-doubling generalized Orlicz spaces

“…In the generalized Orlicz case it is known that solutions with given boudary values exist [15,28,33], minimizers or solutions with given boundary values are locally bounded, satisfy Harnack's inequality and belong to C 0,α loc [9,36,37,55] or C 1,α loc [38,39], quasiminimizers satisfy a reverse Hölder inequality [32], minimizers for the obstacle problem are continuous [41] and the boundary Harnack inequality holds for harmonic functions [12]. Some articles deal with the non-doubling [13] or parabolic [54] case as well as with the Gauss image problem [44].…”