Nonuniformly elliptic problems

Mingione, Guiseppe

doi:10.52843/cassyni.5s09kz

2021

DOI: 10.52843/cassyni.5s09kz

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Nonuniformly elliptic problems

Guiseppe Mingione

Abstract: Regularity for nonuniformly elliptic problems has been developed intensively over the last years, especially due to the interest raised by a few special model examples coming from the Calculus of Variations. I shall first give a brief overview of main results in the variational setting and then I shall present some more recent ones obtained in collaboration with Cristiana De Filippis (Università di Torino).

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2022

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

Hästö

2022

Arch Rational Mech Anal

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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.

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

Hästö

2022

Arch Rational Mech Anal

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Nonuniformly elliptic problems

Cited by 1 publication

References 184 publications

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

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

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