Boundedness of Wolff-type potentials and applications to PDEs

Article Mathematics, Applied

Local Lipschitz continuity for p, q-PDEs with explicit u-dependence

Paolo Marcellini

Summary: This article discusses the local Lipschitz continuity problem of weak solutions to a class of nonlinear elliptic partial differential equations, and presents some novel conditions and dependencies.

NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS (2023)

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Article Mathematics, Applied

Wolff potentials and measure data vectorial problems with Orlicz growth

Iwona Chlebicka et al.

Summary: We study solutions to measure data elliptic systems with Uhlenbeck-type structure. The solutions involve operators of divergence form, depend continuously on the spatial variable, and expose doubling Orlicz growth with respect to the second variable. Pointwise estimates for the solutions are provided in terms of a nonlinear potential of generalized Wolff type. Our study not only retrieves recent sharp results for p-Laplace systems, but also covers the natural scope of operators with similar structure and natural class of Orlicz growth.

CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS (2023)

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Article Mathematics, Interdisciplinary Applications

Local boundedness of weak solutions to elliptic equations with p; q growth

Giovanni Cupini et al.

Summary: This article discusses the conditions for the local boundedness of weak solutions to a class of nonlinear elliptic partial differential equations, and presents novel results on general growth conditions and explicit dependence of the equations on the solution u.

MATHEMATICS IN ENGINEERING (2023)

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Article Mathematics

Controlling monotonicity of nonlinear operators

Michal Borowski et al.

Summary: This paper discusses the monotonicity and growth control of Leray-Lions' operators, including the p-Laplacian, which plays a fundamental role in the theory of existence and regularity of solutions to second order nonlinear PDEs. The paper collects, corrects, and provides known estimates, with special attention to the exposition of proofs and the use of elementary arguments.

EXPOSITIONES MATHEMATICAE (2022)

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Article Mathematics, Applied

Lipschitz Bounds and Nonautonomous Integrals

Cristiana De Filippis et al.

Summary: The study presents a general approach to Lipschitz regularity of solutions for a large class of vector-valued, nonautonomous variational problems exhibiting nonuniform ellipticity, ranging from unbalanced polynomial growth conditions to fast, exponential growth. The sharp results obtained are optimal with respect to all considered data, and also provide new regularity criteria in the classical uniformly elliptic case. Furthermore, a classification of different types of nonuniform ellipticity is given, along with suitable conditions to obtain regularity theorems.

ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS (2021)

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Article Mathematics, Applied