SEMINAR “Contemporary Analysis and Partial Differential Equations” (20.04.2023)

SEMINAR “Contemporary Analysis and Partial Differential Equations” (20.04.2023)





Simone Ciani

University of Bolonga, Italy

Anisotropic De Giorgi Classes and the Regularity of Solutions to Some Anisotropic Elliptic Equations

Thursday, 20 April 2023 at 15:00

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We introduce anisotropic De Giorgi classes through general energy inequalities, to describe an old problem that remained open. We motivate the introduction of such classes by recent results on solutions to anisotropic singular equations of the kind

$$ \sum^{s}_{i=1} \partial_{ii} u + \sum^{N}_{i=s+1} \partial_{i} \left( A_{i} (x, u, \nabla u)\right) = 0, \quad x\in\Omega\subset\subset \mathbb{R}^{N} \text{ for } 1\ge s \ge (N-1)$$

where each operator \(A_{i}\) behaves as the \(p\)-Laplacian with \(1 \lt p \lt 2\). The resolution of this special problem borrows parabolic potential techniques of expansion of positivity to obtain the interior Hölder continuity and some integral and pointwise Harnack inequalities.

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