On optimal $ L^1 $-control in coefficients for quasi-linear Dirichlet boundary value problems with $ BMO $-anisotropic $ p $-Laplacian

Umberto De Maio; Peter I. Kogut; Gabriella Zecca

doi:10.3934/mcrf.2020021

Article Contents

2020, Volume 10, Issue 4: 827-854. Doi: 10.3934/mcrf.2020021

This issue Previous Article Time-inconsistent stochastic optimal control problems: a backward stochastic partial differential equations approach Next Article Stochastic impulse control Problem with state and time dependent cost functions

On optimal $ L^1 $-control in coefficients for quasi-linear Dirichlet boundary value problems with $ BMO $-anisotropic $ p $-Laplacian

1.
Dipartimento di Matematica e Applicazioni 'R.Caccioppoli', Università di Napoli Federico Ⅱ - Complesso Universitario di Monte S. Angelo, via Cintia 80126, Napoli, Italy
2.
Department of Differential Equations, Oles Honchar Dnipro National University, Gagarin av., 72, Dnipro, 49010, Ukraine
3.
Dipartimento di Matematica e Applicazioni 'R.Caccioppoli', Università di Napoli Federico Ⅱ - Complesso Universitario di Monte S. Angelo, via Cintia 80126, Napoli, Italy

^* Corresponding author: Gabriella Zecca
^* Corresponding author: Gabriella Zecca

Received: July 2019

Revised: December 2019

Early access: March 2020

Published: October 2020

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Abstract

We study an optimal control problem for a quasi-linear elliptic equation with anisotropic p-Laplace operator in its principal part and $ L^1 $-control in coefficient of the low-order term. We assume that the matrix of anisotropy belongs to BMO-space. Since we cannot expect to have a solution of the state equation in the classical Sobolev space, we introduce a suitable functional class in which we look for solutions and prove existence of optimal pairs using an approximation procedure and compactness arguments in variable spaces.

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Mathematics Subject Classification: Primary: 49J20; Secondary: 35J95.

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