Publication Date:
2020
Description:
〈p〉Publication date: Available online 8 January 2020〈/p〉
〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉
〈p〉Author(s): S. Giuffrè〈/p〉
〈h5〉Abstract〈/h5〉
〈div〉〈p〉The aim of the paper is to study a gradient constrained problem associated with a linear operator. Two types of problems are investigated. The first one is the equivalence between a non-constant gradient constrained problem and a suitable obstacle problem, where the obstacle solves a Hamilton-Jacobi equation in the viscosity sense. The equivalence result is obtained under a condition on the gradient constraint. The second problem is the existence of Lagrange multipliers. We prove that the non-constant gradient constrained problem admits a Lagrange multiplier, which is a Radon measure if the free term of the equation 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mi〉f〈/mi〉〈mo〉∈〈/mo〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msup〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈mi〉p〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉1〈/mn〉〈/math〉. If 〈em〉f〈/em〉 is a positive constant, we regularize the result, namely we prove that the Lagrange multipliers belong to 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈/math〉.〈/p〉〈/div〉
Print ISSN:
0022-0396
Electronic ISSN:
1090-2732
Topics:
Mathematics
