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In this paper, we study the existence and interior W 2 , p -regularity of the solution, and the regularity of the free boundary ∂ ⁡ { u > ϕ } to the obstacle problem of the porous medium equation, u t = Δ ⁢ u m ( m > 1 ) with the obstacle function ϕ. The penalization method is applied to have the existence and interior regularity. To deal with the interaction between two free boundaries ∂ ⁡ { u > ϕ } and ∂ ⁡ { u > 0 } , we consider two cases on the initial data which make the free boundary ∂ ⁡ { u > ϕ } separate from the free boundary ∂ ⁡ { u > 0 } . Then the problem is converted into the obstacle problem for a fully nonlinear operator. Hence, the C 1 -regularity of the free boundary ∂ ⁡ { u > ϕ } is obtained by the regularity theory of a class of obstacle problems for the general fully nonlinear operator. [ABSTRACT FROM AUTHOR]

We study minimisation problems in L ∞ for general quasiconvex first order functionals, where the class of admissible mappings is constrained by the sublevel sets of another supremal functional and by the zero set of a nonlinear operator. Examples of admissible operators include those expressing pointwise, unilateral, integral isoperimetric, elliptic quasilinear differential, Jacobian and null Lagrangian constraints. Via the method of L p approximations as p → ∞ , we illustrate the existence of a special L ∞ minimiser which solves a divergence PDE system involving certain auxiliary measures as coefficients. This system can be seen as a divergence form counterpart of the Aronsson PDE system which is associated with the constrained L ∞ variational problem. [ABSTRACT FROM AUTHOR]

5 S. N. Armstrong, Principal eigenvalues and an anti-maximum principle for homogeneous fully nonlinear elliptic equations, J. Differential Equations 246 (2009), no. 7, 2958-2987. Keywords: Fully nonlinear operators; principal eigenvalue; Dirichlet problem; half-eigenvalues; uniqueness; 35J60; 35P30; 35B50 EN Fully nonlinear operators principal eigenvalue Dirichlet problem half-eigenvalues uniqueness 35J60 35P30 35B50 673 691 19 10/03/22 20221001 NES 221001 1 Introduction This article contributes to the study of eigenvalue problem of the form remark on the eigenvalue problem in a general smooth unbounded domain. Fully nonlinear operators, uniqueness, 35J60, 35J60, 35B50, principal eigenvalue, Dirichlet problem, half-eigenvalues, 35P30. [Extracted from the article]