An elliptic free boundary arising from the jump of conductivity

• In this paper we consider a quasilinear elliptic PDE, div(A(x,u)∇u)=0, where the underlying physical problem gives rise to a jump for the conductivity A(x,u), across a level surface for u. Our analysis concerns Lipschitz regularity for the solution u, and the regularity of the level surfaces, where A(x,u) has a jump and the solution u does not degenerate. In proving Lipschitz regularity of solutions, we introduce a new and unexpected type of ACF-monotonicity formula with two different operators, that might be of independent interest, and surely can be applied in other related situations. The proof of the monotonicity formula is done through careful computations, and (as a byproduct) a slight generalization to a specific type of variable matrix-valued conductivity is presented.

Ämnesord

NATURVETENSKAP  -- Matematik -- Matematisk analys (hsv//swe)

NATURAL SCIENCES  -- Mathematics -- Mathematical Analysis (hsv//eng)

Nyckelord

Conductivity jump

Free boundary problem

Quasilinear elliptic equation

Mathematical techniques

Nonlinear analysis

Free boundary

Free-boundary problems

Lipschitz regularity

Matrix-valued

Monotonicity

Quasi-linear elliptic

Quasilinear elliptic equations

Linear equations

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ref (ämneskategori)

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