A monotonicity formula for a classical free boundary problem

We construct a monotonicity formula for the free boundary problem of the form $Δu=μ$, where $μ$ is a Radon measure. It implies that the blow up limits of solutions are homogenous functions of degree one. The first formula is new even for classical Laplace operator. Our method of proof uses a careful application of the strong maximum principle.

Publication Details

Journal
ArXiv.org
Published
2025-04-01
Primary Topic
Nonlinear Partial Differential Equations
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article
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article

A monotonicity formula for a classical free boundary problem

ArXiv.org
Nonlinear Partial Differential Equations
article

A monotonicity formula for a classical free boundary problem

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Abstract

We construct a monotonicity formula for the free boundary problem of the form $Δu=μ$, where $μ$ is a Radon measure. It implies that the blow up limits of solutions are homogenous functions of degree one. The first formula is new even for classical Laplace operator. Our method of proof uses a careful application of the strong maximum principle.

ArXiv.org
Openalex Percentile: Top 40%
Nonlinear Partial Differential Equations
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A monotonicity formula for a classical free boundary problem · ArXiv.org (2025) | TGRS Research Map | TGRS