Equivalence between distributional and viscosity solutions for the double-phase equation
Y Fang, C Zhang - Advances in Calculus of Variations, 2022 - degruyter.com
Y Fang, C Zhang
Advances in Calculus of Variations, 2022•degruyter.comWe investigate the different notions of solutions to the double-phase equation-div(| D u| p-
2 D u+ a(x)| D u| q-2 D u)= 0, which is characterized by the fact that both ellipticity
and growth switch between two different types of polynomial according to the position. We
introduce the 𝒜 H(⋅)-harmonic functions of nonlinear potential theory and then show that
𝒜 H(⋅)-harmonic functions coincide with the distributional and viscosity solutions,
respectively. This implies that the distributional and viscosity solutions are exactly the same.
2 D u+ a(x)| D u| q-2 D u)= 0, which is characterized by the fact that both ellipticity
and growth switch between two different types of polynomial according to the position. We
introduce the 𝒜 H(⋅)-harmonic functions of nonlinear potential theory and then show that
𝒜 H(⋅)-harmonic functions coincide with the distributional and viscosity solutions,
respectively. This implies that the distributional and viscosity solutions are exactly the same.
We investigate the different notions of solutions to the double-phase equation-div(| D u| p-2 D u+ a(x)| D u| q-2 D u)= 0, which is characterized by the fact that both ellipticity and growth switch between two different types of polynomial according to the position. We introduce the 𝒜 H(⋅)-harmonic functions of nonlinear potential theory and then show that 𝒜 H(⋅)-harmonic functions coincide with the distributional and viscosity solutions, respectively. This implies that the distributional and viscosity solutions are exactly the same.
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